Alexandru Ioan Cuza University of Ia³i
Faculty of Mathematics
PhD Thesis
BIHARMONICITY AND BICONSERVATIVITY TOPICS
IN THE THEORY OF SUBMANIFOLDS
Advisor, PhD Student,
Prof.Dr. Cezar ONICIUC Simona NISTOR (married BARNA)
IAI, 2017
Contents
Introduction v
1 Preliminaries 1
1.1 Some generalities on Riemannian manifolds . . . . . . . . . . . . . . . . 1
1.2 Some generalities on Riemannian submanifolds . . . . . . . . . . . . . . 7
1.3 CMC surfaces in 3-dimensional space forms . . . . . . . . . . . . . . . . 11
1.4 The energy and bienergy functionals . . . . . . . . . . . . . . . . . . . . 14
1.5 Biharmonic and biconservative submanifolds . . . . . . . . . . . . . . . . 17
2 Biconservative surfaces in 3-dimensional space forms 27
2.1 Biconservativity and minimality in N3(c) . . . . . . . . . . . . . . . . . 27
2.2 An intrinsic characterization of biconservative surfaces in N3(c) . . . . . 35
3 Complete biconservative surfaces in R3 and S3 49
3.1 Complete biconservative surfaces in R3 . . . . . . . . . . . . . . . . . . . 49
3.1.1 Uniqueness of complete biconservative surfaces in R3 . . . . . . . 59
3.2 Complete biconservative surfaces in S3 . . . . . . . . . . . . . . . . . . . 68
4 Biconservative surfaces in arbitrary Riemannian manifolds 107
4.1 More characterizations of biconservative submanifolds . . . . . . . . . . 107
4.2 Properties of biconservative surfaces . . . . . . . . . . . . . . . . . . . . 111
4.3 A Simons type formula for S2 . . . . . . . . . . . . . . . . . . . . . . . . 119
4.3.1 Exemples of submanifolds with ∇AH = 0 . . . . . . . . . . . . . 124
Bibliography 125
iii
Introduction
The introduction is, rstly, concerned with presenting some of the ideas that encouraged
the study of the geometry of biharmonic and biconservative submanifolds and, secondly,
with briey describing the new results that we have obtained and are presented in the
next chapters.
In the last few years the theory of biconservative submanifolds proved to be a very
interesting research topic (see, for example, [15,28,30,31,49,6770]). This theory arose
from the theory of biharmonic submanifolds, but the class of biconservative submanifolds
is richer than the later one. For this reason, we have been focused on the study of
biconservative submanifolds.
Let (Mm, g) and (Nn, h) be two Riemannian manifolds. A biharmonic map, as sug-
gested by J. Eells and J.H. Sampson in [26], is a critical point of the bienergy functional
E2 : C∞(M,N) → R, E2(φ) =
1
2
∫M
|τ(φ)|2 vg,
where τ(φ) is the tension eld of a smooth map φ : M → N , with respect to the xed
metrics g and h. The corresponding Euler-Lagrange equation, obtained by G.Y. Jiang
in [38], is
τ2(φ) = −∆φτ(φ)− traceg RN (dφ, τ(φ))dφ = 0,
where τ2(φ) is the bitension eld of φ, ∆φ = − traceg(∇φ∇φ −∇φ
∇)is the rough
Laplacian dened on sections of φ−1(TN) and RN is the curvature tensor eld of N
given by
RN (X,Y )Z = [∇NX ,∇N
Y ]Z −∇N[X,Y ]Z.
An isometric immersion φ : (Mm, g) → (Nn, h) or, simply, a submanifold M of N , is
called biharmonic if φ is a biharmonic map. Any harmonic map is, clearly, biharmonic.
Therefore, we are interested in studying biharmonic and non-harmonic maps which
are called proper-biharmonic. As a submanifold M of N is minimal if and only if
v
vi Chapter 0. Introduction
φ : (M, g) → (N,h) is a harmonic map, by a proper-biharmonic submanifold we mean
a non-minimal biharmonic submanifold.
Following D. Hilbert ([34]), to an arbitrary functional E we can associate a symmet-
ric tensor eld S of type (1, 1), called the stress-energy tensor, which is conservative, i.e.,
divS = 0, at the critical points of E. In the particular case of the bienergy functional
E2, G.Y. Jiang ([39]) dened the stress-energy tensor S2, also called the stress-bienergy
tensor, by
⟨S2(X), Y ⟩ =1
2|τ(φ)|2⟨X,Y ⟩+ ⟨dφ,∇τ(φ)⟩⟨X,Y ⟩
− ⟨dφ(X),∇Y τ(φ)⟩ − ⟨dφ(Y ),∇Xτ(φ)⟩,
and proved that
divS2 = −⟨τ2(φ), dφ⟩.
Therefore, if φ is biharmonic, i.e., is a critical point of E2, then divS2 = 0. The
variational meaning of S2 was given in [42].
One can see that if φ : (Mm, g) → (Nn, h) is an isometric immersion, then divS2 =
0 if and only if the tangent part of the bitension eld associated to φ vanishes. A
submanifold M is called biconservative if divS2 = 0.
The biconservative submanifolds were studied for the rst time in 1995 by Th.
Hasanis and Th. Vlachos in [33]. In that paper the biconservative hypersurfaces in the
Euclidean space Rn were called H-hypersurfaces, and they were fully classied in R3 and
R4. Actually, the authors were looking for biharmonic hypersurfaces in R4, and their
strategy was to determine rst the hypersurfaces with τ2(φ)⊤ = 0 and then to check if
such hypersurfaces also satisfy τ2(φ)⊥ = 0. They proved that none of the hypersurfaces
with τ2(φ)⊤ = 0 satises τ2(φ)
⊥ = 0, except the minimal ones. We mention that we
prefer to use the term biconservative submanifolds, instead of H-submanifolds as
they are characterized by the vanishing of the divergence of S2 (the authors in [33] did
not use the stress-bienergy tensor).
On the other hand, the study of submanifolds with constant mean curvature, i.e.,
CMC submanifolds, and of minimal submanifolds, has been a very active research topic
in Dierential Geometry for more than 50 years. Recent monographs on these topics
are, for example, [41, 47,48]. There are two ways to develop new research directions:
• to study CMC submanifolds which satisfy some additional geometric hypotheses
(for example, CMC and biharmonicity);
• to study hypersurfaces in space forms, i.e., hypersurfaces in spaces with constant
sectional curvature, which are highly non-CMC.
The study of biconservative surfaces matches with both directions from above.
vii
Indeed, biconservative submanifolds in arbitrary manifolds (and in particular, bicon-
servative surfaces) which are also CMC have some remarkable properties, as we will
see in Chapter 4.
The CMC hypersurfaces in space forms are trivially biconservative, so more in-
teresting is the study of biconservative hypersurfaces which are non-CMC, i.e., with
grad f = 0 at any point of an open subset. Moreover, where grad f = 0, a biconser-
vative surface in a 3-dimensional space form, i.e., a 3-dimensional space with constant
sectional curvature, satises
f∆f + | grad f |2 + 4
3cf2 − f4 = 0,
where ∆ is the Laplace-Beltrami operator on M , i.e., the mean curvature satises a
second order PDE.
We would also want to underline the fact that under the hypothesis of biconservativ-
ity some known results in the theory of submanifolds can be extended to more general
contexts. For example, the generalized Hopf function associated to a CMC biconservat-
ive surface in a Riemannian manifold is holomorphic (compare with the classical results:
the Hopf function associated to a CMC surface in a 3-dimensional space form N3(c)
is holomorphic, and the generalized Hopf function associated to a PMC surface in an
n-dimensional space form is holomorphic). Other example is that a pseudoumbilical
biconservative submanifold φ : Mm → Nn, with m = 4, is CMC, for any ambient
manifold N .
The thesis is organized as follows. In Chapter 1, we establish the notations and
recall some known results. It is divided in ve sections. In the rst section, we recall
properties of symmetric tensor elds of type (1, 1) on a Riemannian manifold, as their
rough Laplacian and their divergence. Then, we give some results about complete
metrics on manifolds and we end this section with few formulas and denitions from
conformal geometry and theory of distributions. In the second section, we present some
basic facts on Riemannian submanifolds: the Gauss, Codazzi and Ricci equations (i.e.,
the basic equations of submanifolds) and the fundamental theorem for submanifolds.
Further, we specialize in the next section on CMC surfaces in 3-dimensional space
forms. We recall here the famous Ricci problem:
Given an abstract surface, which are the necessary and sucient conditions such that it
admits a minimal immersion in N3(c)?
We note that minimal surfaces in spheres, of higher codimension, which satisfy the
Ricci condition were studied, for example, in [73], where the author proved the Lawson's
conjecture (which state that if(M2, g
)is a non-at abstract surface satisfying the Ricci
condition and φ :(M2, g
)→ Sn is a minimal immersion, then φ has a certain form) in
some particular cases.
viii Chapter 0. Introduction
In the fourth section, we present the energy and bienergy functionals E and E2, and
the associated stress-energy tensor S and the stress-bienergy tensor S2. We also give
here the variational meaning of these tensor elds. In the last section, we present some
characterizations formulas for biharmonic and biconservative submanifolds. Then, we
focus on biconservative surfaces in N3(c) with grad f = 0 at any point and recall some
known properties of them.
The main results in this chapter are Theorem 1.42 and Theorem 1.45, obtained
in [15], which describe the properties of biconservative surfaces in 3-dimensional space
forms with grad f = 0 at any point, and Theorem 1.49, obtained in [27], which is a
uniqueness result and says that:
If we have an abstract surface which admits two biconservative immersions in N3(c)
such that the gradients of their mean curvature functions are dierent from zero at any
point, then these two immersions dier by an isometry of N3(c).
Although the major part of the results in this chapter is known, we also present here
few original results as Corollary 1.39, Theorem 1.40 (these two results are presented here
for the rst time), and Theorem 1.49 (this already appeared in [27]).
Chapter 2 contains two sections. We note that the explicit local equations of
biconservative surfaces in a 3-dimensional space form N3(c), with grad f = 0 at any
point, were obtained in [15] and [30]. Moreover, in [15], it is shown that the Gaussian
curvature of a such biconservative surface in a 3-dimensional space form satises the
following equation
(c−K)∆K − | gradK|2 − 8
3K(c−K)2 = 0, (0.1)
that is very similar with that used by G. Ricci-Curbastro [64] in 1895 to characterize,
intrinsically, minimal surfaces in R3.
As we will see in the rst section, we can use this property of biconservative surfaces
to prove results similar to those in [40], [53], or [64], in our context. More precisely,
given an abstract surface which satises intrinsic equation (0.1) with c = 0, by a simple
conformal transformation, it becomes a Ricci surface in R3 (see Theorem 2.4). The
case when c = 0 is dierent. Given an abstract surface which admits a biconservative
immersion in N3(c) with grad f = 0 at any point (thus, a stronger condition than (0.1)),
there exists a conformal transformation of the metric on the surface (this time more
complicated), such that the surface with the new metric is a Ricci surface in N3(c) (see
Theorem 2.7).
An implication of the fact that an abstract surface(M2, g
)admits a biconservative
immersion in N3(c), with grad f = 0 at any point, is that the level curves of K are
ix
circles in M with constant curvature
κ =3| gradK|8(c−K)
, (0.2)
a condition found in [15]. We begin the second section with Theorem 2.10 which gives
some equivalent conditions to (0.2).
Another important result in this chapter is Theorem 2.17 which state the following:
If an abstract surface satisfying c−K > 0, gradK = 0 at any point and (0.2), admits
a biconservative immersion in N3(c), then grad f = 0 (and it is unique).
One of the main results in this thesis is Theorem 2.18 which says that:
An abstract surface admits locally a biconservative embedding in N3(c), with grad f = 0
at any point, if and only if c−K > 0, gradK = 0 at any point and (0.2) holds.
Therefore, even if the notion of biconservative submanifolds belongs, obviously, to
the extrinsic geometry, in the particular case of biconservative surfaces in N3(c), they
admit also an intrinsic characterization.
The chapter ends with Theorem 2.22 that gives other three equivalent conditions
to (0.2). This time, these conditions are expressed in terms of certain isothermal co-
ordinates. This result will be very helpful in the construction process of complete
biconservative surfaces in R3 and in S3, as we will see in the next chapter.
We note that most of the results from this chapter are original and they can be also
found in [27] and [56]. Some results, as Theorem 2.10, Theorem 2.17, Theorem 2.21,
appear here for the rst time.
The goal of Chapter 3 is to construct complete biconservative surfaces in R3 and
S3. We start with the local extrinsic and intrinsic results and extend them to the
global extrinsic and intrinsic results. The local extrinsic problem consists in nding
all biconservative surfaces in 3-dimensional space forms with grad f = 0 at any point
and the local intrinsic problem is to determine all abstract surfaces(M2, g
)satisfying
c−K > 0, gradK = 0 at any point of M , and condition (0.2).
We have seen that the local extrinsic and intrinsic problems are completely solved,
and they are equivalent since given an abstract surface(M2, g
), it admits a biconser-
vative immersion in N3(c) with grad f = 0 at any point of M if and only if c−K > 0,
gradK = 0 at any point ofM and (0.2) holds. Moreover, this immersion is unique, and
thus we have a bijection between the set of biconservative immersions in N3(c), with
grad f = 0 at any point, where c ∈ R is xed, and the set of abstract surfaces which
satisfy the above conditions.
Then, we consider the global problem, again, from extrinsic and intrinsic point of
view. To solve the global extrinsic problem means to determine all complete biconser-
vative surfaces in 3-dimensional space forms which satisfy grad f = 0 at any point of
x Chapter 0. Introduction
an open and dense subset. To solve the global intrinsic problem means to determine
all complete abstract surfaces(M2, g
)which on that open and dense subset satisfy
c−K > 0, gradK = 0 and the level curves of K are the circles in M with curvature κ
given in (0.2).
We cannot say that the global extrinsic and intrinsic are equivalent. As we will see,
the global extrinsic problem implies the global intrinsic problem, but, even if M would
be simply connected, the converse implication we do not know to be true, because we
cannot dene a tensor eld A on the whole M .
The two global problems are not completely solved, in the sense that we construct
examples of complete biconservative surfaces in N3(c), c = 0 and c = 1, but without
proving their uniqueness.
As we said, for the global extrinsic problem, we have asked grad f = 0 at any point
of an open and dense subset. We impose this hypothesis because we belive that
Conjecture 1. Let M2 be a biconservative surface in N3(c). If there exists an open
subset U of M such that grad f = 0 on U , then grad f = 0 on M .
In Theorem 3.16 and in Theorem 3.21 we prove Conjecture 1 in some particular
cases. We suppose that the proof of Conjecture 1 will follow from the analysis of the
PDE obtained in Corollary 1.39. We also believe that
Conjecture 2. The only complete simply connected biconservative surfaces in R3 or S3,with grad f = 0 at any point of an open and dense subset, are those given in Theorem
3.11 and Theorem 3.48, respectively.
In Theorem 3.17 and in Theorem 3.22 we prove Conjecture 2 in some particular
cases.
Summarising the above two conjectures, we can state the following
Conjecture 3. The only complete simply connected non-CMC biconservative surfaces
in R3 or S3 are those given by Theorem 3.11 and Theorem 3.48, respectively.
Moreover, we can also state the following open problem that would follow from the
quality of the biconservative immersion given in Theorem 3.48 of being double periodic
or not.
Open problem. Does there exist a non-CMC biconservative surface in S3 that is
compact?
This chapter has two sections. In the rst section, we consider the global problem
and construct complete biconservative surfaces in R3, with grad f = 0 at any point
of an open dense subset. We determine such surfaces in two ways. One way is to
use the local extrinsic characterization of biconservative surfaces in R3 and to glue
xi
two pieces together in order to obtain a complete biconservative surface (Theorem 3.5).
The other way is more analytic and consists in using the local intrinsic characterization
theorem in order to obtain a complete biconservative immersion from(R2, gC0
)in R3
with grad f = 0 on an open dense subset of R2; here, C0 is a positive constant and
therefore we obtain a one-parameter family of solutions (Theorem 3.11). It is worth
mentioning that, by a simple transformation of the metric gC0 ,(R2,
√−KC0gC0
)is
(intrinsically) isometric to a helicoid (Theorem 3.12).
The rst section ends with a subsection, where we study the uniqueness of complete
biconservative surfaces in R3. One of the main results in this chapter is Theorem 3.20,
which state that:
If S is a compact biconservative regular surface in R3, then S is CMC and thus a round
sphere.
Moreover, we prove a stronger theorem, Theorem 3.22, which is one of the most
important result of this thesis, and says that:
If S is a complete non-CMC biconservative regular surface in R3, then S = SC0, where
SC0is the complete biconservative surface of revolution in R3 given in Theorem 3.5.
In the second section, we consider the global problem of biconservative surfaces in
S3, with grad f = 0 at any point of an open dense subset. As in the R3 case, we use
the local extrinsic classication of biconservative surfaces in S3, but now the gluing
process is not as clear as in R3. Further, we change the point of view and use the
local intrinsic characterization of biconservative surfaces in S3. We construct complete
Riemannian surfaces(R2, gC1,C∗
1
)which admit a biconservative immersion in S3 with
grad f = 0 on an open dense subset of R2 and we show that, up to isometries, there
exists only a one-parameter family of such Riemannian surfaces indexed by C1 (Theorem
3.48). The above construction consists in two steps: rst, we obtain a complete surface
of revolution in R3, whose universal cover is(R2, gC1,C∗
1
), and second, we determine
explicitly the biconservative immersion in S3. Theorem 3.48 is one of the main results
in this thesis.
We note that most of the results from this chapter are original and they were presen-
ted also in [54], [56], and [57]. Moreover, the results in Subsection 3.1.1 are presented
here for the rst time.
In the last chapter, Chapter 4, we use dierent techniques as in the previous
chapters, and study in a uniform manner the properties of biconservative surfaces in
arbitrary Riemannian manifolds.
The chapter is divided in three sections. In the rst section, we present some charac-
terizations of biconservative submanifolds which satisfy additional geometric hypotheses
xii Chapter 0. Introduction
as AH being a Codazzi tensor eld (Proposition 4.7), or the submanifold being PMC,
i.e., having the mean curvature vector eld H parallel in the normal bundle (Proposi-
tion 4.11). We also study the properties of submanifolds with AH parallel, as they are
automatically biconservative (Proposition 4.1).
In the second section, we focus on biconservative surfaces. As biconservative surfaces
are characterized by divS2 = 0, where S2 is a symmetric tensor eld of type (1, 1), some
of their properties will follow from general properties of a symmetric tensor eld of type
(1, 1) with free-divergence as they are presented in Theorem 4.18. In Theorem 4.19, we
nd the link between biconservativity, the property of the shape operator AH to be a
Codazzi tensor eld, the holomorphicity of a generalized Hopf function and the quality
of the surface to have constant mean curvature.
Another result in this section is Theorem 4.26, which gives a description of the
metric and of the shape operator AH for a CMC biconservative surface in an arbitrary
manifold. This description is done in terms of |H| and the principal curvatures of M ,
i.e., the eigenvalues functions of AH .
We end this section by proving that a biconservative surface in an arbitrary manifold,
with constant principal curvatures, can be immersed in N3(c) having either AH or S2,
as shape operator (Theorem 4.31 and Theorem 4.32).
In the last section, we nd the expression of the rough Laplacian∆RS2 of S2 and then
we determine a Simons type formula (Proposition 4.34). A consequence of Proposition
4.34 is Theorem 4.39, which states:
If φ :M2 → Nn is a compact CMC biconservative surface and K ≥ 0, then ∇AH = 0
and M is at or pseudoumbilical.
With a dierent technique we get a similar result in the complete non-compact case
(Theorem 4.41).
Almost all of the results from this chapter are original and they can be also found
in [55]. Some of them are known results, but obtained in a dierent way.
Acknowledgements. Firstly, I would like to express my sincere gratitude to my
advisor Prof.Dr. Cezar Oniciuc for his useful ideas, his patience, motivation and constant
support throughout this thesis. His invaluable guidance helped me in all the time of
research and writing of this thesis.
Beside my advisor, I would like to thank the others members of my PhD committee:
Prof.Dr. Ioan Buc taru, Prof.Dr. Dorel Fetcu and Prof.Dr. R zvan Liµcanu, for their
insightful comments and encouragement.
I am also grateful to the Doctoral School of Faculty of Mathematics, for the nancial
support, and to the Faculty of Mathematics, Alexandru Ioan Cuza University of Ia³i,
xiii
for hospitality and for the opportunity to have access to its infrastructure. My sincere
thanks go to the following Financing Programs and their directors, that supported me
in this research:
• Constant mean curvature and biharmonic submanifolds, PN-II-RU-TE-2014-4-
0004; Director: Prof.Dr. Dorel Fetcu.
• Variational methods with applications to generalized vector optimization problems,
PN-II-RU-TE-2014-4-0019; Director: Prof.Dr. Marius Durea.
• The European Social Fund through Sectoral Operational Programme Human
Resources Development 2007 2013, Towards a New Generation of Elite Re-
searchers through Doctoral Scolarships, POSDRU/187/1.5/S/155397; Director:
Assoc.Prof. Liviu-George Maha.
Last but not the least, I would like to thank my family: my husband, my parents
and my brother, for supporting me spiritually throughout writing this thesis, for their
love and constant motivation.
Chapter 1Preliminaries
In this chapter, we explain the notations and recall some basic results which will be
used throughout the thesis.
We will not dene the fundamental notions of dierentiable and Riemannian geo-
metry, as dierentiable manifolds, vector bundles, linear connections, Riemannian man-
ifolds, etc., as they are supposed to be known.
Conventions. Throughout this thesis, all manifolds, metrics and maps are assumed
to be smooth, i.e. of class C∞, and we will often indicate the various Riemannian
metrics by the same symbol ⟨·, ·⟩. All manifolds and submanifolds are assumed to be
connected and oriented. Also, by a CMC submanifold we understand a submanifold
with constant mean curvature dierent from zero, and a non-CMC hypersurface is a
hypersurface with grad f = 0 at any point of an open subset W of M , where f denotes
the mean curvature function on M , and W is not necessarily the whole M , i.e., M \Wcan be a non-empty set.
Most of the results in this chapter are known and we will mainly follow the mono-
graphs [11, 12, 16, 17, 22, 60, 63], to present them. Though, we note that the results in
Corollary 1.39 and in Theorem 1.40 appear here for the rst time, and the result given
in Theorem 1.49 is also original, but was presented for the rst time in [27].
1.1 Some generalities on Riemannian manifolds
It is well-known that a symmetric tensor eld T of type (1, 1) on a Riemannian manifold
(Mm, g) can be identied with a symmetric tensor eld T of type (0, 2) by
⟨T (X), Y ⟩ = T (X,Y ), X, Y ∈ C(TM),
1
2 Chapter 1. Preliminaries
and, henceforth, we will use the same notation T instead of T .
Denition 1.1. Let (Mm, g) be a Riemannian manifold and ∇ its Levi-Civita connec-
tion. Consider the vector elds on M , X, Z, Yj , j = 1, r, the tensor eld T of type
(1, r), the symmetric tensor eld of type (0, 2), S, and Xii=1,m a local orthonormal
frame eld on M . Then
(i) the divergence of a vector eld is a smooth function given by
divX = trace (Z → ∇ZX)
=
m∑i=1
⟨∇XiX,Xi⟩;
(ii) the divergence of a tensor eld of type (1, r) is a tensor eld of type (0, r) dened
by
(div T ) (Y1, Y2, · · · , Yr) = trace (Z → (∇ZT ) (Y1, Y2, · · · , Yr))
=m∑i=1
⟨(∇XiT ) (Y1, Y2, · · · , Yr) , Xi⟩;
(iii) the divergence of a symmetric tensor eld of type (0, 2) is a tensor eld of type
(0, 1) given by
(divS)(X) =m∑i=1
(∇S) (Xi, Xi, X) =m∑i=1
(∇XiS) (Xi, X) .
Moreover, if T is a symmetric tensor eld of type (1, 1), then
(div T )(X) =
m∑i=1
⟨(∇XiT ) (X), Xi⟩ =m∑i=1
⟨(∇XiT ) (Xi) , X⟩ = ⟨trace(∇T ), X⟩,
i.e., div T = (trace(∇T ))♯ or, equivalently, (div T ) = trace(∇T ), where ♯ and are themusical isomorphisms.
Denition 1.2. Let (Mm, g) be a Riemannian manifold and α be a smooth function
on M . Then
(i) the gradient of a smooth function is a vector eld dened by
gradα =m∑i=1
(Xiα)Xi,
where Xii=1,m is a local orthonormal frame eld on M ;
1.1. Some generalities on Riemannian manifolds 3
(ii) the Hessian of a smooth function is a symmetric tensor eld of type (1, 1) given
by
(Hessα)(X) = ∇X gradα, X ∈ C(TM).
Moreover, we have
⟨(Hessα)(X), Y ⟩ = ⟨(Hessα)(Y ), X⟩ = X(Y α)− (∇XY )α, X, Y ∈ C(TM),
and therefore, we can think the Hessian as a symmetric tensor eld of type (0, 2).
Clearly,
∆α = − trace(Hessα).
Proposition 1.3. Let (Mm, g) be a Riemannian manifold and consider T and S two
symmetric tensor elds of type (1, 1). Then
⟨∆RT, S⟩ = ⟨∇T,∇S⟩ − divZ, (1.1)
with ∆RT = − trace∇2T , Z ∈ C(TM), Z = ⟨∇XiT, S⟩Xi, where Xii=1,m is an
orthonormal local frame eld.
Proposition 1.4. Let (Mm, g) be a Riemannian manifold and consider T a symmetric
tensor eld of type (1, 1) and α a smooth function on M . Then
div (T (gradα)) = ⟨div T, gradα⟩+ ⟨T,Hessα⟩, (1.2)
Proposition 1.5 (The Ricci formula). Let (Mm, g) be a Riemannian manifold and T
a tensor eld of type (1, 1) (not necessarily symmetric). Then(∇2T
)(X,Y, Z)−
(∇2T
)(Y,X,Z) = R(X,Y )T (Z)− T (R(X,Y )Z),
where X,Y, Z ∈ C(TM).
Denition 1.6. A symmetric tensor eld T of type (1, 1) is called a Codazzi tensor
eld if
(∇T )(X,Y ) = (∇T )(Y,X), X, Y ∈ C(TM).
Remark 1.7. If ∇T = 0, then T is a Codazzi tensor eld.
Further, we present some results about the completeness of Riemannian metrics.
Proposition 1.8 ([32]). Let Mm be a manifold and consider g and g two metrics on
it. If (M, g) is complete and g − g is non-negative denite at any point of M , i.e.,
(g − g) (X,X) ≥ 0, for any X ∈ C(TM), then (M, g) is also complete.
4 Chapter 1. Preliminaries
Even if this result is known, we did not nd a proof of it, and this is the reason why
we will present now our proof.
Proof. Let d and d be the distance functions determined by g and g, respectively. Since
(M, g) is complete, (M,d) is a complete metric space, i.e., every Cauchy sequence of
points in M converges to a point in M , with respect to the distance d. If we denote
g1 = g − g, clearly g1(X,X) ≥ 0, for any X ∈ C(TM) and g = g1 + g.
Consider p and q two points in M and γ : [a, b] → M a piece-wise smooth curve
such that γ(a) = p and γ(b) = q. Let a = t0 < t1 < · · · < tn = b be a partition of the
interval [a, b] such that γ[ti,ti+1] is smooth, for any i = 0, n− 1. The length of γ, with
respect to the metric g, is given by
l(γ; g) =n−1∑i=0
∫ ti+1
ti
√g (γ′(τ), γ′(τ)) dτ,
and the length of the same γ, with respect to the metric g, is given by
l (γ; g) =
n−1∑i=0
∫ ti+1
ti
√g (γ′(τ), γ′(τ)) dτ.
Obviously,
l (γ; g) =
n−1∑i=0
∫ ti+1
ti
√g (γ′(τ), γ′(τ)) + g1 (γ′(τ), γ′(τ)) dτ
≥n−1∑i=0
∫ ti+1
ti
√g (γ′(τ), γ′(τ)) dτ
= l(γ; g),
and, then, one obtains
d(p, q) ≤ d(p, q). (1.3)
In order to prove that(M, d
)is a complete metric space, we consider a Cauchy sequence,
(pn)n∈N∗ ⊂ M , with respect to d, i.e., for any ε > 0, there exists nε ∈ N∗, such that
for any m,n ∈ N∗ with m,n > nε, we have d (pm, pn) < ε. From (1.3) it follows that
(pn)n∈N∗ is Cauchy with respect to d. Therefore, since (M,d) is a complete metric space,
there exists p0 ∈M such that (pn)n∈N∗ converges to p0, with respect to the metric d.
On the other side, the topology of the submanifold M coincide with the topology
induced by the metric space (M,d). Thus, d and d determine the same topology on M .
Let ε0 > 0 arbitrary xed. Denote by
Bd (p0; ε0) =q ∈M | d (q, p0) < ε0
.
1.1. Some generalities on Riemannian manifolds 5
Since Bd (p0; ε0) is open and p0 ∈ Bd (p0; ε0), there exists r = rε0 > 0 such that
Bd (p0; r) ⊂ Bd (p0; ε0) .
Now, since (pn)n∈N∗ converges to p0, with respect to the metric d, for r, there exists
nr ∈ N∗ such that for any n > nr we have
pn ∈ Bd (p0; r) ⊂ Bd (p0; ε0) .
Therefore, (pn)n∈N∗ converges to p0, with respect to the metric d, i.e.,(M, d
)is a
complete metric space.
Further, we present a similar result with the above one.
Proposition 1.9. Let Mm be a manifold and consider g and g two metrics on it.
If (M, g) is complete and there exists a > 0 such that g(X,X) ≥ ag(X,X), for any
X ∈ C(TM), then (M, g) is also complete.
Proof. The proof of this result is, in fact, the same as that of Proposition 1.8, with only
one dierence. Here,
l (γ; g) =
n−1∑i=0
∫ ti+1
ti
√g (γ′(τ), γ′(τ)) dτ
≥ an−1∑i=0
∫ ti+1
ti
√g (γ′(τ), γ′(τ)) dτ
= a l(γ; g),
and, therefore
d(p, q) ≤ 1
ad(p, q).
In fact, Proposition 1.9 also appear in [16], where is state the following problem.
Proposition 1.10. Let S and S two regular surfaces and φ : S → S a dieomorphism.
Assume that S is complete and there exists a > 0 such that
Ip (w) ≥ aIφ(p) (dφ(p) (w)) , w ∈ TpS, (1.4)
for any p ∈ S, where I and I denote the rst fundamental form of S and S, respectively.
Then S is complete.
6 Chapter 1. Preliminaries
Proof. We consider i : S → R3 and i : S → R3 the canonical inclusions of S and S in
R3 and ⟨·, ·⟩0 the Euclidean metric on R3. Denote by g = i∗⟨, ⟩0 and g = φ∗ (i∗⟨, ⟩0) themetrics induced by R3 on S, and by φ on S, respectively.
With these notations, our problem can be reformulated as: given a surface S with
g and g two metrics on it, if (S, g) is complete (because S is complete) and
g(X,X) ≥ ag(X,X), X ∈ C(TM),
then (S, g) is complete. This is obviously true, from Proposition 1.9.
From the conformal geometry, we recall the following known formulas.
Let(M2, g
)be a Riemannian surface with the Levi-Civita connection ∇, curvature
tensor eld R, Gaussian curvature K and Laplacian ∆. Consider a new Riemannian
metric g = e2ρg on M , where ρ ∈ C∞(M). Then
• the Levi-Civita connection, with respect to g, is given by
∇XY = ∇XY + (Xρ)Y + (Y ρ)X − g(X,Y ) grad ρ, X, Y ∈ C(TM); (1.5)
• the curvature tensor eld, with respect to g, is equal to
R(X,Y )Z = R(X,Y )Z − (∇dρ)(Y, Z)X + (∇dρ)(X,Z)Y + (Y ρ)(Zρ)X−
− (Xρ)(Zρ)Y −∇g(Y,Z)X−g(X,Z)Y grad ρ−
− | grad ρ|2 (g(Y, Z)X − g(X,Z)Y ) + (Xρ)g(Y, Z) grad ρ−
− (Y ρ)g(X,Z) grad ρ, X, Y, Z ∈ C(TM);
• the Riemann-Christoel tensor eld, with respect to g, is given by
R(X,Y, Z,W ) = e2ρ(R(X,Y, Z,W )− (∇dρ)(Y,W )g(X,Z)+
+ (∇dρ)(X,W )g(Y, Z)− (∇dρ)(X,Z)g(Y,W )+
+ (∇dρ)(Y, Z)g(X,W )− (Xρ)(Wρ)g(Y, Z)+
+ (Y ρ)(Wρ)g(X,Z) + (Xρ)(Zρ)g(Y,W )−
− (Y ρ)(Zρ)g(X,W )− | grad ρ|2g(Y,W )g(X,Z)+
+ | grad ρ|2g(X,W )g(Y, Z)), X, Y, Z,W ∈ C(TM);
• the Gaussian curvature, with respect to g, is determined by
K = e−2ρ(K +∆ρ); (1.6)
1.2. Some generalities on Riemannian submanifolds 7
• the Laplacian, with respect to g, can be computed by
∆α = e−2ρ∆α, α ∈ C∞(M); (1.7)
We end this section with some results concerning the distributions.
Let Nn be a manifold of dimension n = m+ k. Assume that to each point p ∈ N is
assigned anm-dimensional subspace Dp of TpN and in a neighborhood U of each p ∈ N ,
there are m vector elds Xi ∈ C(TU), i = 1,m, such that Xi(p)i=1,m ⊂ TpN is a
basis of Dq, for any q ∈ U . Then, we say that D is a m-plane distribution of dimension
m on N and Xii=1,m is a local frame eld of D. Moreover, the distribution D is called
involutive if there exists a local frame eld Xii=1,m around any point such that
[Xi, Xj ] = αkijXk, i, j ∈ 1,m,
where αkij are local smooth functions.
We note that the above notion does not depend of the chosen local frame eld.
Now, if D is a distribution on N and M is a submanifold of N such that for any
q ∈M , TqM ⊂ Dq, then M is called integral manifold of D.We have to notice that an integral manifold may be of lower dimension than D, and
is not necessary a regular manifold.
Finally, a distribution D onNn of dimensionm, where n = m+k, is called completely
integrable if for each point p ∈ N there exists a local chart (U ;φ) =(U ;x1, · · · , xn
)with φ(U) = Cnε (0), ε > 0, and ∂xii=1,m is a local frame eld on U for D, where∂xi = ∂
∂xiand
∂∂xα
α∈1,n is the natural frame eld corresponding to the local chart
(U ;φ). Note that, in this case, there exists an m-dimensional integral manifold M
through each point q ∈ U such that TqM = Dq, i.e. dimM = m. Indeed, an m-slice
dened by xm+1 = constant, · · · , xn = constant, is an m-dimensional integral manifold
of D. Of course, a completely integrable distribution is involutive since
[∂xi , ∂xj ] = 0, i, j ∈ 1,m.
In fact, we have the following result.
Theorem 1.11 (Frobenius). A distribution D on a manifold N is completely integrable
if and only if it is involutive.
1.2 Some generalities on Riemannian submanifolds
A submanifold of a given Riemannian manifold (Nn, h) is a pair (Mm, φ), where Mm
is a manifold and φ : M → N is an immersion. We always consider on M the induced
8 Chapter 1. Preliminaries
metric g = φ∗h, thus φ : (M, g) → (N,h) is an isometric immersion (for simplicity we
write φ :M → N without mentioning the metrics). We also write φ :M → N , or even
M , instead of (M,φ).
In order to x the notations, we recall the rst order fundamental equations of a sub-
manifold in a Riemannian manifold, as these equations dene the second fundamental
form, the shape operator and the connection in the normal bundle. Let φ :Mm → Nn
be an isometric immersion. For each p ∈ M , Tφ(p)N can be written as the orthogonal
direct sum
Tφ(p)N = dφ(TpM)⊕ dφ(TpM)⊥, (1.8)
and NM =∪p∈M
dφ(TpM)⊥ is referred to as the normal bundle of φ (or of M), in N .
Denote by ∇ and ∇N the Levi-Civita connections on M and N , respectively, and
by ∇φ the induced connection in the pull-back bundle φ−1(TN) =∪p∈M
Tφ(p)N . Taking
into account the decomposition in (1.8), one has the Gauss formula
∇φXdφ(Y ) = dφ(∇XY ) +B(X,Y ), X, Y ∈ C(TM),
where B ∈ C(⊙2T ∗M ⊗NM) is called the second fundamental form of M in N . Here
T ∗M denotes the cotangent bundle of M . The mean curvature vector eld of M in N
is dened by H = (traceB)/m ∈ C(NM), where the trace is considered with respect
to the metric g.
Furthermore, if η ∈ C(NM), then one has the Weingarten formula
∇φXη = −dφ(Aη(X)) +∇⊥
Xη, X ∈ C(TM),
where Aη ∈ C(T ∗M ⊗ TM) is called the shape operator of M in N in the direction
η, and ∇⊥ is the induced connection in the normal bundle. Moreover, ⟨B(X,Y ), η⟩ =⟨Aη(X), Y ⟩, for all X,Y ∈ C(TM), η ∈ C(NM). In the case of hypersurfaces, we
denote f = traceA, where A = Aη and η is the unit normal vector eld, and we have
H = (f/m)η; f is the (m times) mean curvature function.
A submanifold M of N is called a PMC submanifold if H is parallel in the normal
bundle and dierent from zero, and a CMC submanifold if |H| is constant and dierent
from zero.
When confusion is unlikely we locally identify M with its image by φ, X with
dφ(X) and ∇φXdφ(Y ) with ∇N
XY . With this in mind, we can write the Gauss and the
Weingarten formulas as
∇NXY = ∇XY +B(X,Y ),
and
∇NXη = −Aη(X) +∇⊥
Xη.
1.2. Some generalities on Riemannian submanifolds 9
Proposition 1.12 (The Gauss equation). Let φ :Mm → Nn be a submanifold. Then
⟨RN (X,Y )Z,W ⟩ = ⟨R(X,Y )Z,W ⟩ − ⟨B(X,W ), B(Y, Z)⟩+ ⟨B(Y,W ), B(X,Z)⟩,
where X,Y, Z,W ∈ C(TM).
If Mm is a hypersurface in Nm+1(c), then the Gauss equation becomes
R(X,Y )Z = c (⟨Y, Z⟩X − ⟨X,Z⟩Y ) + ⟨A(Y ), Z⟩A(X)− ⟨A(X), Z⟩A(Y )
for any X,Y, Z ∈ C(TM).
Moreover, if M2 is a surface in N3(c), we can rewrite the Gauss equation as
K = detA+ c,
where K is the Gaussian curvature of the surface.
Proposition 1.13 (The Codazzi equation). Let φ :Mm → Nn be a submanifold. Then
(∇XAη) (Y )− (∇YAη) (X) = A∇⊥Xη
(Y )−A∇⊥Y η
(X)−(RN (X,Y )η
)⊤,
where X,Y ∈ C(TM) and η ∈ C(NM), or, equivalently,(∇⊥XB)(Y, Z)−
(∇⊥YB)(X,Z) =
(RN (X,Y )Z
)⊥, X, Y, Z ∈ C(TM).
In particular, if N has constant sectional curvature, i.e., N = Nn(c), where c ∈ R, theCodazzi equation becomes
(∇XAη) (Y )− (∇YAη) (X) = A∇⊥Xη
(Y )−A∇⊥Y η
(X), X, Y ∈ C(TM), η ∈ C(NM),
or, equivalently,(∇⊥XB)(Y, Z) =
(∇⊥YB)(X,Z), X, Y, Z ∈ C(TM).
Moreover, if Mm is a hypersurface in Nm+1(c), then the Codazzi equation becomes
(∇XA) (Y )− (∇YA) (X) = 0.
Using the Codazzi equation, we easily nd the next result.
Proposition 1.14. Let φ :Mm → Nn be a submanifold. Then
trace∇AH =m
2grad
(|H|2
)+ traceA∇⊥
· H(·) + trace
(RN (·,H)·
)⊤. (1.9)
Corollary 1.15. Let φ :Mm → Nn(c) be a submanifold, c ∈ R. Then
trace∇AH =m
2grad
(|H|2
)+ traceA∇⊥
· H(·). (1.10)
10 Chapter 1. Preliminaries
Proposition 1.16 (The Ricci equation). Let φ :Mm → Nn be a submanifold. Then(RN (X,Y )η
)⊥= R⊥(X,Y )η +B (Aη(X), Y )−B (X,Aη(Y )) ,
where X,Y ∈ C(TM) and η ∈ C(NM), or, equivalently,
⟨RN (X,Y )η, ξ⟩ = ⟨R⊥(X,Y )η, ξ⟩ − ⟨[Aη, Aξ]X,Y ⟩,
where X,Y ∈ C(TM), η, ξ ∈ C(NM), and [Aη, Aξ] = AηAξ −AξAη.
Denition 1.17. A hypersurface φ : Mm → Nm+1 is called umbilical if A = (f/m)I,
where I is the identity tensor eld of type (1, 1).
Denition 1.18. A submanifold φ : Mm → Nn is called pseudoumbilical if AH =
|H|2I, where I is the identity tensor eld of type (1, 1).
Theorem 1.19. Let φ :Mm → Nm+1(c) an umbilical hypersurface. Then M is CMC.
Next, we recall the fundamental theorem for submanifolds.
Theorem 1.20. (The fundamental theorem for submanifolds)
(i) Let Mm be a simply connected Riemannian manifold, π : E → M a Riemannian
vector bundle of rank p with a compatible connection ∇, and let B be a symmetric
section of the homeomorphism bundle Hom(TM ×TM,E) ≡ (TM ⊗TM)∗⊗E =
(TM)∗ ⊗ (TM)∗ ⊗ E, i.e., B : C(TM)× C(TM) → E is a C∞(M) bilinear and
symmetric map. Dene, for each local section η of E, a map Aη : C(TM) →C(TM) by
⟨Aη(X), Y ⟩ = ⟨B(X,Y ), η⟩, X, Y ∈ C(TM).
If B and ∇ satisfy the Gauss, Codazzi and Ricci equations for the case of con-
stant sectional curvature c, then there is an isometric immersion φ : Mm →Nn=m+p(c), and a vector bundle isomorphism φ : C(E) → C(NM) along φ, such
that for every X,Y ∈ C(TM) and any η, ξ local sections of E:
⟨φ(η), φ(ξ)⟩ = ⟨η, ξ⟩, φ(B(X,Y )) = B(X,Y ), φ(∇Xη) = ∇⊥X φ(η),
where B and ∇⊥ are the second fundamental form, and the normal connection of
φ, respectively.
(ii) Suppose that φ and ψ are two isometric immersions of a connected manifold Mm
into Nn=m+k(c). Let NMφ, Bφ and ∇⊥φ denote the normal bundle, the second
fundamental form and the normal connection of φ, respectively; and let NMψ,
Bψ and ∇⊥ψ be the corresponding objects for ψ. If there exists a vector bundle
1.3. CMC surfaces in 3-dimensional space forms 11
isomorphism ˜φ : C(NM)φ → C(NM)ψ such that, for every X,Y ∈ C(TM) and
every η, ξ ∈ C(NM)φ:
⟨ ˜φ(η), φ(ξ)⟩ = ⟨η, ξ⟩, ˜φ(Bφ(X,Y )) = Bψ(X,Y ), ˜φ(∇⊥φXη) = ∇⊥
ψX
˜φ(η),
then there is an isometry F : Nn(c) → Nn(c) such that
ψ = F φ and dF |NMφ= ˜φ.
Theorem 1.21. (The fundamental theorem for hypersurfaces)
(i) Let Mm be a simply connected Riemannian manifold and let A : C(TM) →C(TM) be a symmetric tensor eld of type (1, 1) satisfying the Gauss and Codazzi
equations in the case of constant sectional curvature c. Then there is an isometric
immersion φ : Mm → Nm+1(c) such that A = Aη, for some unit normal vector
eld η ∈ C(NM), where Aη denotes the shape operator of the immersion φ.
(ii) Let φ :Mm → Nm+1(c) and ψ :Mm → Nm+1(c) be two connected hypersurfaces,
and let φ : C(NM)φ → C(NM)ψ be one of the two vector bundle isomorphisms.
Suppose that Bψ(X,Y ) = ˜φ(Bφ(X,Y )) or Bψ(X,Y ) = − ˜φ(Bφ(X,Y )), for every
X,Y ∈ C(TM), where Bψ and Bφ denote, respectively, the second fundamental
forms of ψ and φ. Then there exists an isometry F : Nm+1(c) → Nm+1(c) such
that ψ = F φdF |NMφ
= ˜φ or dF |NMφ= − ˜φ
.
1.3 CMC surfaces in 3-dimensional space forms
We recall now, a classical result concerning the existence of CMC surfaces in three-
dimensional space forms, i.e., in 3-dimensional spaces with constant sectional curvature
N3(c).
Theorem 1.22. ([40]) Let φ :(M2, g
)→ N3(c) be a CMC surface. Then |H|2 + c−
K ≥ 0 at any point, and either |H|2 + c −K = 0 everywhere, i.e., M is umbilical, or
|H|2 + c−K = 0 only at isolated points. Moreover, on the set where |H|2 + c−K > 0,
we have
∆log(|H|2 + c−K
)+ 4K = 0, (1.11)
or, equivalently,(|H|2 + c−K
)∆K − | gradK|2 + 4K
(|H|2 + c−K
)2= 0. (1.12)
12 Chapter 1. Preliminaries
Proof. Let φ :(M2, g
)→ N3(c) be a CMC surface and η ∈ C(NM), |η| = 1. Consider
b ∈ C(⊙2T ∗M
)such that
b(X,Y ) = ⟨B(X,Y ), η⟩ = ⟨A(X), Y ⟩, X, Y ∈ C(TM).
Let (U ;u, v) an isothermal chart onM such that g = e2ρ(du2 + dv2
), where ρ = ρ(u, v)
is a smooth function on U . If we denote by
b11 = b (∂u, ∂u) , b12 = b (∂u, ∂v) , b22 = b (∂v, ∂v) ,
the metric g can be rewritten as
g = b11du2 + 2b12dudv + b22dv
2.
The matrix of the shape operator A with respect to ∂u, ∂v is
A = e−2ρ
(b11 b12
b12 b22
).
Since A is a symmetric tensor eld of type (1, 1), traceA = f and f2 = 4|H|2, it followsthat we can consider a smooth function h1 on U such that
b11 = h1 +f
2e2ρ and b22 = −h1 +
f
2e2ρ.
As usual, we denote ∂z = (∂u − i∂v) /2. Then
b (∂z, ∂z) =1
4(b11 − b22 − 2ib12) ,
and substituting the expressions of b11 and b22 from above, we nd
2b (∂z, ∂z) = h1 − ih2,
where h2 = b12.
The matrix of A with respect to ∂u, ∂v becomes
A = e−2ρ
h1 +f2 e
2ρ h2
h2 −h1 + f2 e
2ρ
.
From the Gauss equation, detA = K − c, we get |h|2 = e4ρ(|H|2 + c−K
), where
h = h1 − ih2. It follows that |H|2 + c−K ≥ 0 at any point of U .
By a straightforward computation, it can be seen that the Codazzi equation is
equivalent to h = h1 − ih2 being holomorphic.
1.3. CMC surfaces in 3-dimensional space forms 13
Since the zeros of a non-zero holomorphic function are isolated, we have |H|2 +
c −K = 0 only at such isolated points. Since U was arbitrary chosen, it follows that
|H|2 + c−K ≥ 0 at any point of M and the zeros of |H|2 + c−K are isolated points.
On the set where |H|2+c−K > 0, as h is holomorphic, it follows that ∆log |h|2 = 0
and then
∆log(|H|2 + c−K
)+ 4K = 0.
By a straightforward computation, we obtain
∆log(|H|2 + c−K
)= − 1
|H|2 + c−K
(∆K − 1
|H|2 + c−K| gradK|
),
and using this relation, it is easy to see that (1.11) is equivalent to (1.12).
Remark 1.23. ([64]) If φ :(M2, g
)→ N3(c) is a minimal surface, i.e., H = 0, the
conclusions of Theorem 1.22 also hold.
Remark 1.24. If φ :(M2, g
)→ N3(c) is a CMC surface with constant Gaussian
curvature, then either M is umbilical or M is a at isoparametric surface with no
umbilical points. The isoparametric surfaces in N3(c) are well-known.
We also have a (kind of) converse result of Theorem 1.22.
Theorem 1.25. ([40]) Let(M2, g
)be an abstract surface, c ∈ R and a ∈ R∗
+. Assume
that on M we have a2 + c−K > 0 and
4K +∆ log(a2 + c−K
)= 0.
Then(M2, g
)admits locally a CMC embedding in N3(c) with |H| = a.
Proof. Let (U ;u, v) an isothermal chart on M such that g = e2ρ(du2 + dv2
), where
ρ = ρ(u, v) is a smooth function on U . Assume that U is simply connected (and also
connected). From (1.6), it is easy to see that K = ∆ρ = ∆ log(e4ρ)/4. Substituting K
in
4K +∆ log(a2 + c−K
)= 0,
we obtain ∆log((a2 + c−K
)e4ρ)= 0. Thus, there exists a holomorphic function h
such that
|h|2 =(a2 + c−K
)e4ρ. (1.13)
The function h is unique up to a multiplicative complex number of type eiθ. We consider
h = h1−ih2 and dene a tensor eld A of type (1, 1) on U by A (∂u) =(e−2ρh1 + a
)∂u+
e−2ρh2∂v and A (∂v) = e−2ρh2∂u +(−e−2ρh1 + a
)∂v. Then A, can be written, with
respect to ∂u, ∂v, as
A = e−2ρ
(h1 + e2ρa h2
h2 −h1 + e2ρa
).
14 Chapter 1. Preliminaries
Using (1.13), it is easy to prove that A satises the Gauss equation detA = K − c.
By a straightforward computation, since h is holomorphic, it follows that A satises
the Codazzi equation (∇∂uA) (∂v) = (∇∂vA) (∂u). Thus, there exists an isometric im-
mersion from U in N3(c) having A as its shape operator. Since traceA = 2a and
| traceA| = 2|H|, we get that |H| = a is a constant, i.e, the immersion is CMC.
If fact, as h is determined up to a multiplicative complex number of norm 1, we
have a one-parameter family of CMC immersions, all with |H| = a. We can restrict
the domain U such that these immersions become embeddings.
Remark 1.26. ([64]) If a = 0 in Theorem 1.25, then,(M2, g
)admits locally a minimal
embedding in N3(c).
1.4 The energy and bienergy functionals
A harmonic map φ : (Mm, g) → (Nn, h) between two Riemannian manifolds is a critical
point of the energy functional
E : C∞(M,N) → R, E(φ) =1
2
∫M
|dφ|2 vg,
and it is characterized by the vanishing of its tension eld
τ(φ) = traceg∇dφ.
For general accounts on the theory of harmonic maps we could see, for example, the
monographs [8, 23,25,71] and the papers [24,26].
The idea of the stress-energy tensor associated to a functional comes from D. Hilbert
([34]). Thus, to a given functional E, one can associate a symmetric 2-covariant tensor
eld S such that divS = 0 at the critical points of E. When E is the energy functional,
P. Baird and J. Eells in [4], and A. Sanini in [66], dened the tensor eld
S = e(φ)g − φ∗h =1
2|dφ|2g − φ∗h,
and proved that
divS = −⟨τ(φ), dφ⟩.
Hence, S can be chosen as the stress-energy tensor of the energy functional. It is
worth mentioning that S also has a variational meaning. Indeed, we can x a map
φ :Mm → (Nn, h) and think E as being dened on the set of all Riemannian metrics on
M . The critical points of this new functional are these Riemannian metrics determined
by the vanishing of their stress-energy tensor S.
1.4. The energy and bienergy functionals 15
More precisely, we assume that M is compact and denote by
G = g : g is a Riemannian metric on M .
For a deformation gt of g we consider ω = ddt
∣∣t=0
gt ∈ TgG = C(⊙2T ∗M
)and dene
a new functional
F : G → R, F(g) = E(φ).
Thus, we have the following result.
Theorem 1.27 ([4,66]). Let φ :Mm → (Nn, h) and assume that M is compact. Then
d
dt
∣∣∣∣t=0
F (gt) =1
2
∫M⟨ω, e(φ)g − φ∗h⟩ vg.
Therefore g is a critical point of F if and only if its stress-energy tensor S vanishes.
We mention here that, if φ : (Mm, g) → (Nn, h) is an arbitrary isometric immersion,
then divS = 0.
A natural generalization of harmonic maps is represented by biharmonic maps. A
biharmonic map φ : (Mm, g) → (Nn, h) between two Riemannian manifolds is a critical
point of the bienergy functional
E2 : C∞(M,N) → R, E2(φ) =
1
2
∫M
|τ(φ)|2 vg,
and is characterized by the vanishing of its bitension eld
τ2(φ) = −∆φτ(φ)− traceg RN (dφ, τ(φ))dφ,
where
∆φ = − traceg(∇φ∇φ −∇φ
∇)
is the rough Laplacian of φ−1TN and the curvature tensor eld is
RN (X,Y )Z = ∇NX∇N
Y Z −∇NY ∇N
XZ −∇N[X,Y ]Z, X, Y, Z ∈ C(TM).
We note that the biharmonic equation τ2(φ) = 0 is a fourth-order non-linear elliptic
equation and that any harmonic map is biharmonic. A non-harmonic biharmonic map
is called proper-biharmonic.
The theory of biharmonic maps, which represents the most natural generalization
of biharmonic functions, is an old subject with the origin in the theory of elasticity and
uid mechanics (see, for example, [1,46]). Nowadays, this theory is well developed and
we can see, for example, [2, 3, 57,9, 29,37,44,45,58,61,72].
16 Chapter 1. Preliminaries
In [39], G.Y. Jiang dened the stress-energy tensor S2 of the bienergy (also called
the stress-bienergy tensor) by
⟨S2(X), Y ⟩ =1
2|τ(φ)|2⟨X,Y ⟩+ ⟨dφ,∇τ(φ)⟩⟨X,Y ⟩
− ⟨dφ(X),∇Y τ(φ)⟩ − ⟨dφ(Y ),∇Xτ(φ)⟩,
as it satises
divS2 = −⟨τ2(φ), dφ⟩.
As in the harmonic case, the tensor eld S2 has a variational meaning too. We x
a map φ :Mm → (Nn, h) and dene a new functional
F2 : G → R, F2(g) = E2(φ).
Then we have the following result.
Theorem 1.28 ([42]). Let φ :Mm → (Nn, h) and assume that M is compact. Then
d
dt
∣∣∣∣t=0
F2 (gt) = −1
2
∫M⟨ω, S2⟩ vg,
so g is a critical point of F2 if and only if S2 = 0.
We mention that, if φ : (Mm, g) → (Nn, h) is an isometric immersion, then divS2
does not necessarily vanish.
We end this section with the following properties of the stress-bienergy tensor.
Proposition 1.29. Consider a submanifold φ :Mm → Nn. Then we have:
(i) the stress-bienergy tensor of φ is determined by
S2 = −m2
2|H|2I + 2mAH ; (1.14)
(ii) traceS2 = m2|H|2(2− m
2
);
(iii) the relation between the divergence of S2 and the divergence of AH is given by
divS2 = −m2
2grad
(|H|2
)+ 2m divAH ; (1.15)
(iv) |S2|2 = m4|H|4(m4 − 2
)+ 4m2 |AH |2.
Remark 1.30. From equation (1.15), we see that if divS2 = 0 it does not follow
that divAH automatically vanishes. In fact, only when |H| is constant divS2 = 0 is
equivalent to divAH = 0.
1.5. Biharmonic and biconservative submanifolds 17
1.5 Biharmonic and biconservative submanifolds
A submanifold φ : Mm → Nn is called biharmonic if the isometric immersion φ is a
biharmonic map from (Mm, g) to (Nn, h).
Even if the notion of biharmonicity may be more appropriate for maps than for
submanifolds, as the domain and the codomain metrics are xed and the variation is
made only through maps, the biharmonic submanifolds proved to be interesting objects
(see, for example, [61]).
If divS2 = 0 for a submanifold M in N , then M is called biconservative. Thus, M
is biconservative if and only if the tangent part of its bitension eld vanishes.
We have the following characterization theorem of biharmonic submanifolds, ob-
tained by splitting the bitension eld in the tangent and normal part.
Theorem 1.31. A submanifold Mm of a Riemannian manifold Nn is biharmonic if
and only if
traceA∇⊥· H
(·) + trace∇AH + trace(RN (·,H)·
)⊤= 0
and
∆⊥H + traceB (·, AH(·)) + trace(RN (·,H)·
)⊥= 0,
where ∆⊥ is the Laplacian in the normal bundle.
Various forms of the above result were obtained in [18,42,59]. From here we deduce
some characterization formulas for the biconservativity.
Proposition 1.32. Let φ :Mm → Nn be a submanifold. Then the following conditions
are equivalent:
(i) M is biconservative;
(ii) traceA∇⊥· H
(·) + trace∇AH + trace(RN (·,H)·
)⊤= 0;
(iii) m2 grad
(|H|2
)+ 2 traceA∇⊥
· H(·) + 2 trace
(RN (·,H)·
)⊤= 0;
(iv) 2 trace∇AH − m2 grad
(|H|2
)= 0.
The following properties are immediate.
Proposition 1.33. LetMm be a submanifold of a Riemannian manifold Nn. If ∇AH =
0, then M is biconservative.
Proposition 1.34. Let Mm be a submanifold of a Riemannian manifold Nn. Assume
that N is a space form and M is PMC. Then M is biconservative.
18 Chapter 1. Preliminaries
Proposition 1.35 ([10]). Let Mm be a pseudoumbilical submanifold of a Riemannian
manifold Nn with m = 4. Then M is a CMC submanifold.
If we consider the particular case of hypersurfaces, Theorem 1.31 gives the following
result
Theorem 1.36 ([10, 62]). If Mm is a hypersurface in a Riemannian manifold Nm+1,
then M is biharmonic if and only if
2A(grad f) + f grad f − 2f(RicciN (η)
)⊤= 0,
and
∆f + f |A|2 − f RicciN (η, η) = 0,
where η is the unit normal vector eld of M in N and f is the mean curvature function
of M .
Proposition 1.37. A hypersurface Mm in a space form Nm+1(c) is biconservative if
and only if
A(grad f) = −f2grad f. (1.16)
Corollary 1.38. Any CMC hypersurface in Nm+1(c) is biconservative.
Therefore, biconservative hypersurfaces may be regarded as the next natural topic
to be studied after CMC surfaces.
Considering the divergence in equation (1.16), using the fact that divA = grad f
and equation (1.2) with T = A and α = f , we obtain the following corollary.
Corollary 1.39. Let Mm be a biconservative hypersurface in Nm+1(c). Then
f∆f − 3| grad f |2 − 2⟨A,Hess f⟩ = 0.
Next, we show that the two distributions determined by grad f , where f is the mean
curvature function of a biconservative hypersurface in a space form, are completely
integrable. As a one-dimensional distribution is always integrable, we only must prove
the following result.
Theorem 1.40. Let Mm be a biconservative hypersurface in Nm+1(c) and assume that
grad f = 0 at any point of M . Then, the distribution D orthogonal to that determined
by grad f is completely integrable. Moreover, any integral manifold of D, of maximal
dimension, has at normal connection as a submanifold in Nm+1(c).
1.5. Biharmonic and biconservative submanifolds 19
Proof. Since grad f = 0 at any point of M , there exists the global unit vector eld
X1 = grad f/| grad f |. Since M is a biconservative hypersurface in a space form with
grad f = 0 at any point of M , using (1.16), we easily get A (X1) = −(f/2)X1.
Let Xii=1,m be a local orthonormal frame eld on M . We note that Xkk=2,m is
a local basis of the distribution D orthogonal to that determined by grad f . It is clear
that grad f = (X1f)X1 and Xkf = 0, for any k ∈ 2,m.
As grad f = 0, we can assume that there exists(U ;x1, · · · , xm
)a local chart on M
such that f = x1. We have
grad f = gij∂f
∂xi∂xj = g1j∂xj ,
and then one obtains ⟨grad f, ∂xk⟩ = 0, for any k ∈ 2,m. It follows that ∂x2 , · · · , ∂xmis a local basis for the distribution D and therefore D is completely integrable.
Further, let us denote by P an integral manifold of D, of maximal dimension, i.e. of
dimension m − 1, and by the indices 1 and 2 the objects corresponding to the normal
bundle of P in M , and in N , respectively. Consider the global unit normal vector eld
η of M in N . Clearly X1, η is a unit frame eld in the normal bundle of P in N .
Let Z ∈ C(TP ). It is clear that Z is also a tangent vector eld on M , and using
the Gauss formula (for M in N), one obtains
∇NZX1 = ∇M
Z X1 +B (Z,X1)
= ∇MZ X1 + ⟨B (Z,X1) , η⟩η
= ∇MZ X1 + ⟨A (X1) , Z⟩η,
where B is the second fundamental form of M in N and A is the shape operator of M
in N .
We note that A (Xi) = −(f/2)X1 is a normal vector eld of P in N , and then
∇NZX1 = ∇M
Z X1 ∈ C(TM).
Using the Weingarten formula (for P in N) one also has
∇NZX1 = −A2
X1(Z) +2 ∇⊥
ZX1.
Therefore, as A2X1
(Z) ∈ C(TP ), and then A2X1
(Z) ∈ C(TM), one gets 2∇⊥ZX1 ∈
C(TM). Since 2∇⊥Z , X1 is also a normal vector eld of P in N , it follows that 2∇⊥
ZX1
is collinear with X1. But |X1| = 1, and then⟨2∇⊥
ZX1, X1
⟩= 0. Finally, we obtain
2∇⊥ZX1 = 0. (1.17)
Similarly, we can see that 2∇⊥Zη ∈ C(TM). Indeed, as |η| = 1, we have ∇⊥
Zη = 0
and, from the Weingarten formula (forM in N), one obtains ∇NZ η = −A(Z) ∈ C(TM).
20 Chapter 1. Preliminaries
Now, using the Weingarten formula, but this time, for P in N , it follows that
∇NZ η = −A2
η(Z) +2 ∇⊥
Zη.
As A2η(Z) ∈ C(TM), we come to the conclusion.
Since 2∇⊥Zη ∈ C(TM) and is also a normal vector eld of P in N , one gets that
2∇⊥Zη is collinear with X1. Moreover, as ⟨X1, η⟩ = 0 and 2∇⊥
ZX1 = 0, we obtain
2∇⊥Zη = 0. (1.18)
Using (1.17) and (1.18), we conclude with 2R⊥ (Z1, Z2)σ = 0, for any Z1, Z2 ∈ C(TP )
and σ a normal vector eld of P in N .
Remark 1.41. The above result, which holds for biconservative hypersurfaces in any
space form Nm+1(c), extends the similar result for biconservative hypersurfaces in
Rm+1, obtained in [33].
Next, we study properties of biconservative surfaces in 3-dimensional space forms.
Theorem 1.42 ([15]). Let φ :M2 → N3(c) be a biconservative surface with grad f = 0
at any point of M . Then we have f > 0 and
f∆f + | grad f |2 + 4
3cf2 − f4 = 0, (1.19)
where ∆ is the Laplace-Beltrami operator on M .
Proof. Since grad f = 0 at any point ofM , we can consider X1 = (grad f)/| grad f | andX2 two vector elds such that X1(p), X2(p) is a positively oriented orthonormal basis
at any point p ∈M . In particular, we obtain that M is parallelizable. Using (1.16), we
have A (X1) = −(f/2)X1. Since traceA = f , we get A (X2) = 3f/2. Thus, the matrix
of A with respect to the (global) orthonormal frame eld X1, X2 is
A =
−f2 0
0 3f2
.
From the Gauss equation, K = c+ detA, we obtain
f2 =4
3(c−K). (1.20)
Thus c−K ≥ 0 on M .
Now, from the denitions of X1 and X2, we nd that
grad f = (X1f)X1 and X2f = 0.
1.5. Biharmonic and biconservative submanifolds 21
Further, we consider the connection forms ωji on M dened by ∇Xi = ωjiXj , where
i, j ∈ 1, 2. Obviously, ωji = −ωij . Using the Codazzi equation
∇X1A (X2)−∇X2A (X1) = A [X1, X2] ,
and X2f = 0, we get
4fω12 (X1)X1 +
(3 (X1f)− 4fω1
2 (X2))X2 = 0.
If we assume that there exists a point p0 ∈ M such that f (p0) = 0, from the above
equation, one obtains (X1f) (p0) = 0. As X2f = 0, it follows that (grad f) (p0) = 0,
which is a contradiction. Therefore, f = 0 at any point of M , and we can assume that
f > 0. This leads to c−K > 0 on M,
ω12 (X1) = 0 and ω1
2 (X2) =3X1f
4f, (1.21)
so the Levi-Civita connection on M is given by
∇X1X1 = ∇X1X2 = 0, ∇X2X1 = −3X1f
4fX2, ∇X2X2 =
3X1f
4fX1. (1.22)
By a simple computation, it can be proved that the intrinsic expression for the Gaussian
curvature K of M is
K = X1
(ω12 (X2)
)−(ω12 (X2)
)2.
If we substitute ω12 (X2) (from (1.21)) in the above relation, we can see that
K =12f (X1 (X1f))− 21 (X1f)
2
16f2. (1.23)
Also, substituting (1.23) in (1.20), we get
f (X1 (X1f)) =7
4(X1f)
2 +4
3cf2 − f4.
From (1.22) it is easy to see that
∆f = −2∑i=1
(Xi (Xif)− (∇XiXi) f)
= −X1 (X1f) +3
4
(X1f)2
f.
Now, it is easy to see that
f∆f = −7
4(X1f)
2 − 4
3cf2 + f4 +
3
4(X1f)
2
= (X1f)2 − 4
3cf2 + f4.
22 Chapter 1. Preliminaries
Therefore, the mean curvature function of a non-CMC biconservative surface must
satisfy a second-order partial dierential equation
f∆f + | grad f |2 + 4
3cf2 − f4 = 0.
Remark 1.43. We note that in the above theorem obtained in [15], the authors did
not notice that grad f = 0 at any point of M implies f = 0 at any point.
Further, we can see that around any point of M there exist positively oriented local
coordinates (U ;u, v) such that f = f(u, v) = f(u) and (1.19) is equivalent to
ff ′′ − 7
4
(f ′)2 − 4
3cf2 + f4 = 0, (1.24)
i.e., f must satisfy a second-order ordinary dierential equation.
Indeed, let p0 ∈M be an arbitrary xed point of M and let γ = γ(u) be an integral
curve of X1 with γ(0) = p0. Let ϕ be the ow of X2 and (U ;u, v) positively oriented
local coordinates with p0 ∈ U such that
X(u, v) = ϕγ(u)(v) = ϕ(γ(u), v).
We have
Xu(u, 0) = γ′(u) = X1(γ(u)) = X1(u, 0)
and
Xv(u, v) = ϕ′γ(u)(v) = X2
(ϕγ(u)(v)
)= X2(u, v).
Of course, Xu, Xv is positively oriented. If we write the Riemannian metric g on M
in local coordinates as
g = g11du2 + 2g12dudv + g22dv
2,
we get g22 = |Xv|2 = |X2|2 = 1, and X1 can be expressed with respect to Xu and Xv as
X1 =1
σ(Xu − g12Xv) = σ gradu,
where σ =√g11 − g212 > 0, σ = σ(u, v).
Let f X = f(u, v). Since X2f = 0, we nd that
f(u, v) = f(u, 0) = f(u), (u, v) ∈ U.
It can be proved that
[X1, X2] =3 (X1f)
4fX2,
and therefore X2 (X1f) = X1 (X2f)− [X1, X2] f = 0.
1.5. Biharmonic and biconservative submanifolds 23
On the other hand we have
X2 (X1f) = Xv
(1
σf ′)
= Xv
(1
σ
)f ′ (1.25)
= 0. (1.26)
We recall that
grad f = (X1f)X1 =
(1
σf ′)X1 = 0
at any point of U , and then f ′ = 0 at any point of U . Therefore, from equation (1.25),
Xv (1/σ) = 0, i.e., σ = σ(u). Since g11(u, 0) = 1, and g12(u, 0) = 0, we have σ = 1, i.e.,
on U
X1 = Xu − g12Xv = gradu. (1.27)
In [15] an equivalent expression for (1.19) it was found, i.e.,
f (X1X1f) =7
4(X1f)
2 +4c
3f2 − f4.
Therefore, using (1.27), relation (1.19) is equivalent to (1.24).
Remark 1.44. If φ : M2 → N3(c) is a non-CMC biharmonic surface, then, there
exists an open subset U such that grad f = 0 at any point of U , and f satises the
following system ∆f = f
(2c− |A|2
)A(grad f) = −f
2 grad f
,
on M . As we have seen, this system implies that, on U∆f = f
(2c− |A|2
)f∆f + | grad f |2 + 4
3cf2 − f4 = 0
,
which, in fact, is an ODE system. We getff ′′ − 3
4 (f′)2 + 2cf2 − 5
2f4 = 0
ff ′′ − 74 (f
′)2 − 43cf
2 + f4 = 0
. (1.28)
As an immediate consequence, we have(f ′)2
+10
3cf2 − 7
2f4 = 0,
and combining this with the rst integral(f ′)2
= 2f4 − 8cf2 + αf3/2
24 Chapter 1. Preliminaries
of the rst equation in (1.28), where α ∈ R is a constant, we obtain
3
2f5/2 +
14
3cf1/2 − α = 0.
If we denote f = f1/2, we get 3f5/2 + 14cf/3− α = 0. Thus, f satises a polynomial
equation with constant coecients, so f has to be a constant and then, f is also constant,
i.e., grad f = 0 on U (in fact, f has to be zero). Therefore, we come to a contradiction
and thus any biharmonic surface in N3(c) has to be CMC or minimal. From the rst
system we see that we can have proper-biharmonic surfaces in N3(c) only if c > 0 (see
[19,20] for c = 0, and [13,14] for c = ±1).
We can also note that relation (1.19) (which is extrinsic), together with (1.20), allows
us to nd an intrinsic relation that (M, g) must satisfy. More precisely, we have the
following result
Theorem 1.45 ([15]). Let φ :M2 → N3(c) be a biconservative surface with grad f = 0
at any point of M . Then the Gaussian curvature K satises
(i)
K = detA+ c = −3f2
4+ c; (1.29)
(ii) c−K > 0, gradK = 0 on M , and its level curves are circles in M with constant
curvature
κ =3| gradK|8(c−K)
;
(iii)
(c−K)∆K − | gradK|2 − 8
3K(c−K)2 = 0, (1.30)
where ∆ is the Laplace-Beltrami operator on M .
Proof. Formula (1.29) is just (1.20), which we have already proved.
In order to prove (ii), we rst note that from grad f = 0 on M one has f > 0 at
any point of M , as we have already seen, and from (1.29), we obtain c − K > 0 and
gradK = 0 at any point of M . We dene X1 = gradK/| gradK|. It is easy to see that
X1 = −X1, where X1 = grad f/| grad f |. Now, we dene the vector eld X2 = −X2
such thatX1, X2
is a positively oriented global frame eld.
We have already seen that X2f = 0. Thus, X2K = 0, i.e., the integral curves of X2
are the level curves of K. We note thatX2,−X1
is positively oriented.
From (1.22), it follows that
∇X2X2 = − 3X1K
8(c−K)X1 = −3| gradK|
8(c−K)X1
1.5. Biharmonic and biconservative submanifolds 25
and[X1, X2
]K = X1
(X2K
)− X2
(X1K
)= −X2
(X1K
)=(∇X1
X2
)(K)−
(∇X2
X1
)(K) = (∇X1X2) (K)− (∇X2X1) (K).
Using (1.22), one obtains X2
(X1K
)= 0. Now, it is easy to see that we have
X2
(3(X1K
)/(8(c−K))
)= 0, i.e.,
κ =3X1K
8(c−K)
is constant along the level curves of K. If us consider Γ = Γ(v) an integral curve of X2,
i.e., X2(Γ(v)) = Γ′(v), for any v, since∣∣∣X2
∣∣∣ = 1, it follows that Γ is a curve parametrized
by arc-length. As Γ′,− X1
∣∣∣Γ is positively oriented, and
∇Γ′Γ′ = −κ|Γ X1
∣∣∣Γ
and ∇Γ′X1 = κ|ΓΓ′,
where κ|Γ is a constant, it follows that Γ is a circle with constant curvature κ.
The last item follows easily using (1.29) and (1.19).
Remark 1.46. Using the global orthonormal frame eld X1, X2 and the local co-
ordinates (u, v) from above, we can rewrite equation (1.30) as
24(c−K)X1 (X1K) + 33 (X1K)2 + 64K(c−K)2 = 0 (1.31)
or, equivalently,
24(c−K(u))K ′′(u) + 33(K ′(u)
)2+ 64K(u)(c−K(u))2 = 0. (1.32)
Remark 1.47. By standard transformations, equation (1.32) can be rewritten as a rst
order, nonhomogeneous, linear dierential equation and then, from the classical ODE
theory we can nd the rst integral(K ′)2 = 64
3K3 − 640
9cK2 + C(c−K)11/4 +
704
9c2K − 256
9c3,
where C ∈ R is a constant.
Corollary 1.48. Let φ :M2 → N3(c) be a biconservative surface. Assume that one of
the following holds
(i) c−K ≤ 0 on M ,
or
26 Chapter 1. Preliminaries
(ii) K is constant on M .
Then M is a CMC submanifold.
While the existence of biconservative immersions in 3-dimensional space forms with
the gradient of the mean curvature function dierent from zero at any point will be
proved in the next chapter, the uniqueness is treated here.
Theorem 1.49 ([27]). Let(M2, g
)be an abstract surface and c ∈ R a constant. If M
admits two biconservative immersions in N3(c) such that the gradients of their mean
curvature functions are dierent from zero at any point of M , then the two immersions
dier by an isometry of N3(c).
Proof. Let φ :(M2, g
)→ N3(c) be a biconservative immersion as in the statement of the
theorem. Then, we can assume that its mean curvature function is f = 2√(c−K)/3,
i.e., f depends only on the abstract surface(M2, g
). Dene X1 = grad f/| grad f |. We
have already seen that X1 = gradK/| gradK| satises X1 = −X1.
It is then easy to check that
A(X1
)= −
√c−K
3X1 and A
(X2
)=√3(c−K)X2,
whereX1, X2
is a positively oriented global orthonormal frame eld. It follows that
the shape operator of the immersion φ does not depend on φ, but only on the surface(M2, g
)and we can conclude.
Chapter 2Biconservative surfaces in
3-dimensional space forms
In this chapter, we focus on two issues: rst, we consider biconservative surfaces(M2, g
)in a 3-dimensional space formN3(c), with mean curvature function f satisfying grad f =0 at any point, and determine a certain Riemannian metric gr onM such that
(M2, gr
)is a Ricci surface in N3(c); second, we obtain the intrinsic necessary and sucient
conditions for an abstract surface to be locally embedded in N3(c) as a non-CMC
biconservative surface.
Most of the results in this chapter are original and they were presented in [27] and
[56]. There are also few results which are presented here for the rst time (see Theorem
2.10, Theorem 2.17, Theorem 2.21).
2.1 Biconservativity and minimality in N 3(c)
An abstract surface(M2, g
)with Gaussian curvature K is said to satisfy the Ricci
condition with respect to c (or simply the Ricci condition) if c−K > 0 and the metric√c−Kg is at, where c ∈ R is a constant. In this case,
(M2, g
)is called a Ricci
surface with respect to c (or simply a Ricci surface). As we will see further, in this
chapter, when c = 0, a surface satisfying the Ricci condition can be locally isometrically
embedded in R3 as a minimal surface. Actually, there exists a one-parameter family of
such embeddings. H. B. Lawson in [40] generalized this result by showing that the Ricci
condition is an intrinsic characterization of minimal surfaces in space forms N3(c), with
constant sectional curvature c (see also [65]).
In the following, we will see that the Ricci condition, as stated above, is equivalent
to an equation that looks very much like equation (1.30), satised by the Gaussian
curvature of a non-CMC biconservative surface in a space form N3(c). Then, a natural
27
28 Chapter 2. Biconservative surfaces in 3-dimensional space forms
question is whether there exists a simple way to transform surfaces satisfying (1.30) in
Ricci surfaces in N3(c). As it will turn out, the answer to this question is armative.
The following proposition points out some equivalent characterizations of Ricci sur-
faces.
Proposition 2.1. Let(M2, g
)be an abstract surface such that its Gaussian curvature
K satises c −K > 0, where c ∈ R is a constant. Then, the following conditions are
equivalent:
(i) K satises
(c−K)∆K − | gradK|2 − 4K(c−K)2 = 0; (2.1)
(ii) K satises
∆log(c−K) + 4K = 0; (2.2)
(iii) the metric√c−Kg is at.
Moreover, when c = 0, we also have a fourth equivalent condition:
(iv) the metric (−K)g has constant Gaussian curvature equal to 1.
Proof. First, we prove that
∆log(c−K) =(K − c)∆K + | gradK|2
(c−K)2. (2.3)
Let us consider p ∈M an arbitrary point and X1, X2 a local orthonormal frame eld
that is geodesic around p. Then, at p we have
∆log(c−K) =−2∑i=1
Xi (Xi log(c−K))
=− ∆K
c−K+
2∑i=1
(Xi(K))2
(c−K)2
=− ∆K
c−K+
| gradK|2
(c−K)2
=(K − c)∆K + | gradK|2
(c−K)2.
Obviously, since p was arbitrary xed in M , relation (2.3) holds globally.
Now it is easy to see that (i) and (ii) are equivalent. If we assume that (i) holds and
we substitute (c−K)∆K from (2.3) in (2.1), we nd (2.2). Conversely, if (ii) holds, we
substitute ∆log(c−K) from (2.2) in (2.3) and obtain (2.1).
2.1. Biconservativity and minimality in N3(c) 29
Next, in the same way as in [53], we consider a family of Riemannian metrics on M
given by gr = (c −K)rg, where r ∈ R is a constant. From equation (1.6), one obtains
that the Gaussian curvature curvature Kr of gr is given by
Kr = (c−K)−r(K +
1
2∆ log(c−K)r
). (2.4)
If (ii) holds, then substituting (2.2) in (2.4), we get Kr = (1− 2r)(c−K)−rK. Now, if
we consider the particular cases, r = 1/2 or r = 1, it follows that (ii) implies (iii) and
(iv). Conversely, it is easy to see, from (2.4), that (iii) implies (ii) and also, if c = 0,
(iv) implies (ii).
Remark 2.2. Proposition 2.1 was rst proved in the case when c = 0 in [53].
Working exactly as in the proof of Proposition 2.1 we get the following result.
Proposition 2.3. Let(M2, g
)be an abstract surface such that its Gaussian curvature
K satises c −K > 0, where c ∈ R is a constant. Then, the following conditions are
equivalent:
(i) K satises equation (1.30);
(ii) ∆log(c−K) + 83K = 0;
(iii) the metric (c−K)3/4g is at.
Moreover, when c = 0, we also have a fourth equivalent condition:
(iv) the metric (−K)g has constant Gaussian curvature equal to 1/3.
Now, we can state our rst main result.
Theorem 2.4. Let(M2, g
)be an abstract surface with negative Gaussian curvature K
that satises
K∆K + | gradK|2 + 8
3K3 = 0. (2.5)
Then(M2,
√−Kg
)is a Ricci surface in R3.
Proof. From Proposition 2.1, one can see that suces to show that there exists a
Riemannian metric on M , conformally equivalent to g, that satises (2.2).
In order to nd such a metric, let us consider again the metrics gr = (−K)rg, with
r ∈ R. Since K satises (2.5), and therefore equation (1.30) for c = 0, from Proposition
2.3, it follows that
∆log(−K) = −8
3K. (2.6)
30 Chapter 2. Biconservative surfaces in 3-dimensional space forms
Now, substituting (2.6) in equation (2.4) corresponding to c = 0, one obtains that the
Gaussian curvature curvature Kr of gr is given by
Kr = (−K)−r(K +
1
2∆ log(−K)r
)= −3− 4r
3(−K)1−r.
Assume that 3− 4r > 0, i.e., Kr < 0, and then, using equations (1.7) and (2.6), we can
compute
∆r log(−Kr) = ∆r log
(3− 4r
3(−K)1−r
)= (1− r)∆r log(−K)
= (1− r)(−K)−r∆log(−K)
=8(1− r)
3(−K)1−r,
where ∆r is the Laplacian of gr. Now, equation (2.2) becomes
0 = ∆r log(−Kr) + 4Kr
=8(1− r)
3(−K)1−r − 4(3− 4r)
3(−K)1−r =
4(2r − 1)
3(−K)1−r
and we get that r = 1/2.
We just have proved that(M2, g1/2 =
√−Kg
)is a Ricci surface with Gaussian
curvature K1/2 = −(1/3)√−K < 0.
From Theorem 1.45 and Theorem 2.4, one obtains the following corollary.
Corollary 2.5. Let(M2, g
)be a biconservative surface in R3, where g is the induced
metric on M . If (grad f)(p) = 0 at any point p ∈ M , then(M2,
√−Kg
)is a Ricci
surface.
Remark 2.6. In the same way as in Theorem 2.4, one can show that if(M2, g
)is a
Ricci surface in R3 with negative Gaussian curvature K, then the Gaussian curvature
of(M2, (−K)−1g
)is negative and satises equation (2.5).
Although the method used to prove Theorem 2.4 does not work in the case of non-
at space forms, it is still possible to extend this result to the case of space forms, as
shown by the following theorem.
Theorem 2.7. Let(M2, g
)be a biconservative surface in a space form N3(c) with
induced metric g and Gaussian curvature K. If (grad f)(p) = 0 at any point p ∈ M ,
then, on an open dense subset,(M2, (c−K)rg
)is a Ricci surface in N3(c), where r is
a locally dened function that satises
K +∆
(1
4log (c−Kr) +
r
2log(c−K)
)= 0, (2.7)
2.1. Biconservativity and minimality in N3(c) 31
with the Gaussian curvature Kr of (c−K)rg given by
Kr = (c−K)−r(3− 4r
3K +
1
2log(c−K)∆r + (c−K)−1g(grad r, gradK)
).
Proof. Let us consider a family of Riemannian metrics gr = (c−K)rg on M , this time
r being a function on M . From (1.6), we have that the Gaussian curvature Kr of gr is
given by
Kr = (c−K)−r(K +
1
2∆(r log(c−K))
), (2.8)
where K is the Gaussian curvature of g. Since M is biconservative, then relation (1.30)
holds, and using Proposition 2.3, and the well-known property (see, for example [17])
∆(αβ) = (∆α)β + α(∆β)− 2g(gradα, gradβ),
where α, β are smooth functions on M , we obtain that
∆(r log(c−K)) = r∆log(c−K) + log(c−K)∆r − 2g(grad r, grad log(c−K))
= −83rK + log(c−K)∆r + 2(c−K)−1g(grad r, gradK),
(2.9)
If we substitute the above expression in (2.8) it follows that
Kr = (c−K)−r(3− 4r
3K +
1
2log(c−K)∆r + (c−K)−1g(grad r, gradK)
). (2.10)
Next, assume that (c−Kr) (p) > 0 at any point p ∈M and consider a new Riemannian
metric g on M given by
g =√c−Krgr =
√c−Kr(c−K)rg
= e2ρg.
From the denition of g, one obtains
ρ =1
4log (c−Kr) +
r
2log(c−K). (2.11)
We denote by K the Gaussian curvature corresponding to g and ask that it vanishes.
From equation (1.6) we have
K =√c−Kr(c−K)−r(K +∆ρ).
Therefore, K = 0 is equivalent to
K +∆ρ = 0. (2.12)
32 Chapter 2. Biconservative surfaces in 3-dimensional space forms
Now, if we consider a (arbitrary) local orthonormal frame eld X1, X2 on M , we can
see that
∆log (c−Kr) =−2∑i=1
Xi (Xi (log (c−Kr)))− (∇XiXi) (log (c−Kr))
=−2∑i=1
(c−Kr)
−1 (−Xi (Xi (Kr)) + (∇XiXi) (Kr))
− (c−Kr)−2 (Xi (Kr))
2
=(c−Kr)
−2((Kr − c)∆Kr + |gradKr|2
). (2.13)
Using (2.11), (2.13), (2.9), and then (2.10) we get on M
∆ρ =1
4∆ log (c−Kr) +
1
2∆(r log(c−K))
=1
4(c−Kr)
−2 ((Kr − c)∆Kr + | gradKr|2)−4
3rK +
1
2log(c−K)∆r
+ (c−K)−1g(grad r, gradK)
=1
4(c−Kr)
−2
(Kr − c)∆
((c−K)−r
(3− 4r
3K +
1
2log(c−K)∆r
+ (c−K)−1g(grad r, gradK)
))+
∣∣∣∣ grad((c−K)−r(3− 4r
3K +
1
2log(c−K)∆r
+ (c−K)−1g(grad r, gradK)
))∣∣∣∣2− 4
3rK +
1
2log(c−K)∆r + (c−K)−1g(grad r, gradK).
Equation (2.12) is a fourth order PDE in r. The leading term is
−1
8(c−Kr)
−1 log(c−K)∆2r
and all the other derivatives of r are of order less or equal to three.
Now, let us consider X1 = (grad f)/| grad f |. Since grad f = 0 at any point of M ,
then X1, X2 is a global orthonormal frame eld on M , as in Theorem 1.42. We recall
that, in the same theorem, it was proved that X2f = 0, which implies, using (1.20),
that also X2K = 0.
Assuming that r is a function on M such that X2r = 0, from formulas in (1.22),
it easily follows that [X1, X2] (K) = 0 and [X1, X2] (r) = 0. Therefore, we also have
X2 (X1K) = 0 and X2 (X1r) = 0.
Obviously, from (1.29), we have gradK = 0 at any point of M . We note that the
function log(c−K) cannot vanish on an open subset of M . Indeed, if we assume that
2.1. Biconservativity and minimality in N3(c) 33
log(c−K) = 0 on an open subset, it follows that grad(log(c−K)) = 0 on that subset,
where also gradK = 0, which is a contradiction.
Now, away from the points where log(c −K) = 0, using the above equation, then
(1.22) and (1.29), equation (2.12) can be written as
∆2r = F (r,X1r,X1(X1r), X1(X1(X1r))), (2.14)
where the coecients in the expression F in the right hand side are smooth functions
depending on K, X1(K), X1(X1K), and X1(X1(X1K)). In fact, these coecients
depend only on K and X1K, as (1.31) holds.
Let us consider a point p0 ∈ M , where log(c − K) = 0, and γ = γ(u) an integral
curve of X1 with γ(0) = p0. Let ϕ be the ow of X2 and, on a neighborhood U ⊂M of
p0, dene a local parametrization of M ,
X(u, v) = ϕγ(u)(v) = ϕ(γ(u), v).
We have X(u, 0) = ϕγ(u)(0) = γ(u),
Xu(u, 0) = ∂u(u, 0) = γ′(u) = X1(γ(u)) = X1(u, 0),
and
Xv(u, v) = ∂v(u, v) = ϕ′γ(u)(v) = X2(ϕγ(u)(v)) = X2(u, v),
for any u and v. Assume now that u ∈ I, where I is an open interval containing 0 such
that log(c−K(u)) = 0, for any u ∈ I.
By hypothesis, we have X2r = 0, which means that r = r(u) on U . Moreover, from
(1.27), we have X1 = Xu − g12Xv. Thus, on U , X1(X1r) = r′′(u), X1 (X1 (X1r)) =
r′′′(u), and X1 (X1 (X1 (X1r))) = r(iv)(u), respectively. Moreover, the same formulas
hold if we take K instead of r. Therefore, on U , equation (2.14) becomes
r(iv)(u) = F (u, r(u), r′(u), r′′(u), r′′′(u)). (2.15)
We note that the coecients in the expression of F in the right hand side are smooth
functions depending only on K and K ′(u), as shown by (1.32).
The initial conditions follow from (c−Kr) (p0) > 0, i.e., from
(c−K(0))−r(0)(3− 4r(0)
3K(0) +
1
2log(c−K(0))
(− r′′(0) +
3f ′(0)
4f(0)r′(0)
)+ (c−K(0))−1r′(0)K ′(0)
)< c.
We can choose r(0), r′(0), and r′′(0) such that the above inequality is satised. Indeed,
consider a smooth function H : I × R3 → R dened by
H(u, ξ1, ξ2, ξ3
)=(c−K(u))−ξ
1
(3− 4ξ1
3K(u)
+1
2log(c−K(u))
(− ξ3 +
3f ′(u)
4f(u)ξ2)+ (c−K(u))−1ξ2K ′(u)
).
34 Chapter 2. Biconservative surfaces in 3-dimensional space forms
Then, as the limit of H, when(u, ξ1, ξ2, ξ3
)goes to 0, is K(0) which is less than c, then
there exists an open subset D ⊂ R4 containing 0,
D = (−ε, ε)× (−ε, ε)× (−ε, ε)× (−ε, ε) = (−ε, ε)4 ,
with (−ε, ε) ⊂ I, such that H(u, ξ1, ξ2, ξ3
)< c, for any
(u, ξ1, ξ2, ξ3
)∈ D.
Now, from the ODE's theory, we know that equation (2.15), with the given initial
conditions of type (0, r0, r
′0, r
′′0 , r
′′′0
)∈ D × R,
has a unique solution. More precisely, we denote
ξ1 = r(u), ξ2 = r′(u), ξ3 = r′′(u), ξ4 = r′′′(u)
and consider
G : D × R = (−ε, ε)× (−ε, ε)3 × R → R4,
dened by
G(u, ξ1, ξ2, ξ3, ξ4
)=(ξ2, ξ3, ξ4, F
(u, ξ1, ξ2, ξ3, ξ4
)).
Equation (2.15) is equivalent to
ξ′(u) = G
(u, ξ(u)
). (2.16)
The function G is smooth and therefore, for the initial condition(0, r0, r
′0, r
′′0 , r
′′′0
)∈ D × R,
there exists a unique local solution ξ = ξ(u) dened around 0. Thus, equation (2.15)
has a unique local solution r = r(u).
This means that there exists a at Riemannian metric g =√c−Krgr on (a smaller)
U , and then we use [40, Theorem 8] to conclude that, on the open dense subset, our
surface (M2, g) can be locally conformally embedded in N3(c) as a minimal surface.
Remark 2.8. It is straightforward to verify that, when c = 0, the only constant solution
of equation (2.7) is r = 1/2. When c > 0, we note that r = 3/4 is a solution of (2.7).
Therefore,(M2, (c−K)3/4g
)is a at surface and then, trivially, a Ricci surface with
respect to c > 0; it can be immersed in the Euclidean 3-dimensional sphere of radius
1/√c as the minimal Cliord torus.
Remark 2.9. As we will see in the next section (see Theorem 2.18), the hypotheses of
Theorem 2.7 can be replaced by:
Let(M2, g
)be an abstract surface and c ∈ R a constant. Assume that c−K > 0,
gradK = 0 at any point of M , and the level curves of K are circles in M with constant
curvature
κ =3| gradK|8(c−K)
.
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 35
2.2 An intrinsic characterization of biconservative
surfaces in N 3(c)
While any of the equivalent conditions in Proposition 2.1 characterizes intrinsically min-
imal surfaces in 3-dimensional space forms N3(c), the similar conditions in Proposition
2.3 alone fail to do the same in the case of biconservative surfaces. In this section, we
will nd the intrinsic necessary and sucient conditions for an abstract surface to be
locally embedded in N3(c) as a non-CMC biconservative surface.
According to Theorem 1.45, (ii), we present some conditions equivalent to the fact
that the level curves of the Gaussian curvature are circles.
Theorem 2.10. Let(M2, g
)be an abstract surface with Gaussian curvature K satis-
fying c−K(p) > 0 and (gradK)(p) = 0 at any point p ∈M , where c ∈ R is a constant.
Let X1 = (gradK)/| gradK| and X2 ∈ C(TM) be two vector elds on M such that
X1(p), X2(p) is a positively oriented basis at any point p ∈ M . Then, the following
conditions are equivalent:
(i) the level curves of K are circles in M with constant curvature
κ =3| gradK|8(c−K)
=3X1K
8(c−K);
(ii)
X2 (X1K) = 0 and ∇X2X2 =−3X1K
8(c−K)X1;
(iii)
∇X1X1 = ∇X1X2 = 0, ∇X2X2 = − 3X1K
8(c−K)X1, ∇X2X1 =
3X1K
8(c−K)X2.
Proof. In order to prove (i) ⇒ (ii), we note that from the denition of X1, X2 we
have X2K = 0, i.e., the integral curves of X2 are the level curves of K. Since κ is
constant along the integral curves of X2, one obtains X2 (X1K) = 0. Let us now
consider Γ = Γ(v) an integral curve of X2, i.e., X2(Γ(v)) = Γ′(v), for any v. It follows
that Γ is a circle in M with curvature κ. Since |X2| = 1, we also get that Γ is a curve
parametrized by arc-length. As Γ′,− X1|Γ is positively oriented, one obtains
∇Γ′Γ′ = ∇X2X2 = −κ|Γ X1|Γ .
The proof of (ii) ⇒ (i) can be found in Theorem 1.45, (ii).
To prove that (iii) implies (ii) it is enough to note that [X1, X2]K = −X2 (X1K)
and also [X1, X2]K = − (∇X2X1) (K) = 0. Therefore, we have X2 (X1K) = 0.
36 Chapter 2. Biconservative surfaces in 3-dimensional space forms
In order to show the converse implication, i.e., (ii) implies (iii), we note that from the
expression of∇X2X2, one obtains, as we have already seen,∇X2X1 = 3 (X1K)X2/(8(c−K)). Thus, [X1, X2]K = (∇X1X2) (K) = 0, which is equivalent to ⟨∇X1X2, gradK⟩ =0, i.e., ⟨∇X1X2, X1⟩ = 0. Then ∇X1X2 = 0. Now, it is easy to see that ∇X1X1 =
−⟨X1,∇X1X2⟩ = 0.
Remark 2.11. The integral curves of X2 are circles in M with constant curvature
κ =3X1K
8(c−K)=
3| gradK|8(c−K)
and the integral curves of X1 are geodesics of M .
The next result gives a description of the metrics for which the level curves of K are
circles.
Theorem 2.12. Let(M2, g
)be an abstract surface with Gaussian curvature K satis-
fying (gradK)(p) = 0 and c−K(p) > 0 at any point p ∈M , where c ∈ R is a constant.
Let X1 = (gradK)/| gradK| and X2 ∈ C(TM) be two vector elds on M such that
X1(p), X2(p) is a positively oriented orthonormal basis at any point p ∈ M . If the
level curves of K are circles in M with constant curvature
κ =3X1K
8(c−K)=
3| gradK|8(c−K)
,
then, for any point p0 ∈ M , there exists a positively oriented parametrization X =
X(u, v) of M in a neighborhood U ⊂M of p0 such that
(i) the curve u → X(u, 0) is an integral curve of X1 with X(0, 0) = p0 and v →X(u, v) is an integral curve of X2, for any u and v;
(ii) K(u, v) = (K X)(u, v) = (K X)(u, 0) = K(u), for any (u, v);
(iii) for any pair (u, v), we have
g11(u, v) =9
64
(K ′(u)
c−K(u)
)2
v2 + 1,
g12(u, v) = − 3K ′(u)
8(c−K(u))v, g22(u, v) = 1;
(iv) the Gaussian curvature K = K(u) satises
24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0.
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 37
Proof. Let p0 be a xed point in M , γ = γ(u) an integral curve of X1, i.e., X1(γ(u)) =
γ′(u), with γ(0) = p0, and ϕ the ow of X2. Consider again
X(u, v) = ϕγ(u)(v) = ϕ(γ(u), v).
As we have already seen, X(u, 0) = ϕγ(u)(0) = γ(u),
Xu(u, 0) = ∂u(u, 0) = γ′(u) = X1(γ(u)) = X1(u, 0),
and
Xv(u, v) = ∂v(u, v) = ϕ′γ(u)(v) = X2(ϕγ(u)(v)) = X2(u, v),
for any u and v.
Since Xv(u, v) = X2(u, v), it follows that |Xv(u, v)|2 = 1, which means that
g22(u, v) = 1. (2.17)
We also have, for any u,
g11(u, 0) = |Xu(u, 0)|2 = 1 and g12(u, 0) = g (Xu(u, 0), Xv(u, 0)) = 0. (2.18)
Next, we nd the expression of X1 with respect to Xu = ∂u and Xv = ∂v. We
write X1 = α∂u + β∂v, where α and β are smooth functions such that α(u, 0) = 1 and
β(u, 0) = 0. Using (2.17), it follows that
1 = g (X1, X1) = α2g11 + 2αβg12 + β2g22 = α2g11 + 2αβg12 + β2,
and
0 = g (X1, X2) = αg12 + βg22 = αg12 + β.
From the second equation, one obtains β = −αg12 and, replacing in the rst one, we
get 1 = α2(g11 − g212
). Let us denote σ(u, v) =
√g11 − g212 > 0 and then we have
X1 =1
σ(∂u − g12∂v) . (2.19)
It is easy to see that α = 1/σ and β = −g12/σ. Next, we note that, from the denition
of X1 and X2, since X1(K) = | grad(K)|, one obtains X2K = 0, i.e., the integral curves
v → ϕγ(u)(v) of X2 are the level curves of K, which means that v → K(ϕγ(u)(v)) is a
constant function. Also, identifying K with K X, we can write K = K(u, v). Since
X2K = 0, it actually follows that K(u, v) = K(u, 0) = K(u), for any pair (u, v). The
level curves v → ϕγ(u)(v) of K are parametrized by arc-length and, by hypothesis, are
circles with constant curvature κ = 3X1K/(8(c −K)), which means, also using (2.19)
and the fact that κ(u, v) = κ(u), that
X1K =8
3κ(c−K) =
8
3κ(u)(c−K(u)) (2.20)
=1
σK ′ =
1
σ(u, v)K ′(u),
38 Chapter 2. Biconservative surfaces in 3-dimensional space forms
which implies that X2 (X1K) = 0 and
σ(u, v) =K ′(u)
8κ(u)(c−K(u))= σ(u) = 1,
for any u and v. Hence X1 = ∂u − g12∂v = gradu.
Now, let us x the parameter u. As X2,−X1 is positively oriented, we have
∇ϕ′γ(u)
(v)ϕ′γ(u)(v) = ∇X2X2 = κ (−X1)
= Γ122∂u + Γ2
22∂v
and then
κ = g(∇ϕ′
γ(u)(v)ϕ
′γ(u)(v),−X1
)= g
(Γ122∂u + Γ2
22∂v,−∂u + g12∂v)
= −Γ122
(g11 − g212
)= −Γ1
22,
where Γkij are the Christoel symbols. Thus, κ(u) = −Γ122(u, v), for any v, and for any
u.
Using the denition of Γ122 and 1 = σ2 = g11 − g212, we have, for any u and v
Γ122 =
1
2g11(∂g21∂v
+∂g21∂v
− ∂g22∂u
)+
1
2g12(∂g22∂v
+∂g22∂v
− ∂g22∂v
)= g11
∂g12∂v
=∂g12∂v
.
So, κ = −∂g12∂v . From equation (2.20), it follows that
K ′(u) = −8
3
∂g12∂v
(c−K(u)),
for any u and v, which leads to
∂g12∂v
= − 3K ′(u)
8(c−K(u))=
3
8(log(c−K(u)))′
and, therefore,
g12(u, v) = − 3K ′(u)
8(c−K(u))v + θ(u).
But, from (2.18), we know that g12(u, 0) = 0, which implies θ(u) = 0, and we conclude
with
g12(u, v) = − 3K ′(u)
8(c−K(u))v, (2.21)
for any u and v.
Finally, since 1 = σ2 = g11 − g212, we nd
g11(u, v) =9
64
(K ′(u)
c−K(u)
)2
v2 + 1. (2.22)
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 39
By a straightforward computation one gets the expressions of the Christoel symbols
Γ111 = −33
29
(K′(u)c−K(u)
)3v2, Γ1
12 = Γ121 = −Γ2
22 =32
26
(K′(u)c−K(u)
)2v,
Γ212 = Γ2
21 =33
29
(K′(u)c−K(u)
)3v2, Γ1
22 = − 323
K′(u)c−K(u)
(2.23)
and
Γ211 = − 3
23
(33
29
(K ′(u)
c−K(u)
)4
v3 +K ′′(u)
c−K(u)v +
11
23
(K ′(u)
c−K(u)
)2
v
). (2.24)
Using these expressions, we reobtain the formulas for the Levi-Civita connection
∇X1X1 = ∇X1X2 = 0, ∇X2X2 = − 3X1K
8(c−K)X1, ∇X2X1 =
3X1K
8(c−K)X2.
Now, from (2.23) and (2.24), one sees, after a straightforward computation, that the
Gauss equation of(M2, g
)is
K = − 1
g11
(Γ212
)u−(Γ211
)v+ Γ1
12Γ211 + Γ2
12Γ212 − Γ2
11Γ222 − Γ1
11Γ212
,
where(Γkij
)u=
∂Γkij
∂u , is equivalent to
24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0. (2.25)
Remark 2.13. Using equation (2.25), we can rewrite (2.24) in a simpler way as
Γ211 =
34
212
(K ′(u)
c−K(u)
)4
v3 +Kv.
Remark 2.14. It is easy to verify that, in the hypotheses of Theorem 2.12, equation
24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0
can be written as
(c−K)∆K − | gradK|2 − 8
3K(c−K)2 = 0.
Remark 2.15. Considering the change of coordinates (u, v) →(u, (c−K)3/8v
)= (u, s) in Theorem 2.12, we obtain, after a straightforward computation, a simpler
expression
g = du2 + (c−K)−3/4ds2
40 Chapter 2. Biconservative surfaces in 3-dimensional space forms
for the Riemannian metric on the surface. Moreover, if we consider a second change of
coordinates (u, s) →(∫ u
u0(c−K(τ))3/8 dτ, s
)= (u, s), then the metric g can be written
as
g = (c−K(u))−3/4(du2 + ds2
),
where K(u) = K(u(u)), which means that (u, s) are isothermal coordinates on the
surface.
The converse of Theorem 2.12 is the following result that ensures the existence of
surfaces(M2, g
)such that the level curves of the Gaussian curvature are circles. It can
be proved by a straightforward computation.
Theorem 2.16. Let D be an open subset of R2 = Ouv and c ∈ R a constant. Consider
K = K(u) a function on D such that K ′(u) > 0 and c−K(u) > 0, for any u, and
24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0.
Dene a Riemannian metric g = g11du2 + 2g12dudv + g22dv
2 on D by
g11(u, v) =9
64
(K ′(u)
c−K(u)
)2
v2 + 1,
g12(u, v) = − 3K ′(u)
8(c−K(u))v, g22(u, v) = 1.
Then, K is the Gaussian curvature of g and its level curves v → (u0, v) are circles in
(D, g) with curvature κ = 3K ′(u)/(8(c−K(u))).
Theorem 2.17. Let(M2, g
)be an abstract surface and c ∈ R a constant. Assume that
c−K > 0 and gradK = 0 at any point of M , and the level curves of K are circles in
M with constant curvature
κ =3| gradK|8(c−K)
.
If there exists a biconservative immersion φ :(M2, g
)→ N3(c), then grad f = 0, f > 0
at any point of M , and φ is unique.
Proof. Assume that there exists a biconservative immersion φ :(M2, g
)→ N3(c). First,
we prove that grad f = 0 on an open dense subset of M . Indeed, if we assume that
W = p ∈M | (grad f)(p) = 0
is not dense, then we have that grad f = 0 on M \W , which is an open, non-empty set.
Let us denote by V a connected component of M \W . We note that V is also open in
M . In Remark 2.14, we have seen that K satisfy
24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0. (2.26)
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 41
On the other hand, as φ is CMC on V , using the local coordinates (u, v) as above,
equation (1.12) can be rewritten as
8(c−K)K ′′ +
(8(c−K)
|H|2 + c−K+ 3
)(K ′)2 + 32K(c−K) = 0. (2.27)
Combining equations (2.26) and (2.27), one obtains
3|H|2(K ′)2 − 4K(c−K)
((|H|2 + c−K
)2+ 2|H|2
(|H|2 + c−K
))= 0. (2.28)
Now, we recall that in Remark 1.47 is given a rst integral of
24(c−K)K ′′ + 33(K ′)2 + 64K(c−K)2 = 0.
The rst integral is(K ′)2 = 64
3K3 − 640
9cK2 + C(c−K)11/4 +
704
9c2K − 256
9c3,
where C ∈ R is a constant.
Substituting (K ′)2 in (2.28), we obtain that K has to satisfy a fourth order polyno-
mial equation with constant coecients, with the leading term 4K4, so K has to be a
constant and this is a contradiction.
Thus, grad f = 0 on W , which is an open dense subset of M . From the Gauss
equation, K = c+ detA, we obtain on W
f2 =4
3(c−K).
As W is dense in M , it follows that, in fact, the above relation holds on whole M .
Therefore, since c−K > 0 and gradK = 0 on M , one obtains f > 0 and grad f = 0 at
any point of M .
Finally, the uniqueness of φ follows from Theorem 1.49.
We are now ready to prove the main result of this section, which provides an in-
trinsic characterization of non-CMC biconservative surfaces in a 3-dimensional space
form N3(c). Basically, the intrinsic conditions given in Theorem 1.45, (ii), ensure the
existence of a non-CMC biconservative immersion in N3(c).
Theorem 2.18. Let (M2, g) be an abstract surface and c ∈ R a constant. Then M can
be locally isometrically embedded in a space form N3(c) as a biconservative surface with
the gradient of the mean curvature dierent from zero at any point p ∈M if and only if
the Gaussian curvature K satises c−K(p) > 0, (gradK)(p) = 0, and its level curves
are circles in M with constant curvature
κ =3| gradK|8(c−K)
.
42 Chapter 2. Biconservative surfaces in 3-dimensional space forms
Proof. The direct implication was proved in Theorem 1.45, (ii).
To prove the converse, let us consider X1 = (gradK)/| gradK| and X2 ∈C(TM) two vector elds such that X1(p), X2(p) is a positively oriented orthonor-
mal basis at any point p ∈ M . From Theorem 2.12 we have seen that the Levi-Civita
connection on (M2, g) is given by
∇X1X1 = ∇X1X2 = 0, ∇X2X2 = − 3X1K
8(c−K)X1, ∇X2X1 =
3X1K
8(c−K)X2.
Now, consider f = (2/√3)√c−K > 0 and, since X2K = 0, we easily get
grad f = − X1K√3(c−K)
X1 = − gradK√3(c−K)
.
Dene X1 = (grad f)/| grad f | = −X1 and X2 = −X2 and then
∇X1X1 = ∇X1
X2 = 0,
and
∇X2X2 = ∇X2X2 = − 3X1K
8(c−K)X1, ∇X2
X1 = ∇X2X1 =3X1K
8(c−K)X2.
Since (X1f)/f = −(X1K)/(2(c−K)), one obtains
∇X2X2 =
3X1f
4fX1 and ∇X2
X1 = −3X1f
4fX2.
Let us now consider a tensor eld A of type (1, 1) on M dened by
AX1 = −f2X1 and AX2 =
3f
2X2.
It is straightforward to verify that A satises the Gauss equation
K = c+ detA
and the Codazzi equation
(∇X1A)X2 = (∇X2
A)X1,
which means that the surface M can be locally isometrically embedded in N3(c) with
A as its shape operator. Moreover, from the denition of A, it is easy to see that
A(grad f) = −f2grad f,
which shows, using (1.16), that M is a biconservative surface in N3(c).
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 43
Remark 2.19. If we consider the local coordinates (u, v) corresponding to the frame
eld X1, X2, as in the proof of Theorem 2.12, then (−u,−v) represent the local
coordinates corresponding to the frame eldX1, X2
, as we have seen in the proof of
(1.24).
Remark 2.20. If the surfaceM in Theorem 2.18 is simply connected, then the theorem
holds globally, but, in this case, instead of a local isometric embedding we have a global
isometric immersion.
We note that, unlike in the minimal immersions case, if M satises the hypotheses
in Theorem 2.18, then there exists a unique biconservative immersion in N3(c) (up to
an isometry of N3(c)), and not a one-parameter family.
Using Theorem 2.17 and Theorem 2.18 we can state the following result.
Theorem 2.21. Let(M2, g
)be an abstract surface and c ∈ R a constant. Assume that
c−K > 0 and gradK = 0 at any point of M , and the level curves of K are circles in
M with constant curvature
κ =3| gradK|8(c−K)
.
Then, there exists a unique biconservative immersion φ :(M2, g
)→ N3(c). Moreover,
the gradient of its mean curvature function is dierent from zero at any point of M .
Using local isothermal coordinates, we can nd some more intrinsic characteriza-
tions of biconservative surfaces in N3(c). These characterizations give some explicit
expressions for the metric g.
Theorem 2.22. Let(M2, g
)be an abstract surface with Gaussian curvature K satis-
fying c−K(p) > 0 and (gradK)(p) = 0 at any point p ∈M , where c ∈ R is a constant.
Let X1 = (gradK)/| gradK| and X2 ∈ C(TM) be two vector elds on M such that
X1(p), X2(p) is a positively oriented basis at any point p ∈ M . Then, the following
conditions are equivalent:
(i) the level curves of K are circles in M with constant curvature
κ =3| gradK|8(c−K)
=3X1K
8(c−K);
(ii) the metric g can be written locally, as g = (c −K)−3/4(du2 + dv2
), where (u, v)
are local coordinates positively oriented, K = K(u), and K ′ > 0;
(iii) the metric g can be written locally, as g = e2ρ(du2 + dv2
), where (u, v) are local
coordinates positively oriented, and ρ = ρ(u) satises the equation
ρ′′ = e−2ρ/3 − ce2ρ (2.29)
44 Chapter 2. Biconservative surfaces in 3-dimensional space forms
and the condition ρ′ > 0; moreover, the solutions of the above equation, u = u(ρ),
are
u =
∫ ρ
ρ0
dτ√−3e−2τ/3 − ce2τ + a
+ u0,
where ρ is in some open interval I, ρ0 ∈ I and a, u0 ∈ R are constants;
(iv) the metric g can be written locally, as g = e2ρ(du2 + dv2
), where (u, v) are local
coordinates positively oriented, and ρ = ρ(u) satises the equation
3ρ′′′ + 2ρ′ρ′′ + 8ce2ρρ′ = 0 (2.30)
and the conditions ρ′ > 0 and c+e−2ρρ′′ > 0; moreover, the solutions of the above
equation, u = u(ρ), are
u =
∫ ρ
ρ0
dτ√−3be−2τ/3 − ce2τ + a
+ u0,
where ρ is in some open interval I, ρ0 ∈ I and a, b, u0 ∈ R are constants, b > 0.
Proof. The implication (i) ⇒ (ii) was already proved in Remark 2.15.
To prove (ii) ⇒ (i), consider a local orthonormal frame eld positively oriented
Y1, Y2 by Y1 = (c−K)3/8∂u and Y2 = (c−K)3/8∂v. It is easy to see that gradK = (c−K)3/8K ′(u)Y1 and, then | gradK| = (c −K)3/8K ′(u). Since X1 = (gradK)/| gradK|,we obtain X1 = Y1 and, therefore X2 = Y2. By direct computation, from the denitions
of Y1 and Y2 one gets
X2 (X1K) = Y2 (Y1K) = (c−K)3/8∂v
((c−K)3/8K ′
)= 0
and
∇X2X2 = ∇Y2Y2 = (c−K)3/4∇∂v∂v.
Further, to write ∇∂v∂v with respect to ∂u and ∂v, we compute the Christoel symbols
Γ122 = − 3
8(c−K)K ′ and Γ2
22 = 0
and obtain
∇∂v∂v = Γ122∂u + Γ2
22∂v = − 3K ′
8(c−K)∂u.
Therefore
∇X2X2 = − 3K ′
8(c−K)1/4∂u = −3(c−K)3/8K ′
8(c−K)(c−K)3/8∂u
= − 3Y1K
8(c−K)Y1 = − 3X1K
8(c−K)X1.
Now we use Theorem 2.10 to conclude.
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 45
To prove that (ii) implies (iii), consider a smooth function ρ = ρ(u) such that
(c −K)−3/4 = e2ρ. It follows that K = c − e−8ρ/3. On the other hand, from (1.6) we
have K = −e−2ρρ′′. Therefore,
ρ′′ = e−2ρ/3 − ce2ρ (2.31)
and K = −e−8ρ/3 + c. Since K ′ = 8ρ′e−8ρ/3/3 > 0, one obtains ρ′ > 0.
In order to solve equation (2.31), rst we multiply the equation by 2ρ′ and then,
integrating, one obtains (ρ′)2
= −3e−2ρ/3 − ce2ρ + a,
where a ∈ R is a constant. Since ρ′ > 0, we get
dρ
du=√−3e−2ρ/3 − ce2ρ + a,
which leads to
u =
∫ ρ
ρ0
dτ√−3e−2τ/3 − ce2τ + a
+ u0,
where ρ is in some open interval I, ρ0 ∈ I and u0 ∈ R is a constant.
To prove (iii) ⇒ (ii), dene ρ = −(3 log(c−K))/8 and, we only have to show that
this function satises equation (2.29) and ρ′ > 0 if and only if K ′ > 0. As we have
seen the metric g can be written as g = (c−K)−3/4(du2 + dv2
). By a straightforward
computation, from (1.6), one obtains
K = −e−2ρρ′′
= −e−2ρ3(K ′′(c−K) + (K ′)2
)8(c−K)2
= −3(K ′′(c−K) + (K ′)2
)8(c−K)5/4
.
It follows that K has to satisfy
3K ′′(c−K) + 3(K ′)2 + 8K(c−K)5/4 = 0.
Using the above relation it is easy to see that ρ = −(3 log(c − K))/8 is a solution of
(2.29).
From the denition of ρ, one gets that ρ′ > 0 if and only if K ′ > 0.
In order to show (iii) ⇒ (iv), consider the change of coordinates (u, v) = (αu, αv),
where α > 0 is a real constant. Then, the metric g can be written as
g = e2(ρ(u)+logα)(du2 + dv2
),
46 Chapter 2. Biconservative surfaces in 3-dimensional space forms
where ρ(u) = ρ (u (u)). We denote ϕ (u) = ρ(u) + logα and by a direct computation
obtain that (2.29) is equivalent to
ϕ′′ (u) = α8/3e−2ϕ(u)/3 − ce2ϕ(u). (2.32)
Multiplying the above relation by e2ϕ(u)/3 and then dierentiating, we get that ϕ satises
3ϕ′′′ (u) + 2ϕ′ (u)ϕ′′ (u) + 8ce2ϕ(u)ϕ′ (u) = 0. (2.33)
Next, we solve the above equation. First, multiply it by e2ϕ/3/3 and then integrate, to
get that
ϕ′′ (u) = be2ϕ(u)/3 − ce2ϕ(u), (2.34)
where b ∈ R is a constant.
From (2.32), one obtains b = α8/3, which shows that b > 0. We can see that b has to
be positive also using the hypothesis c−K > 0. More precisely, from (1.6) we have that
K (u) = −be−8ϕ(u)/3 + c and, since c−K > 0 at any point, one gets that the constant
b has to be positive. Since K ′ (u) = 8beϕ(u)/3/3 > 0 and b > 0, it follows that ϕ′ > 0.
Moreover, multiplying by 2ϕ′ and then integrating equation (2.34), one obtains(ϕ′)2
= −3be−2ϕ/3 − ce2ϕ + a,
where a ∈ R.As ϕ′ > 0, we have
u =
∫ ϕ
ϕ0
dτ√−3be−2τ/3 − ce2τ + a
+ u0,
where ϕ is in some open interval I, ϕ0 ∈ I and a, b, u0 ∈ R are constants, b > 0. We
note that if c > 0, then a > 0.
Denote ϕ by ρ, u by u and v by v, we come to the conclusion.
To prove the last implication, (iv) ⇒ (iii), we rst note that equation (2.30) involve,
as we have already seen, that
ρ′′ = be−2ρ/3− ce2ρ, (2.35)
where b > 0 is a real constant. Consider the change of coordinates (u, v) =(b−3/8u, b−3/8v
)and rewrite the metric g as
g = e2(ρ(u)−(3/8) log b)(du2 + dv2
),
where ρ(u) = ρ (u (u)). Denote ϕ (u) = ρ(u)− (3 log b)/8 and by a direct computation,
one obtains that (2.35) is equivalent to
ϕ′′ (u) = e−2ϕ(u)/3 − ce2ϕ(u),
an equation that was already solved when we proved (ii) ⇒ (iii). Therefore, we come
to the conclusion.
2.2. An intrinsic characterization of biconservative surfaces in N3(c) 47
Remark 2.23. If condition (ii) is satised, then K has to satisfy
3K ′′(c−K) + 3(K ′)2 + 8K(c−K)5/4 = 0
and, if c > 0, then(M2, (c−K)3/4g
)is a at surface and, trivially, a Ricci surface with
respect to c.
Remark 2.24. We have the following properties of the solutions of (2.30):
(i) the parameter b in the expression of the solution of (2.30) is not essential (and so
only the parameter a counts). Thus, we have a one-parameter family of solutions;
(ii) if ρ is a solution of (2.30), for some c, then ρ + α, where α is a real constant, is
also a solution of (2.30) for ce2α;
(iii) when c = 0, we note that if ρ is a solution of (2.30), then also ρ + constant is
a solution of the same equation, i.e, condition (i) from Theorem 2.22 is invariant
under the homothetic tranformations of the metric g. Then, we see that equation
(2.30) is invariant under ane changes of parameter u = αu + β, where α > 0.
Therefore, we solve equation (2.30) up to this change of parameter and an additive
constant of the solution ρ. The additive constant is the parameter that counts.
Chapter 3Complete biconservative
surfaces in R3 and S3
In this chapter, we extend the local classication results for biconservative surfaces in
N3(c), with c = 0 and c = 1, to global results, i.e., we construct complete biconservative
surfaces, with grad f = 0 at any point of on an open dense subset. Also, we study the
uniqueness of such surfaces in R3.
Most of the results presented here are original and they are also presented in [54],
[56], and [57]. Moreover, the results in Subsection 3.1.1 are presented here for the rst
time.
3.1 Complete biconservative surfaces in R3
In this section we construct, from extrinsic point of view, complete biconservative sur-
faces in R3 with grad f = 0 at any point of an open dense subset, and, from intrinsic
point of view, we construct a complete abstract surface(M2, g
)with K < 0 everywhere
and gradK = 0 at any point of an open dense subset ofM , that admits a biconservative
immersion in R3, dened on the whole M , with grad f = 0 on the open dense subset.
First, we recall a local extrinsic result which provides a characterization of biconser-
vative surfaces in R3.
Theorem 3.1 ([33]). Let M2 be a surface in R3 with (grad f)(p) = 0 for any p ∈ M .
Then, M is biconservative if and only if, locally, it is a surface of revolution, and the
curvature κ = κ(u) of the prole curve σ = σ(u), |σ′(u)| = 1, is a positive solution of
the following ODE
κ′′κ =7
4
(κ′)2 − 4κ4.
49
50 Chapter 3. Complete biconservative surfaces in R3 and S3
In [15] there was found the local explicit parametric equation of a biconservative
surface in R3.
Theorem 3.2 ([15]). Let M2 be a biconservative surface in R3 with (grad f)(p) = 0
for any p ∈M . Then, locally, the surface can be parametrized by
XC0(ρ, v) =
(ρ cos v, ρ sin v, uC0
(ρ)),
where
uC0(ρ) =
3
2C0
(ρ1/3
√C0ρ2/3 − 1 +
1√C0
log
(√C0ρ
1/3 +
√C0ρ2/3 − 1
))
with C0 a positive constant and ρ ∈(C
−3/20 ,∞
).
We denote by SC0the image XC0
((C
−3/20 ,∞
)× R
). We note that any two such
surfaces are not locally isometric, so we have a one-parameter family of biconservative
surfaces in R3. These surfaces are not complete.
We dene the boundary of SC0by SC0
\ SC0, where SC0
is the closure of SC0in
R3.
The boundary of SC0is the circle
(C
−3/20 cos v, C
−3/20 sin v, 0
), which lies in the
Oxy plane. At a boundary point, the tangent plane to SC0is parallel to Oz. Moreover,
along the boundary, the mean curvature function is constant fC0=(2C
3/20
)/3 and
grad fC0= 0.
Proposition 3.3. Let SC0and SC′
0. Assume that we can glue them along a curve at the
level of C∞ smoothness. Then SC0and SC′
0coincide or one of them is the symmetric
of the another with respect to the plane where the common boundary lies.
Proof. We consider SC0and SC′
0determined by
XC0(ρ, v) = ρ cos v e1 + ρ sin v e2 + uC0
(ρ) e3,
and
XC′0(ρ, v) = (ρ cos v + a1) f1 + (ρ sin v + a1) f2 +
(uC′
0(ρ) + a3
)f3,
where uC0(ρ), uC′
0(ρ) are given in Theorem 3.2, e1, e2, e3 is the canonical basis in R3,
f1, f2, f3is a positively oriented orthonormal basis of R3 and a1, a2, a3 ∈ R. Assume
that we can glue SC0and SC′
0along a curve γ = γ(s), γ′(s) = 0, for any s, at the level
of C∞ smoothness. In this case we have
γ(s) ∈ SC0∩ SC′
0
ηC0(γ(s)) || ηC′
0(γ(s))
HC0(γ(s)) = HC′
0(γ(s))(
grad∣∣∣HC0
∣∣∣ ) (γ(s)) =
(grad
∣∣∣HC′0
∣∣∣ ) (γ(s))
, (3.1)
3.1. Complete biconservative surfaces in R3 51
for any s, where the mean curvature vector eld HC0is given by HC0
= fC0ηC0
/2. For
SC0we have
ηC0(ρ, v) =
XC0,ρ×XC0,v∣∣∣XC0,ρ×XC0,v
∣∣∣= − 1√
C0ρ1/3cos v e1 −
1√C0ρ1/3
sin v e2 +
√C0ρ2/3 − 1
C0ρ2/3e3
and the mean curvature function
fC0(ρ, v) =
(1 +
(u′C0(ρ))2)−3/2
u′′C0(ρ) +
u′C0(ρ)
(1 +
(u′C0(ρ))2)
ρ
=
2
3√C0ρ4/3
> 0.
It follows that fC0(ρ, v) = fC0
(ρ), fC0= 2
∣∣∣HC0
∣∣∣, and(grad fC0
)(ρ, v) =
1
1 +(u′C0(ρ))2 f ′C0
(ρ) XC0,ρ(ρ, v)
= − 8
9C3/20 ρ3
((C0ρ
2/3 − 1)cos v e1 +
(C0ρ
2/3 − 1)sin v e2+
+
√C0ρ2/3 − 1 e3
).
Similar formulas hold for SC′0. Now, let us consider
(ρ1(s), v1(s)) =(X−1C0
γ)(s) and (ρ2(s), v2(s)) =
(X−1C′
0
γ)(s).
We can rewrite (3.1) as
XC0(ρ1(s), v1(s)) = XC′
0(ρ2(s), v2(s))
ηC1 (ρ1(s), v1(s)) = ηC′0(ρ2(s), v2(s))
fC0(ρ1(s), v1(s)) = fC′
0(ρ2(s), v2(s))(
grad fC0
)(ρ1(s), v1(s)) = (grad fC′
0) (ρ2(s), v2(s))
, (3.2)
for any s, where ρ1(s) ≥ C−3/20 and ρ2(s) ≥
(C ′0
)−3/2.
First, we can notice that C0ρ2/31 (s)− 1 = 0 if and only if C ′
0ρ2/32 (s)− 1 = 0. Next,
we consider two cases.
In the rst case, when C0ρ2/31 (s)−1 = 0 for any s, by a straightforward computation,
from the third relation of (3.2), we can see that C0 = C ′0 and ρ1(s) = ρ2(s) = C
−3/20 ,
52 Chapter 3. Complete biconservative surfaces in R3 and S3
for any s. Moreover, uC0(ρ1(s)) = 0 and uC′
0(ρ2(s)) = 0. Then, from the rst relation
we get a1 = a2 = a3 = 0 and ⟨e1, f3⟩ = ⟨e2, f3⟩ = 0, i.e., e3 = ±f3. Therefore, SC0
and SC′0coincide or one of them is the symmetric of another with respect to the ane
plane where the common boundary lies.
In the second case, we suppose that there exists s0 such that C0ρ2/31 (s0) − 1 = 0.
It follows that also C ′0ρ
2/32 (s0) − 1 = 0. Thus, we get that C0ρ
2/31 (s) − 1 > 0 and
C ′0ρ
2/32 (s) − 1 > 0 around s0. By direct computation, from (3.2), we obtain C0 = C ′
0,
a1 = a2 = a3 = 0, ρ1(s) = ρ2(s) around s0, and ⟨e3, f3⟩ = 1, i.e., e3 = f3. Therefore,
in this case SC0and SC′
0coincide.
However, we must then check that we have a smooth gluing.
Theorem 3.4. If φ : M2 → R3 is a biconservative surface with grad f = 0 at any
point, then there exists a unique C0 such that φ(M) ⊂ SC0.
Proof. From Theorem 3.2, it is easy to see that any point of M admits an open neigh-
borhood which is an open subset of some SC0. Let us consider p0 ∈ M . Then, using
Proposition 3.3, it follows that there exists a unique C0 such that φ(U) ⊂ SC0, where
U is an open neighborhood of p0. If V denotes the set of all points of M such that
they admit open neighborhoods which are open subsets of that SC0, then the set V is
non-empty, open and closed in M . Indeed, it is clear that V is non-empty and open. In
order to prove that V is closed, we x a point q0 ∈ V and note that, from Theorem 3.2,
there exists an open neighborhood W of q0 such that W is an open subset of some SC′0.
If C0 = C ′0, we obtain that q0 ∈ V , so the set V is closed. If we assume that C0 = C ′
0,
since W ∩ V is non-empty, open and W ∩ V ⊂ SC0∩ SC′
0, using again Proposition 3.3,
it follows that SC0= SC′
0and V is a closed set.
Thus, as M is connected, it follows that V =M .
In order to obtain a complete biconservative surface in R3, we can expect to glue
along the boundary two biconservative surfaces of type SC0corresponding to the same
C0 (the two constants have to be the same) and symmetric to each other, at the level
of C∞ smoothness.
We have the following global extrinsic result.
Theorem 3.5 ([49, 54]). If we consider the symmetry of Graf uC0, with respect to the
Oρ(= Ox) axis, we get a smooth, complete, biconservative surface SC0in R3. Moreover,
its mean curvature function fC0is positive and grad fC0
is dierent from zero at any
point of an open dense subset of SC0.
Proof. Obviously, limρC
−3/20
uC0(ρ) = 0. As u′
C0(ρ) > 0 for any ρ ∈
(C
−3/20 ,∞
), we
can think ρ as a function of u and
XC0(u, v) =
(ρC0
(u) cos v, ρC0(u) sin v, u
), u ∈ (0,∞).
3.1. Complete biconservative surfaces in R3 53
If we consider the symmetry of Graf uC0, when ρ ∈
(C
−3/20 ,∞
)with respect to the
Oρ = Ox axis, we get a smooth complete biconservative surface SC0in R3, given by
XC0(u, v) =
(xC0
(u) cos v, xC0(u) sin v, u
), (u, v) ∈ R,
where
xC0(u) =
ρC0
(u), u > 0
C−3/20 , u = 0
ρC0(−u), u < 0
is a smooth function. Moreover, the curvature function f is positive and grad f is
dierent from zero at any point of an open dense subset of SC0. We note that grad fC0
vanishes only along the boundary of SC0.
Remark 3.6. The prole curve σC0=(ρ, 0, uC0
(ρ))
≡(ρ, uC0
(ρ))can be repara-
metrized as
σC0(θ) =
(σ1C0(θ), σ2
C0(θ))
= C−3/20
((θ + 1)3/2, 32
(√θ2 + θ + log
(√θ +
√θ + 1
))), θ > 0,
(3.3)
and now XC0= XC0
(θ, v).
Remark 3.7. The boundary of SC0coincide with the boundary of SC0
as a subset of
SC0, i.e., the intersection between the closure of SC0
in SC0and the closure of SC0
\SC0
in SC0.
Proposition 3.8. The homothety of R3, (x, y, z) → C0(x, y, z), renders S1 onto SC−2/30
.
For the sake of completeness we represent in Figures 3.1, 3.2 and 3.3 the surfaces
SC0, SC0
, and the prole curve of SC0, respectively, when C0 = 91/3 (we will see that
this constant corresponds to the constant C0 = 1).
Now, we change the point of view and construct, from intrinsic point of view, com-
plete biconservative surfaces in R3 with grad f = 0 on an open and dense subset. Using
local intrinsic characterization (Theorem 2.22), for c = 0, one gets the next result.
Proposition 3.9. Let(M2, g
)be a Riemannian surface with Gaussian curvature K
satisfying (gradK)(p) = 0 and K(p) < 0 at any point p ∈ M . Consider X1 =
gradK/| gradK| and X2 ∈ C(TM) be two vector elds on M such that X1(p), X2(p)is a positively oriented orthonormal basis at any point p ∈M . Then X2 (X1K) = 0 and
∇X2X2 = (3 (X1K) /(8K))X1 if and only if the Riemannian metric g can be locally
written as
gC0(u, v) = C0 (coshu)6 (du2 + dv2), u > 0,
where C0 ∈ R is a positive constant.
54 Chapter 3. Complete biconservative surfaces in R3 and S3
Figure 3.1: The surface SC0. Figure 3.2: The complete surface SC0
.
Figure 3.3: The prole curve of SC0.
Proof. For c = 0, equation (2.30) becomes
3ρ′′′(u) + 2ρ′(u)ρ′′(u) = 0, (3.4)
with initial conditions ρ′ > 0 and ρ′′ > 0. We note that since K = −e−2ρ(u)ρ′′(u) < 0,
it is clear that ρ′′(u) > 0 for any u.
By a straightforward computation, we get the unique solution of (3.4)
ρ(u) = a
∫ u
u′0
1− e−2a(τ+u0)/3
1 + e−2a(τ+u0)/3dτ + b1, u ∈ I, (3.5)
where a, b1, u0 ∈ R, I is an open interval and u′0 ∈ I is arbitrary xed.
Next, we can assume that K ′(u) > 0 and we compute the integral in (3.5). First,
we prove that K ′(u) > 0 if and only if u+ u0 > 0.
3.1. Complete biconservative surfaces in R3 55
Since
K(u) = −e−2ρ(u)ρ′′(u), u ∈ I, (3.6)
we have that
K ′(u) = e−2ρ(u)(2ρ′(u)ρ′′(u)− ρ′′′(u)
)> 0, u ∈ I,
if and only if
2ρ′(u)ρ′′(u)− ρ′′′(u) > 0, u ∈ I. (3.7)
From (3.5) we get
ρ′′′(u) = −8a3e−2a(u+u0)/3
(1− e−2a(u+u0)/3
)9(1 + e−2a(u+u0)/3
)3 .
If we replace the rst, the second and the third derivatives of ρ in (3.7), we obtain
that K ′(u) > 0 if and only if a3(1− e−2a(u+u0)/3
)> 0. It is easy to check that this is
equivalent to u+ u0 > 0 if either a > 0 or a < 0.
Therefore, the solution is
ρ(u) = a
∫ u
u′0
1− e−2a(τ+u0)/3
1 + e−2a(τ+u0)/3dτ + b1, u ∈ I, u+ u0 > 0,
where b1, u0 ∈ R, a ∈ R∗, I is an open interval and u′0 ∈ I is arbitrary xed.
Then, we denote by I the integral in (3.5) and, in order to compute it, we consider
some changes of variables. First, if we denote by s = −2a (τ + u0) /3, we obtain
I =(u− u′0
)+
3
a
∫ −2a(u+u′0)/3
−2a(u0+u′0)/3
es
1 + esds.
We continue with an other substitution t = es and one gets
I =(u− u′0
)+
3
a
(log(1 + e−2a(u+u′0)/3
)− log
(1 + e−2a(u0+u′0)/3
)).
If we substitute this expression of I in (3.5), it follows that
ρ(u) = 3 log(1 + e−2a(u+u′0)/3
)+ au+ b2, u ∈ I, u+ u0 > 0,
where b2, u0 ∈ R, a ∈ R∗.
Now, we consider two cases: if a > 0 and if a < 0. In the rst case, we make the
change of coordinates (u, v) = (3u/a− u0, 3v/a) and in the second one, we consider the
change of coordinates (u, v) = (−3u/a− u0,−3v/a). We note that in both situations,
one gets
ρ (u) = ρ (u (u)) = 3 log (cosh u) + b,
56 Chapter 3. Complete biconservative surfaces in R3 and S3
where b ∈ R, and since g =(9/a2
)e2ρ(u)
(du2 + dv2
), we nd
gC0 = C0 (cosh u)6 (du2 + dv2
),
where (W ; u, v) is an isothermal chart positively oriented, u > 0, and C0 ∈ R is a
positive constant.
Remark 3.10. We note that, when c = 0, we have a one-parameter family of solutions
of equation (2.30), i.e., gC0 = C0(coshu)6(du2 + dv2
), C0 being a positive constant.
Concerning the complete biconservative surfaces in R3, with grad f = 0 at any point
of an open dense subset, we have the next global intrinsic result.
Theorem 3.11. Let(R2, gC0 = C0 (coshu)
6 (du2 + dv2))
be a surface, where C0 ∈ Ris a positive constant. Then we have:
(i) the metric on R2 is complete;
(ii) the Gaussian curvature is given by
KC0(u, v) = KC0(u) = − 3
C0 (coshu)8 < 0, K ′
C0(u) =
24 sinhu
C0 (coshu)9 ,
and therefore gradKC0 = 0 at any point of R2 \Ov;
(iii) the immersion φC0 :(R2, gC0
)→ R3 given by
φC0(u, v) =(σ1C0
(u) cos(3v), σ1C0(u) sin(3v), σ2C0
(u))
is biconservative in R3, where
σ1C0(u) =
√C0
3(coshu)3 , σ2C0
(u) =
√C0
2
(1
2sinh(2u) + u
), u ∈ R.
Proof. In order to prove (i), we use Proposition 1.8.
Consider g0 = du2 + dv2 the Euclidean metric on R2, which is complete. Then,
denote by g the Riemannian metric g = (coshu)6g0, and note that
g − g0 =((coshu)6 − 1
)g0
is non-negative denite at any point of R2. Therefore g is also complete and since
gC0 = C0g, it follows that(R2, gC0
)is complete.
To prove (ii), we consider the formula (3.6), with φ(u) = log(√
C0 (coshu)3)and
obtain that the Gaussian curvature KC0(u, v) is equal to
KC0(u, v) = KC0(u) = − 3
C0 (coshu)8
3.1. Complete biconservative surfaces in R3 57
and
K ′C0(u) =
24
C0
sinhu
(coshu)9.
Therefore, K ′C0(u) > 0 if and only if u > 0, K ′
C0(u) < 0 if and only if u < 0, and
K ′C0(0) = 0. Since
(gradKC0) (u, v) =1
C0e−6 log(coshu)K ′
C0(u)∂u,
we have gradKC0 = 0 at any point of R2 \Ov, which is an open dense subset of R2.
We begin the proof of (iii), recalling that, from Remark 3.6, we have the result
which says that if we consider a biconservative surface in R3, with non-constant mean
curvature, then, locally, it is a surface of revolution with the prole curve
σ+C0(θ) =
(σ1C0(θ), σ2
C0(θ))
= C−3/20
((θ + 1)3/2,
3
2
[√θ2 + θ + log(
√θ +
√θ + 1)
]), θ > 0,
and which admits the local parametrization
X+C0(θ, v) = C
−3/20
((θ + 1)3/2 cos v, (θ + 1)3/2 sin v,
3
2
[√θ2 + θ + log(
√θ +
√θ + 1)
]), θ > 0, v ∈ R.
To compute the metric on this surface, we rst need the coecients of the rst
fundamental form
E+C0(θ, v) =
1
C30
9(θ + 1)2
4θ, F+
C0(θ, v) = 0, G+
C0(θ, v) =
1
C30
(θ + 1)3.
Thus, the Riemannian metric on this surface is
g+C0(θ, v) =
1
C30
(9(θ + 1)2
4θdθ2 + (θ + 1)3dv2
).
If we consider the change of coordinates (θ, v) =((sinhu)2 , 3v
), where u = 0, ones
obtains
g+C0(u, v) =
9
C30
(coshu)6(du2 + dv2
).
Since C0 is an arbitrary positive constant, we can consider C0 = (9/C0)1/3, where C0
is the positive constant corresponding to gC0 , and therefore g+C0
= gC0 .
Then, we dene φC0 as: for u > 0, φC0(u, v) is obtained by rotating the prole
curve
σ+(9
C0
)1/3(u) =
(σ1(
9C0
)1/3 (u) , σ2(
9C0
)1/3 (u)
)
=
√C0
3
((coshu)
3,3
2(sinhu coshu+ log (sinhu+ coshu))
)=
√C0
3
((coshu)
3,3
2
(1
2sinh 2u+ u
)), u > 0,
58 Chapter 3. Complete biconservative surfaces in R3 and S3
and for u < 0, φC0(u, v) is obtained by rotating the prole curve
σ−(9
C0
)1/3(u) =
(σ1(
9C0
)1/3 (−u) ,−σ2(9
C0
)1/3 (−u)
)
=C
1/20
3
((coshu)
3,3
2
(1
2sinh 2u+ u
)), u < 0.
Now, it is easy to see that we have a biconservative immersion, in fact a biconser-
vative embedding from the whole(R2, gC0
)in R3, given by
X(9C0
)1/3(u, v) =
√C0
3
((coshu)3 cos 3v, (coshu)3 sin 3v,
3
2
(1
2sinh 2u+ u
)).
By simple transformations of the metric,(R2, gC0
)becomes a Ricci surface or a
surface with constant Gaussian curvature.
Theorem 3.12. Consider the surface(R2, gC0
). Then
(R2,
√−KC0gC0
)is complete,
satises the Ricci condition and can be minimally immersed in R3 as a helicoid or a
catenoid.
Proposition 3.13. Consider the surface(R2, gC0
). Then
(R2,−KC0gC0
)has constant
Gaussian curvature 1/3 and it is not complete. Moreover,(R2,−KC0gC0
)is the uni-
versal cover of the surface of revolution in R3 given by
Z(u, v) =
(α(u) cosh
(√3
av
), α(u) sinh
(√3
av
), β(u)
), (u, v) ∈ R2,
where a ∈ (0,√3] and
α(u) =a
coshu, β(u) =
∫ u
0
√(3− a2) cosh2 τ + a2
cosh2 τdτ .
Remark 3.14. When a =√3, the immersion Z has only umbilical points and the
image Z(R2)is the round sphere of radius
√3, without the North and the South poles.
Moreover, if a ∈ (0,√3), then Z has no umbilical points.
Concerning the biharmonic surfaces in R3 we have the following non-existence result.
Theorem 3.15 ([19, 20]). There exists no proper-biharmonic surface in R3.
3.1. Complete biconservative surfaces in R3 59
3.1.1 Uniqueness of complete biconservative sur-
faces in R3
In this subsection, we give some uniqueness results concerning Theorem 3.11, under
some additional assumptions.
Theorem 3.16. Let φ :M2 → R3 be a non-CMC surface. Assume that
W = p ∈M | (grad f)(p) = 0
has only one connected component, M \W has non-empty interior and the boundaries,
in M , of Int(M \W ) and M \W coincide, i.e., ∂M Int(M \W ) = ∂M (M \W ). Then
M cannot be biconservative.
Proof. Let us consider an arbitrary point p0 ∈ ∂MW = ∂M (M \W ) = ∂M Int(M \W ).
Since p0 ∈M , it follows that there exists an open subset U0 ofM , such that p0 ∈ U0 and
φ|U0: U0 → R3 is an embedding. Thus, we can identify U0 with its image φ (U0) ⊂ R3
and then U0 can be seen as a regular surface in R3.
We note that p0 ∈ ∂MW∩U0 leads to the existence of a sequence(p1n)n∈N∗ ⊂W∩U0,
p1n = p0, for any n ∈ N∗, which converges to p0, with respect to the intrinsic distance
function dM on M , and, similarly, from p0 ∈ ∂M Int(M \W ) ∩ U0 it follows that there
exists a sequence(p2n)n∈N∗ ⊂ Int(M \W )∩U0, p
2n = p0, for any n ∈ N∗, which converges
to p0, with respect to dM . It is clear that we can identify p1n = φ(p1n)and p2n = φ
(p2n).
Now, since W is connected and grad f = 0 at any point of W , from Theorem 3.4,
one obtains that there exists a unique C0 such that φ(W ) is an open set in SC0. Then,
as W ∩ U0 is open in W , it is clear that φ (W ∩ U0) is open in SC0. In fact, using the
identication φ (W ∩ U0) = W ∩ U0, we have W ∩ U0 open in SC0. We recall that SC0
is open in the complete surface SC0, and then W ∩ U0 is also open in SC0
.
Further, we denote by d0 the distance function on R3 and by dSC0
the intrinsic
distance function on SC0. Obviously, the convergence of the sequence
(p1n)to p0, with
respect to the distance dM , implies the convergence of(p1n)to p0, with respect to the
distance d0.
The sequence(p1n)was chosen in W ∩ U0, so
(p1n)⊂ SC0
. As SC0is a closed set in
R3, we obtain that p0 ∈ SC0. Therefore,
(p1n)converges to p0 also with respect to the
distance dSC0
.
We have already seen that W ∩ U0 is open in U0 and also in SC0. Then, the mean
curvature functions f and fC0corresponding to M and SC0
, respectively, coincide on
60 Chapter 3. Complete biconservative surfaces in R3 and S3
W ∩ U0. Therefore, for any n ∈ N∗ one has
f(p1n)= fC0
(p1n)
(grad f)(p1n)=(grad fC0
) (p1n)∣∣(grad f) (p1n)∣∣ = ∣∣∣(grad fC0
) (p1n)∣∣∣
(∆f)(p1n)=(∆fC0
) (p1n) . (3.8)
From the convergence of(p1n)to the same p0, with respect to the both distance functions
dM and dSC0
, and from the third equation in (3.8), one gets
|(grad f) (p0)| =∣∣∣(grad fC0
)(p0)
∣∣∣ .As p0 ∈ ∂MW , we have (grad f) (p0) = 0 and then
(grad fC0
)(p0) = 0, that means p0
belongs to the boundary, in R3, of SC0. Thus, fC0
(p0) = 2C3/20 /3 = 0.
Using the rst equation in (3.8) and the convergence of(p1n)to p0, with respect to
dM and dSC0
, we obtain f (p0) = fC0(p0).
Assume that M is biconservative. From (1.19) applied for c = 0, one has
f(p1n)(∆f)
(p1n)+∣∣(grad f) (p1n)∣∣2 − f4
(p1n)= 0,
for any n ∈ N∗. We may pass to the limit with respect to the distance dM in the above
equation and obtain
f (p0) (∆f) (p0) + |(grad f) (p0)|2 − f4 (p0) = 0.
According to the above observations, this is equivalent to
fC0(p0) (∆f) (p0)− f4
C0(p0) = 0. (3.9)
We also have seen that there exists a sequence(p2n)n∈N∗ ⊂ Int(M \ W ) ∩ U0 which
converges to p0, with respect to the distance dM . Since grad f = 0 at any point of
Int(M \W )∩U0 and Int(M \W )∩U0 is open inM , it is easy to see that (grad f)(p2n)= 0
and (∆f)(p2n)= 0, for any n ∈ N∗. Passing to the limit with respect to the distance
dM in the above two relations we get (grad f) (p0) = 0 and (∆f) (p0) = 0.
Substituting (∆f) (p0) = 0 in (3.9), one obtains fC0(p0) = 0. Therefore we have a
contradiction.
Theorem 3.17. Let φ :M2 → R3 be a biconservative surface. Assume that
W = p ∈M | (grad f)(p) = 0
3.1. Complete biconservative surfaces in R3 61
is dense and it has two connected components, W1 and W2. Assume that the boundaries
of W1 and W2 in W coincide and their common boundary is a smooth curve in M .
Then, there exists a unique C0 such that φ(M) ⊂ SC0. Moreover, if M is complete
and simply connected, then up to isometries of the domain and codomain, φ is the map
given in Theorem 3.11.
Proof. Let us consider p0 ∈ ∂MW1 = ∂MW2. There exists an open set U0 in M ,
such that p0 ∈ U0 and φ|U0: U0 → R3 is an embedding. Thus, we can identify
U0 = φ (U0) ⊂ R3.
SinceW1 andW2 are connected and grad f = 0 at any point of them, from Theorem
3.4, one obtains that there exist C0 and C ′0 such that φ (W1) is an open subset of SC0
and φ (W2) is an open subset of SC′0.
It is clear that U0 ∩W1 is open in W1 and then φ (U0 ∩W1) = U0 ∩W1 is open in
SC0, and analogous, U0 ∩W2 is open in W2 and then φ (U0 ∩W2) = U0 ∩W2 is open in
SC′0.
We note that, as W is dense in M , one has M = W ∪ ∂MW = W1 ∪W2 ∪ ∂MW1.
Therefore,
U0 = U0 ∩M
= (U0 ∩W1) ∪ (U0 ∩W2) ∪(U0 ∩ ∂MW1
).
We consider U0 ∩ ∂MW1 as the image of γ : I → U0, γ′(s) = 0, for any s ∈ I. It is
clear that γ(s) ∈ SC0and
(grad fC0
)(γ(s)) = (grad f)(γ(s)) = 0, i.e., γ(s) belongs to
the boundary of SC0, for any s ∈ I, which is a circle of radius C
−3/20 . With the same
argument, γ(s) belongs to the boundary of SC′0, for any s ∈ I, which is a circle of radius
C′−3/20 . Therefore, C0 = C ′
0 and φ(M) ⊂ SC0.
If M is complete, φ : M → SC0is a covering space with the projection φ and thus
φ(M) = SC0. Moreover, if M is also simply connected, then M is a universal covering
of SC0with the projection φ.
The map φC0 :(R2, gC0
)→ SC0
given in Theorem 3.11 is also a universal covering
projection and therefore there exists an isometry Θ between (M, g) and(R2, gC0
)such
that φC0 Θ = φ.
In the following, we restrict ourself to the case when φ :M2 → R3 is an embedding,
i.e., S = φ(M) is a regular surface in R3. We start with the next result.
Theorem 3.18. Let S a complete biconservative regular surface in R3. Denote by
W = p ∈ S | (grad f)(p) = 0
62 Chapter 3. Complete biconservative surfaces in R3 and S3
and assume that W is non-empty and W0 is a connected component of W . Then, there
exists a unique C0 > 0 such that W0 = SC0. Moreover, the closure of W0 in S coincides
with the closure of SC0in SC0
.
Proof. We note that since W0 is a connected component of W and W is an open subset
of S, then W0 is closed and also open in W . We also have that W0 is a maximal
connected subset of W with respect to the inclusion. It is clear that W0 is also open in
S.
We denote by ∂SW0 the boundary of W0 in S and prove that (grad f) (q) = 0, for
any q ∈ ∂SW0. In order to show this, we assume that there exists a point q0 ∈ ∂SW0
such that (grad f) (q0) = 0. Then, it follows that one has an open ball B2 (q0; r0) in S,
r0 > 0, such that grad f is dierent from zero at any point of it.
Obviously, q0 belongs to the closure ofW0 in S, and then B2 (q0; r0)∩W0 = ∅. SinceB2 (q0; r0) and W0 are connected sets, we get that B2 (q0; r0) ∪W0 is also connected.
Moreover, as grad f is dierent from zero at any point of B2 (q0; r0) it follows that
B2 (q0; r0) ⊂W and, therefore B2 (q0; r0) ∪W0 is a connected subset of W . Now, from
the maximality of W0 in W , we have B2 (q0; r0) ∪ W0 = W0, i.e., B2 (q0; r0) ⊂ W0.
Clearly, we obtain that q0 ∈W0, and this is false because q0 ∈ ∂SW0 and W0 is open S.
Thus, one has (grad f) (q) = 0, for any q ∈ ∂SW0.
It is easy to note that since W0 is connected and grad f = 0 at any point of W0,
from Theorem 3.4, one obtains that there exists a unique C0 such that W0 is open in
SC0. Moreover, we will prove that, in this case, W0 = SC0
.
Let us consider σC0: (0,∞) → R2 the prole curve of SC0
, σC0(0,∞) ⊂ SC0
. We can
reparametrize σC0by arc-length, such that the new curve, denoted also by σC0
= σC0(θ),
has the same orientation as the initial one, is dened on (0,∞) and in zero has the same
limit point(C
−3/20 , 0
)on the boundary of SC0
. The new curve σC0is a parametrized
geodesic of SC0and we recall that
(grad fC0
)(σC0
(θ))= 0, for any θ > 0.
Next, we will prove that σC0(0,∞) ⊂ W0. Clearly, there exists a point θ0 ∈ (0,∞)
such that σC0(θ0) ∈ W0. Since σC0
is continuous and W0 is open in SC0, it follows
that exists ε0 > 0 such that (θ0 − ε0, θ0 + ε0) ⊂ (0,∞) and σC0(θ0 − ε0, θ0 + ε0) ⊂W0.
Assume that σC0(0,∞) ⊂W0, i.e., there exists θ
′ ∈ (0,∞) \ (θ0 − ε0, θ0 + ε0) such that
σC0(θ′) ∈W0.
Assume that θ′ ≥ θ0 + ε0. Denote
Ω =θ | θ > θ0, σC0
(θ) ∈W0
and θ1 = inf Ω.
We note that θ1 ≥ θ0 + ε0. Indeed, if we assume that θ1 < θ0 + ε0, since θ1 = inf Ω,
it follows that there exists θ2 ∈ Ω such that θ1 ≤ θ2 < θ0 + ε0. It is easy to see that
3.1. Complete biconservative surfaces in R3 63
θ2 ∈ (θ0, θ0 + ε0) ⊂ (θ0 − ε0, θ0 + ε0), thus σC0(θ2) ∈ W0. But, this is a contradiction
because θ2 ∈ Ω, and then σC0(θ2) ∈W0.
Next, we show that σC0(θ1) ∈ W0. Indeed, if σC0
(θ1) ∈ W0, it follows that there
exists ε1 > 0 such that (θ1 − ε1, θ1 + ε1) ⊂ (0,∞) and σC0(θ1 − ε1, θ1 + ε1) ⊂ W0.
Therefore, we obtain a contradiction because that implies the non-existence of a se-
quence from Ω which converges to θ1, and so θ1 cannot be an inmum of Ω.
Moreover, we have σC0(θ) ∈ W0 for any θ ∈ [θ0, θ1). We have already seen that
σC0(θ0) ∈W0. Assume that there exists θ ∈ (θ0, θ1) such that σC0
(θ)∈W0. It follows
that θ ∈ Ω. But θ < θ1, so we obtain a contradiction with the fact that θ1 is an inmum.
Since σC0(θ) ∈W0 for any θ ∈ [θ0, θ1), it is clear that σC0
(θ1) belongs to the closure
of W0 in SC0, denoted by W0
SC0 . As σC0(θ1) ∈W0 and W0 is open in SC0
, one obtains
that σC0(θ1) ∈ ∂
SC0W0, i.e. σC0(θ1) belongs to the boundary of W0 in SC0
.
We have seen that σC0(θ) ∈W0, for any θ ∈ (θ0 − ε0, θ1). Now, sinceW0 is open in S
and σC0is a parametrized geodesic of SC0
, we get that σC0dened of (θ0 − ε0, θ1) is also
a parametrized geodesic of S. As S is complete, we can consider a parametrized geodesic
σS dened on whole R, such that σS∣∣(θ0−ε0,θ1) = σC0
∣∣∣(θ0−ε0,θ1)
. It is clear that, since
σS and σC0are continuous on R and on (0,∞), respectively, that σS (θ1) = σC0
(θ1).
Now, we note that σS (θ1) ∈ ∂SW0, (σS (θ1) ∈W0
Sand σS (θ1) = σC0
(θ1) ∈W0).
Therefore (grad f)(σS (θ1)
)= (grad f)
(σC0
(θ1))= 0.
Further, we use the fact that sinceW0 is an open set in both S and SC0, then we have
equality between the mean curvature functions of S and of SC0at every point of W0,
i.e., f |W0= fC0
∣∣∣W0
, and between their gradients, i.e., (grad f)|W0=(grad fC0
)∣∣∣W0
.
Clearly,
(grad f)(σS(θ)
)=(grad fC0
)(σC0
(θ)), θ ∈ (θ0 − ε0, θ1) ,
and, then, ∣∣(grad f) (σS(θ))∣∣ = ∣∣∣(grad fC0
)(σC0
(θ))∣∣∣ , θ ∈ (θ0 − ε0, θ1) .
We may pass to the limit in the above equation and obtain∣∣(grad f) (σS (θ1))∣∣ = ∣∣∣(grad fC0
)(σC0
(θ1))∣∣∣ .
Therefore, one gets a contradiction, because we have already seen that
(grad f)(σS (θ1)
)= (grad f)
(σC0
(θ1))= 0,
and(grad fC0
)(σC0
(θ1))= 0, since σC0
(θ1) ∈ SC0.
Thus, σC0(θ) ∈W0, for any θ ≥ θ0 + ε0.
64 Chapter 3. Complete biconservative surfaces in R3 and S3
In the same way, we can prove that σC0(θ) ∈W0, also for any θ ≤ θ0 − ε0.
Finally, we obtain that σC0(0,∞) ⊂W0.
Now, we recall that SC0is open in SC0
, and then W0 is also open in S ∩ SC0. Since
SC0is complete, we can consider a parametrized geodesic σC0
dened on whole R, suchthat σC0
∣∣∣(0,∞)
= σC0= σS
∣∣(0,∞)
. Obviously, σS(0) = σC0(0), thus σC0
∣∣∣[0,∞)
= σS∣∣[0,∞)
and the closure of W0 in S coincides with the closure of SC0in SC0
.
Further, we consider γ the curve parametrized by arc-length, dened on the whole
R, which gives the boundary of SC0in SC0
. Clearly, γ is a parametrized geodesic in
SC0. According to the above observations, it follows that there exist a, b ∈ R with
a < b, such that γ(a, b) belongs also to S. Moreover, γ dened on (a, b) is also geodesic
in S because, along it, the normal vector eld to S coincide with the normal vector
eld to SC0and it is collinear with the principal unit normal vector of γ|(a,b). Since
S is complete, we can consider γS : R → S a parametrized geodesic on S such that
γS∣∣(a,b)
= γ|(a,b).We note that the maximal interval which contains (a, b) and has the property that
its image by γ is contained in S is R. Indeed, assume that there exists b′, b ≤ b′ < ∞,
such that γ(a, b′) ⊂ S and γ(b′) /∈ S. It is now easy to see that, as γS∣∣(a,b′)
= γ|(a,b′),γ(b′) = γS(b′) ∈ S, and thus we get a contradiction.
Therefore, W0 = SC0.
Remark 3.19. The proof of Theorem 3.18 can be summarizing in Figure 3.4, where
the yellow region represents the connected componentW0 of W in S, and the surface of
revolution represented with the color green is the corresponding SC0(given by Theorem
3.4). It is suggested that, in fact, all the meridians of SC0which intersect W0 are
contained in W0 and then, as the boundary of W0 in S has to be the whole circle which
gives the boundary of SC0in SC0
, W0 = SC0.
A rst consequence of the above result is the following theorem.
Theorem 3.20. Let S be a biconservative regular surface in R3. Assume that S is
compact. Then S is CMC, and therefore a round sphere.
Proof. Assume that S is non-CMC, i.e., W = p ∈ S | (grad f)(p) = 0 is a non-empty
subset of S. Let us consider W0 a connected component of W . From Theorem 3.18,
it follows that there exists C0 > 0 such that W0 = SC0and the closure of W0 in S
coincides with the closure of SC0in SC0
. Since SC0is unbounded in R3, we obtain that
S is unbounded and this is a contradiction with the compactness of S.
The last part follows from the famous theorem of Alexandrov (see, for example
[52]).
Other consequence of Theorem 3.18 is the next result.
3.1. Complete biconservative surfaces in R3 65
Figure 3.4: The idea of the proof of Theorem 3.18.
Theorem 3.21. Let S be a complete regular surface in R3. Assume that
W = p ∈ S | (grad f)(p) = 0
is non-empty and is connected. Then S cannot be biconservative.
Proof. Assume that S is biconservative. From Theorem 3.18 it follows that there exists
C0 > 0 such that W = SC0and the closure of W in S coincides with the closure of SC0
in SC0.
It is easy to see that W is not dense in S, because if we assume that it is, then S =
WS= S
SC0
C0. This means that S is a surface with boundary and this is a contradiction
66 Chapter 3. Complete biconservative surfaces in R3 and S3
with the regularity of S. Therefore, S \WSis non-empty and open in S. We note that
∂SW = ∂S(S \W )
= S \WS ∩WS
= (S \W ) ∩WS
=(S \ Int
(W
S))
∩WS
= S \WSS
∩WS
= ∂S(S \WS
)and ∂SW is a circle of radius C
−3/20 .
Further, let us consider p0 ∈ ∂SW = ∂S(S \WS
). Then there exists a sequence(
p1n)n∈N∗ in W , with p1n = p0, for any n ∈ N∗ which converges to p0, with respect to the
distance function dS on S, and another sequence(p2n)n∈N∗ in S \WS
, with p2n = p0, for
any n ∈ N∗ which converges to p0, with respect to the same dS . Since W and S \WS
are open in S, grad f is dierent from zero at any point of W and grad f vanishes at
any point of S \WS, we can use the same argument as in the proof of Theorem 3.16,
to obtain a contradiction.
Therefore, our assumption is false, and S is not biconservative.
Theorem 3.22. Let S be a complete biconservative regular surface in R3. If S is
non-CMC, then S = SC0.
Proof. Since S is non-CMC, then
W = p ∈ S | (grad f)(p) = 0
is non-empty. Let us consider W0 a connected component of W . Then there exists a
unique C0 such that W0 = SC0and W0
S= SC0
SC0 . We denote, the surface SC0by S+
C0.
Let p0 ∈ ∂SW0, i.e., p0 is a point on the circle of radius C−3/20 . We have three cases.
First, assume that there exists ε0 > 0 such that grad f vanishes at any point of
B2 (p0; ε0) \(B2 (p0; ε0) ∩W0
S). Then there exists a sequence
(p1n)n∈N∗ in W0, with
p1n = p0, for any n ∈ N∗ which converges to p0, with respect to the distance function dS
on S, and another sequence(p2n)n∈N∗ in B
2 (p0; ε0)\(B2 (p0; ε0) ∩W0
S), with p2n = p0,
for any n ∈ N∗ which converges to p0, with respect to the same dS . Since W0 and
B2 (p0; ε0) \(B2 (p0; ε0) ∩W0
S)are open in S, grad f is dierent from zero at any
point of W0 and grad f vanishes at any point of B2 (p0; ε0) \(B2 (p0; ε0) ∩W0
S), we
obtain a contradiction as in the proof of Theorem 3.16.
3.1. Complete biconservative surfaces in R3 67
Second, let us consider that there exists ε0 > 0 such that grad f is dierent from zero
at any point ofB2 (p0; ε0)\(B2 (p0; ε0) ∩W0
S). ThenB2 (p0; ε0)\
(B2 (p0; ε0) ∩W0
S)⊂
S−C0
⊂ S, where S−C0
is the surface obtained by symmetry of S+C0
with respect to the
plane where its boundary lies. Clearly, SC0⊂ S. Since SC0
is complete, then it cannot
be extendible, and thus SC0= S.
In the last case, assume that for any εn > 0, in B2 (p0; εn) \(B2 (p0; εn) ∩W0
S)
there exists at least a point p1n such that (grad f)(p1n)= 0 and at least a point p2n such
that (grad f)(p2n)= 0.
Let us consider an arbitrary ε1 > 0. Then there exists U1 an open subset of S which
contains p21, which is connected, U1 ⊂ B2 (p0; ε1) \(B2 (p0; ε1) ∩W0
S)
and grad f
does not vanish at any point of U1. If we consider the connected component of W
which contains U1, one can notice that this is a surface SC10⊂ S and SC1
0
S= SC1
0
SC10 .
Moreover, SC10∩ S+
C0= ∅. We note that the boundaries of any two such connected
components coincide or they are disjoint.
Also, it is easy to see that the boundary of SC10does not intersect the boundary of
S+C0. Indeed, if these two boundaries would intersect, they would coincide and SC1
0=
S−C0. Therefore, we obtain a contradiction with our assumption.
Next, we can consider ε2 > 0 such that B2 (p0; ε2) \(B2 (p0; ε2) ∩W0
S)does not
intersect the boundary of SC10. With the same argument, one obtains another surface
SC20⊂ S such that SC2
0∩ S+
C0= ∅ and SC2
0∩ SC1
0= ∅. Moreover, SC2
0
SC20 ∩ S+
C0
SC0 = ∅
and SC20
SC20 ∩SC1
0
SC10 = ∅. We continue the reasoning and obtain a sequence of surfaces(
SCn0
)n∈N∗
⊂ S, which are disjoint two by two and SCn0∩ SC0
= ∅, for any n ∈ N∗.
Their boundaries are disjoint two by two and they do not intersect the boundary of
S+C0. For any n ∈ N∗, the boundary of SCn
0is a circle of radius
(Cn0
)−3/2= 2/
(3fCn
0
),
where fCn0is the mean curvature functions of SCn
0and is evaluated at some point of
the boundary of SCn0. But fCn
0= f on SCn
0
SCn0 , where f is the mean curvature function
of S. As f is continuous, the radius(Cn0
)−3/2converges to the radius of the circle
that gives the boundary of S+C0, which is
(C0
)−3/2= 2/
(3fC0
), where fC0
is the mean
curvature function of SC0and is evaluated at some point of the boundary of SC0
.
Therefore, we obtain a contradiction with the fact that S is a regular surface.
68 Chapter 3. Complete biconservative surfaces in R3 and S3
3.2 Complete biconservative surfaces in S3
As in the previous section, we consider the global problem for biconservative surfaces in
S3, i.e., our aim is to construct complete biconservative surfaces in S3, with grad f = 0
at any point of an open and dense subset.
We start with the following local extrinsic result.
Theorem 3.23 ([15]). Let M2 be a biconservative surface in S3 with (grad f)(p) = 0
at any point p ∈M . Then, locally, the surface, viewed in R4, can be parametrized by
YC1(u, v) = σ(u) +
4κ(u)−3/4
3√C1
(f1(cos v − 1) + f2 sin v
), (3.10)
where C1 ∈(64/
(35/4
),∞)is a positive constant; f1, f2 ∈ R4 are two constant or-
thonormal vectors; σ(u) is a curve parametrized by arc-length that satises
⟨σ(u), f1⟩ =4κ(u)−3/4
3√C1
, ⟨σ(u), f2⟩ = 0, (3.11)
and, as a curve in S2, its curvature κ = κ(u) is a positive non-constant solution of the
following ODE
κ′′κ =7
4
(κ′)2
+4
3κ2 − 4κ4 (3.12)
such that (κ′)2
= −16
9κ2 − 16κ4 + C1κ
7/2. (3.13)
Remark 3.24. The curve σ lies in the totally geodesic S2 = S3 ∩ Π, where Π is the
linear hyperspace of R4 orthogonal to f2.
Remark 3.25. The constant C1 determines uniquely the curvature κ, up to a transla-
tion of u, and then κ, f1 and f2 determine uniquely the curve σ.
In the following, we prove, following a slightly dierent method from that in [15],
that such a curve σ exists and nd a more explicit expression for (3.10).
Replacing (3.13) in (3.12), since κ′ = 0, we get
κ′′ = −16
9κ− 32κ3 +
7
4C1κ
5/2.
We consider f1 = e3 and f2 = e4, where e1, e2, e3, e4 is the canonical basis of R4.
From (3.11) it follows that σ can be written as
σ(u) =
(x(u), y(u),
4
3√C1
κ(u)−3/4, 0
).
3.2. Complete biconservative surfaces in S3 69
Using polar coordinates, we have x(u) = R(u) cosµ(u) and y(u) = R(u) sinµ(u), with
R(u) > 0.
Since σ(u) ⊂ S3, R2 = x2 + y2 and R > 0, we get κ >(16/
(9C1
))2/3and
R =
√1− 16
9C1
κ−3/2. (3.14)
As κ′(u) = 0, we can view u as a function of κ, and considering R = R(u(κ)) and
µ = µ(u(κ)), by a straightforward computation, it follows that σ is explicitly given by
σ(κ) =
(R cosµ,R sinµ,
4
3√C1
κ−3/4, 0
),
where R is given by (3.14) and
µ(κ) = ±
108
∫ κ
κ0
√C1τ
3/4(−16 + 9C1τ3/2
)√9C1τ3/2 − 16 (1 + 9τ2)
dτ + c1
,
where c1 is a real constant.
If we use the formula of σ in (3.10), we get
YC1(κ, v) =
(√1−
(4
3√C1
κ−3/4
)2
cosµ(κ),
√1−
(4
3√C1
κ−3/4
)2
sinµ(κ),
4
3√C1
κ−3/4 cos v, 4
3√C1
κ−3/4 sin v
).
(3.15)
Next, we have to determine the maximum domain for YC1. From (3.13), we ask
that −16κ2/9− 16κ4 + C1κ7/2 > 0. Since κ > 0, it is enough to nd the interval where
−16/9− 16κ2 + C1κ3/2 > 0. We denote by
L(κ) = −16
9− 16κ2 + C1κ
3/2, κ > 0.
We can see that if C1 > 64/(35/4), one obtains that there exist exactly two κ01 ∈(0,(3C1/64
)2)and κ02 ∈
((3C1/64
)2,∞)
such that L(κ01) = L(κ02) = 0 and
L(κ) > 0 for any κ ∈ (κ01, κ02).
We note that κ01 >(16/
(9C1
))2/3.
Therefore, the domain of YC1is (κ01, κ02)×R, where κ01 and κ02 are the vanishing
points of L, with 0 < κ01 < κ02.
We note that an alternative expression for YC1was given in [30].
Remark 3.26. We can choose c1 = 0 in the above expression of µ, by considering a
linear orthogonal transformation of R4.
70 Chapter 3. Complete biconservative surfaces in R3 and S3
We denote by
µ0(κ) = 108
∫ κ
κ0
√C1τ
3/4(−16 + 9C1τ3/2
)√9C1τ3/2 − 16 (1 + 9τ2)
dτ,
and, therefore, µ(κ) = ± (µ0(κ) + c1).
The following remark can be proved in a similar way as Lemma 3.35.
Remark 3.27. We have
limκκ01
µ0(κ) = µ0,−1 > −∞ and limκκ02
µ0(κ) = µ0,1 <∞.
Remark 3.28. For simplicity, we choose κ0 = (3C1/64)2.
If we denote S±C1,c1
the image of YC1, then we note that the boundary of S±
C1,c1is made up from two circles and along the boundary, the mean curvature function
is constant (two dierent constants) and its gradient vanishes. More precisely, the
boundary of S±C1,c1
is given by the curves(√1−
(4
3√C1
κ−3/401
)2
cosµ (κ01) ,
√1−
(4
3√C1
κ−3/401
)2
sinµ (κ01) ,
4
3√C1
κ−3/401 cos v, 4
3√C1
κ−3/401 sin v
)and (√
1−(
4
3√C1
κ−3/402
)2
cosµ (κ02) ,
√1−
(4
3√C1
κ−3/402
)2
sinµ (κ02) ,
4
3√C1
κ−3/402 cos v, 4
3√C1
κ−3/402 sin v
),
where µ (κ01) = limκκ01 µ(κ) and µ (κ02) = limκκ02 µ(κ) are real numbers.
These curves are circles in ane planes in R4 parallel to the Ox3x4 plane and their
radii are(4κ
−3/401
)/(3√C1
)and
(4κ
−3/402
)/(3√C1
), respectively.
At a boundary point, using the coordinates (µ, v), we get that the tangent plane to
the closure of S±C1,c1
in R4 is spanned by a vector which is tangent to the corresponding
circle and by−
√1−
(4
3√C1
κ−3/40i
)2
sinµ (κ0i) ,
√1−
(4
3√C1
κ−3/40i
)2
cosµ (κ0i) , 0, 0
,
where i = 1 or i = 2.
In a similar way to the proof of Theorem 3.4, we can obtain the next result.
3.2. Complete biconservative surfaces in S3 71
Theorem 3.29. If φ : M2 → S3 is a biconservative surface with grad f = 0 at any
point, then there exists a unique C1 such that φ(M) ⊂ S±C1,c1
.
Thus, in order to construct, from extrinsic point of view, a complete biconservative
surface in S3, we can expect to glue along the boundary two biconservative surfaces of
type S±C1,c1,k
and S±C1,c1,l
, corresponding to the same C1, where k, l ∈ Z. In fact, if we
want to glue two surfaces corresponding to C1 and C ′1 along the boundary, then these
constants have to coincide and there is no ambiguity concerning along which circle of
the boundary we should glue the two pieces.
Geometrically, we start with a piece of type S+C1,0
corresponding to c1,0 = 0 and to
the sign +, and then consider T1(S+C1,0
), where T1 is a linear orthogonal transformation
of R4 that acts on R2 = span e1, e2 as an axial symmetry with respect to the line
determined by√√√√1−
(4
3√C1
κ−3/402
)2
cosµ0,1,
√√√√1−
(4
3√C1
κ−3/402
)2
sinµ0,1
and the origin, and leaves invariant span e3, e4. Moreover, T1 is a symmetry with
respect to the 3-dimensional subspace spanned by the vectors√√√√1−
(4
3√C1
κ−3/402
)2
cosµ0,1,
√√√√1−
(4
3√C1
κ−3/402
)2
sinµ0,1, 0, 0
,
e3 and e4. Of course, T1 leaves invariant the upper circle from the boundary of S+C1,0
.
The matrix of T1 is cos c1,1 sin c1,1 0 0
sin c1,1 − cos c1,1 0 0
0 0 1 0
0 0 0 1
.
We perform this process innitely many times. But it is dicult to conclude from
here that we get a complete biconservative surface in S3. From this process we obtain
a closed subset of S3 with self-intersections, but we cannot see if the surface that
we have obtained is the image of an isometric immersion (by composing with Tk, thedomain of YC1
does not change).
Of course, as a subset, the surface is complete with respect to the induced distance
from S3, but we want to show that the surface is complete with respect to the intrinsic
distance.
This construction of the complete biconservative surfaces in S3 can be illustrated in
R3 using the stereographic projection of S3, as in Figures 3.5 and 3.6.
72 Chapter 3. Complete biconservative surfaces in R3 and S3
Figure 3.5: Using the stereographic projection from the North pole.
Figure 3.6: Using the stereographic projection from (1, 0, 0, 0).
Further, as in the R3 case, we change the point of view and use the local intrinsic
characterization of the biconservative surfaces in S3.Another way to see that in the c = 0 case we have only a one-parameter family
of solutions of equation (2.30) is to rewrite the metric g in certain non-isothermal
coordinates. Further, we consider only the c = 1 case.
Proposition 3.30. Let(M2, g
)be an abstract surface with g = e2ρ(u)(du2+dv2), where
u = u(ρ) satises
u =
∫ ρ
ρ0
dτ√−3be−2τ/3 − e2τ + a
+ u0,
where ρ is in some open interval I, a, b ∈ R are positive constants, and u0 ∈ R is a
constant. Then(M2, g
)is isometric to(
DC1 , gC1 =3
ξ2(−ξ8/3 + 3C1ξ2 − 3
)dξ2 + 1
ξ2dθ2
),
where DC1 = (ξ01, ξ02)×R, C1 ∈(4/(33/2
),∞)is a positive constant, and ξ01 and ξ02
are the positive vanishing points of −ξ8/3 + 3C1ξ2 − 3, with 0 < ξ01 < ξ02.
3.2. Complete biconservative surfaces in S3 73
Proof. Since
u = u(ρ) =
∫ ρ
ρ0
dτ√−3be−2τ/3 − e2τ + a
+ u0,
we have that
du =1√
−3be−e−2ρ/3 − e2ρ + a
dρ,
and the metric g(u, v) = e2ρ(u)(du2 + dv2) can be rewritten as
g(ρ, v) =e2ρ
−3be−e−2ρ/3 − e2ρ + a
dρ2 + e2ρdv2.
if we consider the change of coordinates (ρ, v) =(log(33/4b3/8/ξ
), v), one obtains that
g(ξ, v) =1
ξ2
(3
−ξ8/3 + 3−1/2ab−3/4ξ2 − 3dξ2 + 33/2b3/4dv2
).
Now, considering another change of coordinates (ξ, v) =(ξ, 3−3/4b−3/8θ
)and denoting
C1 = 3−3/2ab−3/4 > 0, we obtain
g(ξ, θ) =1
ξ2
(3
−ξ8/3 + 3C1ξ2 − 3dξ2 + dθ2
),
for every ξ ∈ J , where J is an open interval such that −ξ8/3 + 3C1ξ2 − 3 > 0, for any
positive ξ ∈ J and C1 is a positive constant.
Next, we determine the interval J . If we denote
T (ξ) = −ξ8/3 + 3C1ξ2 − 3, ξ > 0,
we can see that its derivative is T ′(ξ) = −8ξ5/3/3 + 6C1ξ and it vanishes for ξ =
(9C1/4)3/2. The value of T at this critical point is 37C4
1/44 − 3. It is easy to see
that T is strictly increasing on(0, (9C1/4)
3/2), strictly decreasing on
((9C1/4)
3/2 ,∞)
and limξ→0 T (ξ) = −3, limξ→∞ T (ξ) = −∞. We ask T to have positive values, so we
have to determine when 37C41/4
4 − 3 > 0. It is easy to see that 37C41/4
4 − 3 > 0 if
C1 ∈(4/(33/2
),∞).
Therefore, T (ξ) > 0 for any ξ ∈ (ξ01, ξ02), where T (ξ01) = T (ξ02) = 0,
ξ01 ∈
(0,
(9
4C1
)3/2)
and ξ02 ∈
((9
4C1
)3/2
,∞
)(3.16)
are the only positive vanishing points of T and C1 ∈(4/(33/2
),∞).
Thus,(M2, g
)is isometric to(DC1 , gC1 =
3
ξ2(−ξ8/3 + 3C1ξ2 − 3
)dξ2 + 1
ξ2dθ2
),
where DC1 = (ξ01, ξ02) × R, C1 ∈(4/(33/2
),∞), and ξ01 and ξ02 are the vanishing
points of −ξ8/3 + 3C1ξ2 − 3, with 0 < ξ01 < ξ02.
74 Chapter 3. Complete biconservative surfaces in R3 and S3
The surface (DC1 , gC1) is not complete but it has the following properties.
Theorem 3.31. Consider (DC1 , gC1). Then, we have
(i) KC1(ξ, θ) = K(ξ, θ),
1−K(ξ, θ) =1
9ξ8/3 > 0, K ′(ξ) = − 8
27ξ5/3
and gradK = 0 at any point of DC1;
(ii) the immersion ϕC1 : (DC1 , gC1) → S3 given by
ϕC1(ξ, θ) =
(√1− 1
C1ξ2cos ζ(ξ),
√1− 1
C1ξ2sin ζ(ξ),
cos(√C1θ)√
C1ξ,sin(
√C1θ)√C1ξ
),
is biconservative in S3, where
ζ(ξ) = ±
∫ ξ
ξ00
√C1τ
4/3
(−1 + C1τ2)√−τ8/3 + 3C1τ2 − 3
dτ + c1
,
with c1 ∈ R a constant and ξ00 ∈ (ξ01, ξ02).
Proof. Consider the Riemannian metric
gC1 =3
ξ2(−ξ8/3 + 3C1ξ2 − 3)dξ2 +
1
ξ2dθ2
on DC1 with coecients given by
EC1 =3
ξ2(−ξ8/3 + 3C1ξ2 − 3), FC1 = 0, GC1 =
1
ξ2. (3.17)
Using the formula of the Gaussian curvature
K(ξ, θ) = − 1
2√EG
(∂
∂ξ
(Gξ√EG
)+∂
∂θ
(Eθ√EG
)),
we obtain that KC1 is given by
KC1(ξ, θ) = KC1(ξ) = −1
9ξ8/3 + 1
and
K ′C1(ξ) = − 8
27ξ5/3.
Therefore, K ′C1(ξ) < 0 at any ξ ∈ (ξ01, ξ02). Since
(gradKC1)(ξ, θ) =ξ2(−ξ8/3 + 3C1ξ
2 − 3)
3K ′C1(ξ)∂ξ,
we have that |(gradKC1) (ξ, θ)| = 0 for any (ξ, θ) ∈ DC1 .
3.2. Complete biconservative surfaces in S3 75
To prove (ii), let us rst recall that, if M2 is a biconservative surface in S3, withgrad f = 0 at any point of M , then, as we have already seen in (3.15), M can be locally
parameterized by
YC1(κ, v) =
(√1−
(4
3√C1
κ−3/4
)2
cosµ(κ),
√1−
(4
3√C1
κ−3/4
)2
sinµ(κ),
4
3√C1
κ−3/4 cos v, 4
3√C1
κ−3/4 sin v
),
for any (κ, v) ∈ (κ01, κ02)×R, where κ01 and κ02 are the vanishing points of −16κ2/9−
16κ4 + C1κ7/2, κ01 ∈
(0,(3C1/64
)2), κ02 ∈
((3C1/64
)2,∞), C1 > 64/
(35/4
), and
µ(κ) = ±
108
∫ κ
κ0
√C1τ
3/4(−16 + 9C1τ3/2
)√9C1τ3/2 − 16 (1 + 9τ2)
dτ + c1
,
where c1 is a real constant.
In order to compute the metric on this surface, we need the coecients of the rst
fundamental form
EC1(κ, v) =
81C1κ3/2 − 144
κ2(9C1κ3/2 − 16
)(9C1κ3/2 − 16 (1 + 9κ2)
) ,FC1
(κ, v) = 0,
GC1(κ, v) =
16
9C1κ3/2.
Thus, the Riemannian metric is given by
gC1(κ, v) =
81C1κ3/2 − 144
κ2(9C1κ3/2 − 16
)(9C1κ3/2 − 16 (1 + 9κ2)
)dκ2 + 16
9C1κ3/2dv2.
We write C1 as C1 = 16 ·31/4C1, where C1 ∈ R∗+, and we know that C1 > 64/
(35/4
),
which implies C1 > 4/(33/2
). Therefore, we can choose C1 to be exactly the positive
constant from the metric (DC1 , gC1).
We note that we can consider the change of coordinates
(κ, v) =(3−3/2ξ4/3,
√C1θ
),
where ξ and θ are the coordinates on the domain DC1 . Indeed, we have
−ξ8/3 + 3C1ξ2 − 3 =
27
16κ2
(−16
9κ2 − 16κ4 + C1κ
7/2
)
76 Chapter 3. Complete biconservative surfaces in R3 and S3
and, therefore, the vanishing points ξ01 and ξ02 of−ξ8/3+3C1ξ2−3 are the corresponding
points to κ01 and κ02, i.e., ξ01 = 39/8κ3/401 and ξ02 = 39/8κ
3/402 .
Thus, we get the expression of the initial metric
gC1(ξ, θ) =3
ξ2(−ξ8/3 + 3C1ξ2 − 3
)dξ2 + 1
ξ2dθ2, (ξ, θ) ∈ DC1 .
Then, we dene ϕC1 as
ϕC1(ξ, θ) = Y31/4·16C1
(3−3/2ξ4/3,
√C1θ
).
Therefore,
ϕC1(ξ, θ) =
(√1− 1
C1ξ2cos ζ,
√1− 1
C1ξ2sin ζ,
cos(√C1θ)√
C1ξ,sin(
√C1θ)√C1ξ
),
for any ξ ∈ (ξ01, ξ02) and θ ∈ R, where ζ = µ(κ(ξ)) is given by
ζ(ξ) = ±
∫ ξ
ξ00
√C1τ
4/3
(−1 + C1τ2)√−τ8/3 + 3C1τ2 − 3
dτ + c1
,
where c1 is a real constant and we write ±c1 from esthetic reasons, as we will see
later.
Remark 3.32. For simplicity, we choose ξ00 = (9C1/4)3/2.
Remark 3.33. We note that the expression of the Gaussian curvature of (DC1 , gC1)
does not depend on C1. More precisely,
KC1(ξ, θ) = −1
9ξ8/3 + 1.
But, if we change further the coordinates (ξ, θ) =(ξ01 + ξ (ξ02 − ξ01) , θ
), then we x
the domain, i.e., (DC1 , gC1) is isometric to ((0, 1), gC1) and C1 appears in the expression
of KC1
(ξ, θ).
Remark 3.34. Since (gradKC1) (ξ, θ) = −(8ξ11/3
(−ξ8/3 + 3C1ξ
2 − 3)/81)∂ξ for any
(ξ, θ) ∈ DC1 , we get that
limξξ01
(gradKC1) (ξ, θ) = limξξ02
(gradKC1) (ξ, θ) = 0.
We denote by
ζ0(ξ) =
∫ ξ
ξ00
√C1τ
4/3
(−1 + C1τ2)√
−τ8/3 + 3C1τ2 − 3dτ
and we state the the following lemma that we will use later.
3.2. Complete biconservative surfaces in S3 77
Lemma 3.35. We have
limξξ01
ζ0(ξ) = ζ0,−1 > −∞ and limξξ02
ζ0(ξ) = ζ0,1 <∞.
Proof. Let us dene the continuous functions
H(ξ) =√C1ξ
4/3 and G(ξ) =1
−1 + C1ξ2,
for any ξ ∈ [ξ01, ξ02] and C1 ∈(4/33/2,∞
). It easy to see that these functions have a
maximum and a minimum, which we denote by MH and mH respectively, for H, and
with MG and mG respectively, for G. Of course, mH , mG, MH , and MG are positive
constants.
Let us assume that ξ < ξ00. Then ζ0(ξ) < 0 for any ξ ∈ (ξ01, ξ00) and it is easy to
see that
ζ0(ξ) ≥MHMG
∫ ξ
ξ00
1√−τ8/3 + 3C1τ2 − 3
dτ.
Thus, to show that limξξ01 ζ0(ξ) is nite it suces to prove that
limξξ01
∫ ξ
ξ00
1√−τ8/3 + 3C1τ2 − 3
dτ > −∞.
To prove this, we consider the smooth function T (ξ) = −ξ8/3 + 3C1ξ2 − 3, for any
ξ ∈ (0,∞). We have
T (ξ) = T (ξ01) + T ′ (ξ01) (ξ − ξ01) + (ξ − ξ01)µ1(ξ), ξ > 0, (3.18)
where µ1 is a continuous function on (0,∞) such that limξ→ξ01 µ1(ξ) = 0.
We replace T (ξ01) = 0 and T ′ (ξ01) = −8ξ5/301 /3 + 6C1ξ01 in (3.18) and obtain
T (ξ) =
(−8
3ξ5/301 + 6C1ξ01
)(ξ − ξ01) + (ξ − ξ01)µ1(ξ)
= (ξ − ξ01)
(−8
3ξ5/301 + 6C1ξ01 + µ1(ξ)
), ξ > 0.
We note that(−8ξ
5/301 /3 + 6C1ξ01
)(ξ − ξ01) > 0 for ξ ∈ (ξ01, ξ00). Indeed, this is
equivalent to −8ξ5/301 /3 + 6C1ξ01 > 0, i.e., ξ01 < (9C1/4)
3/2, which, as we have already
seen, it is true.
Now, we consider β1(ξ) = 1/
√(−8ξ
5/301 /3 + 6C1ξ01
)(ξ − ξ01) and γ(ξ) = 1/
√T (ξ),
for any ξ ∈ (ξ01, ξ00). Since limξξ01 (γ(ξ)/β1(ξ)) = 1 ∈ (0,∞) and
limξξ01
∫ ξ
ξ00
β1(τ)dτ = limε→0
∫ ξ01−ε
ξ00
β1(τ) dτ = −2
√ξ00 − ξ01
−83ξ
5/301 + 6C1ξ01
∈ R∗−,
78 Chapter 3. Complete biconservative surfaces in R3 and S3
we get that limξξ01
∫ ξξ00γ(τ)dτ > −∞.
Therefore, limξξ01 ζ0(ξ) = ζ0,−1 > −∞.
Further, let us assume that ξ ≥ ξ00. Then ζ0(ξ) ≥ 0 for any ξ ∈ [ξ00, ξ02) and it is
also easy to see that
ζ0(ξ) ≤MHMG
∫ ξ
ξ00
1√−τ8/3 + 3C1τ2 − 3
dτ.
Thus, to prove that limξξ02 ζ0(ξ) is nite it suces to show that
limξξ02
∫ ξ
ξ00
1√−τ8/3 + 3C1τ2 − 3
dτ <∞.
In order to prove this, we again consider the smooth function T (ξ) = −ξ8/3+3Cξ2− 3,
for any ξ ∈ (0,∞) and we have
T (ξ) = T (ξ02) + T ′ (ξ02) (ξ − ξ02) + (ξ − ξ02)µ2(ξ), ξ > 0, (3.19)
where µ2 is a continuous function on (0,∞) such that limξ→ξ02 µ2(ξ) = 0.
We also have T (ξ02) = 0 and T ′ (ξ02) = −8ξ5/302 /3 + 6C1ξ02, and replacing in (3.19)
one obtains
T (ξ) =
(−8
3ξ5/302 + 6C1ξ02
)(ξ − ξ02) + (ξ − ξ02)µ2(ξ)
= (ξ − ξ02)
(−8
3ξ5/302 + 6C1ξ02 + µ2(ξ)
), ξ > 0.
We note that(−8ξ
5/302 /3 + 6C1ξ02
)(ξ − ξ02) > 0 for ξ ∈ [ξ00, ξ02). Indeed, this is
equivalent to −8ξ5/302 /3 + 6C1ξ02 < 0, i.e., ξ02 > (9C1/4)
3/2, which we have seen that is
true.
Now, we consider β2(ξ) = 1/
√(−8ξ
5/302 /3 + 6C1ξ02
)(ξ − ξ02) and again γ(ξ) =
1/√T (ξ), for any ξ ∈ [ξ00, ξ02). Since limξξ02 (γ(ξ)/β2(ξ)) = 1 ∈ (0,∞) and
limξξ02
∫ ξ
ξ00
β2(τ)dτ = limε→0
∫ ξ02−ε
ξ00
β2(τ)dτ = 2
√ξ00 − ξ02
−83ξ
5/302 + 6C1ξ02
∈ R∗+,
we get that limξξ02
∫ ξξ00γ(τ)dτ <∞, which shows that limξξ02 ζ0(ξ) = ζ0,1 <∞.
Remark 3.36. The immersion ϕC1 depends on the sign ± and on the constant c1 in
the expression of ζ. As the classication is up to isometries of S3, the sign and the
constant are not important, but they play an important role in the gluing process.
The following result shows that we do have a one-parameter family of Riemannian
surfaces (DC1 , gC1).
3.2. Complete biconservative surfaces in S3 79
Proposition 3.37. Let us consider(DC1 , gC1 =
3
ξ2(−ξ8/3 + 3C1ξ2 − 3
)dξ2 + 1
ξ2dθ2
)and DC′
1, gC′
1=
3
ξ2(−ξ8/3 + 3C ′
1ξ2 − 3
)dξ2 + 1
ξ2dθ2
.
The surfaces (DC1 , gC1) and(DC′
1, gC′
1
)are isometric if and only if C1 = C ′
1 and the
isometry is Θ(ξ, θ) = (ξ,±θ + constant). Therefore, we have a one-parameter family of
surfaces.
Proof. Assume that there exists an isometry Θ : (DC1 , gC1) →(DC′
1, gC′
1
)and denote
Θ(ξ, θ) =(Θ1(ξ, θ),Θ2(ξ, θ)
). As we have seen in Theorem 3.31, the Gaussian curvature
of (DC1 , gC1) is K(ξ, θ) = −ξ8/3/9 + 1 and the Gaussian curvature of(DC′
1, gC′
1
)is
K(ξ, θ) = −ξ8/3/9 + 1.
Since Θ is an isometry, we have that K(Θ(ξ, θ)) = K(ξ, θ) and, taking into account
the above expressions of the curvatures, we get Θ1(ξ, θ) = ξ > 0. Therefore, Θ(ξ, θ) =(ξ,Θ2(ξ, θ)
).
Next, from(Θ∗gC′
1
)(∂ξ, ∂ξ) = gC1 (∂ξ, ∂ξ), i.e., gC′
1(Θ∗∂ξ,Θ∗∂ξ) = gC1 (∂ξ, ∂ξ),
using (3.17), we nd
3
−ξ8/3 + 3C1ξ2 − 3=
3
−ξ8/3 + 3C ′1ξ
2 − 3+
(∂Θ2
∂ξ
)2
. (3.20)
Similarly, from(Θ∗gC′
1
)(∂ξ, ∂θ) = gC1 (∂ξ, ∂θ) and
(Θ∗gC′
1
)(∂θ, ∂θ) = gC1 (∂θ, ∂θ), us-
ing (3.17), we get
0 =∂Θ2
∂ξ· ∂Θ
2
∂θand
∂Θ2
∂θ= ±1. (3.21)
From (3.21) one obtains ∂Θ2
∂ξ = 0. Now, using (3.20), it follows that C1 = C ′1. Since
∂Θ2
∂ξ = 0 and ∂Θ2
∂θ = ±1, we have Θ(ξ, θ) = (ξ,±θ+ a1), where a1 is a real constant.
The construction, from intrinsic point of view, of complete biconservative surfaces
in S3 consists in two steps, and the key idea is to notice that (DC1 , gC1) is, locally and
intrinsically, isometric to a surface of revolution in R3.
The rst step is to construct a complete surface of revolution in R3 which on an
open dense subset is locally isometric to (DC1 , gC1). We start with the next result.
Theorem 3.38. Let us consider (DC1 , gC1) as above. Then (DC1 , gC1) is the universal
cover of the surface of revolution in R3 given by
ψC1,C∗1(ξ, θ) =
(χ(ξ) cos
θ
C∗1
, χ(ξ) sinθ
C∗1
, ν(ξ)
), (3.22)
80 Chapter 3. Complete biconservative surfaces in R3 and S3
where χ(ξ) = C∗1/ξ,
ν(ξ) = ±
∫ ξ
ξ00
√3τ2 − (C∗
1 )2 (−τ8/3 + 3C1τ2 − 3
)τ4(−τ8/3 + 3C1τ2 − 3
) dτ + c∗1, (3.23)
C∗1 ∈
(0,√(
33/2)/(33/2C1 − 4
) )is a positive constant and c∗1 ∈ R is constant.
Proof. In fact, we can prove that if (DC1 , gC1) is (locally and intrinsically) isometric to
a surface of revolution, then it has to be of form (3.22). To show this, let us consider
ψ(ξ, θ)=(χ(ξ)cos θ, χ
(ξ)sin θ, ν
(ξ))
,(ξ, θ)∈ D,
a surface of revolution, where D is an open set in R2 and Θ : (DC1 , gC1) →(D, g
)an
isometry, where
g(ξ, θ)=
((χ′(ξ))2
+(ν ′(ξ))2)
dξ2 +(χ(ξ))2
dθ2.
We assume that χ(ξ)> 0 for any ξ.
Next, we proceed in the same way as in the proof of Proposition 3.37. From
K(Θ(ξ, θ)) = K(ξ, θ), we get Θ1(ξ, θ) = Θ1(ξ). In order to simplify the notations,
we write Θ1 = ξ and Θ2 = θ, so that ξ(ξ, θ) = ξ(ξ). As Θ∗g = gC1 , we get(∂θ
∂θ
)2 (χ(ξ(ξ)
))2=
1
ξ2(3.24)
and∂θ
∂θ
∂θ
∂ξ
(χ(ξ(ξ)
))2= 0. (3.25)
From (3.24), one has ∂θ∂θ = 0, and then, from (3.25), it follows that ∂θ
∂ξ = 0. Thus we
have θ(ξ, θ) = θ(θ). Again from (3.24), one obtains(∂θ∂θ
)2= 1/
(ξ2(χ(ξ(ξ)
))2).
Since the left hand term depends only on θ and the right hand term depends only on
ξ, it follows that
χ(ξ(ξ)
)=C∗
ξ, (3.26)
where C∗ ∈ R∗+, and
θ(θ) =θ
C∗ + a0,
where a0 ∈ R. In the following, we shall consider a0 = 0.
Hence, we obtain((χ ξ
)′(ξ)
)2
+
((ν ξ
)′(ξ)
)2
=3
ξ2(−ξ8/3 + 3C1ξ2 − 3
)
3.2. Complete biconservative surfaces in S3 81
and, from (3.26), one has((ν ξ
)′(ξ)
)2
=3ξ2 − (C∗
1 )2 (−ξ8/3 + 3C1ξ
2 − 3)
ξ2(−ξ8/3 + 3C1ξ2 − 3
) . (3.27)
Next, we have to nd the conditions to be satised by the positive constant C∗1 , such
that 3ξ2 − (C∗1 )
2 (−ξ8/3 + 3C1ξ2 − 3
)> 0 for any ξ ∈ (ξ01, ξ02), where C1 > 4/
(33/2
)is xed.
Let us denote
P (ξ) = 3ξ2 − (C∗1 )
2(−ξ8/3 + 3C1ξ
2 − 3), ξ ∈ [ξ01, ξ02] ,
We ask that P (ξ) > 0, for any ξ ∈ (ξ01, ξ02). The rst derivative of P is
P ′(ξ) = ξ
(8
3(C∗
1 )2 ξ2/3 − 6C1 (C
∗1 )
2 − 6
)and, we note that P ′(ξ) = 0 if and only if ξ2/3 = 9
(C1 (C
∗1 )
2 − 1)/(4 (C∗
1 )2).
If we choose C∗1 such that C1 (C
∗1 )
2 − 1 ≤ 0, i.e., C∗1 ∈
(0, 1/
√C1
), then P ′(ξ) = 0,
for any ξ ∈ (ξ01, ξ02). Since P (ξ01) = 3ξ201 > 0 and P (ξ02) = 3ξ202 > 0, we get that
P (ξ) > 0, for any ξ ∈ (ξ01, ξ02).
Now we study the case when C∗1 > 1/
√C1. We recall that in the proof of Proposition
3.30 we have seen that T (ξ) = −ξ8/3+3C1ξ2−3 is strictly increasing on
(0, (9C1/4)
3/2)
and T (ξ01) = 0, where ξ01 ∈(0, (9C1/4)
3/2). Here, we can take (0,∞) as the domain
of denition of T .
We note that the critical point(9(C1 (C
∗1 )
2 − 1)/(4 (C∗
1 )2))3/2
for P is in the
interval(0, (9C1/4)
3/2), for any C1 > 4/
(33/2
), and
T
9
(C1 (C
∗1 )
2 − 1)
4 (C∗1 )
2
3/2 = −
9(C1 (C
∗1 )
2 − 1)
4 (C∗1 )
2
4
+ 3C1
9(C1 (C
∗1 )
2 − 1)
4 (C∗1 )
2
3
− 3.
To simplify the expression, we denote (C∗1 )
2 by A and we have A > 1/C1. Now, we
dene a new function
S(A) = −(9(C1A− 1)
4A
)4
+ 3C1
(9(C1A− 1)
4A
)3
− 3, A >1
C1.
Using standard computations, we get that S is a strictly increasing function,
limA→1/C1
S(A) = −3 and limA→∞
S(A) =3
4
(9
4
)3
C41 + 3 > 0.
We denote by A ∈ (1/C1,∞) the point where S vanishes. Thus, S(A) < 0, for any
A ∈(1/C1, A
)and S(A) ≥ 0, for any A ∈ [A,∞). We split our study in two cases as
A ∈(1/C1, A
)and A ∈
[A,∞
).
82 Chapter 3. Complete biconservative surfaces in R3 and S3
If A ∈(1/C1, A
), since S is a strictly increasing function on (1/C1,∞), we have
T
((9(C1A− 1)
4A
)3/2)
= S(A) < S(A) = 0 = T (ξ01),
and therefore, since T is strictly increasing on the interval(0, (9C1/4)
3/2), and(
9(C1A− 1)
4A
)3/2
∈
(0,
(9C1
4
)3/2),
we get (9 (C1A− 1) /(4A))3/2 < ξ01. Thus, the point at which the rst derivative of P
vanishes is outside of the domain (ξ01, ξ02). As above, P is a strictly increasing function,
with positive values, for every ξ ∈ (ξ01, ξ02).
If A ≥ A, since S is a strictly increasing function on (1/C1,∞), we have
T
((9(C1A− 1)
4A
)3/2)
= S(A) ≥ S(A) = 0 = T (ξ01),
and therefore, since T is strictly increasing on the interval(0, (9C1/4)
3/2], and(
9(C1A− 1)
(4A)
)3/2
∈
(0,
(9C1
4
)3/2),
we get (9(C1A− 1)/(4A))3/2 ≥ ξ01.
We also have ξ02 > (9C1/4)3/2 > (9(C1A− 1)/(4A))3/2 and thus, we get that
(9(C1A− 1)/(4A))3/2 ∈ (ξ01, ξ02).
We want P to have positive values for any ξ ∈ (ξ01, ξ02), and, since the val-
ues of P at ξ01 and at ξ02 are 3ξ201 > 0 and 3ξ202 > 0, respectively, we have to
ask that P((9(C1A− 1)/(4A))3/2
)> 0. It is easy to see that this is equivalent to
A < 33/2/(33/2C1 − 4
).
Therefore, for A ∈[A, 33/2/
(33/2C1 − 4
)), we get that P (ξ) > 0, for any ξ ∈
(ξ01, ξ02).
Consequently, since A = (C∗1 )
2, for C∗1 ∈
(0,√
33/2/(33/2C1 − 4
)), we have P (ξ) >
0 for any ξ ∈ (ξ01, ξ02) and(ν ξ
)(ξ) = ±
∫ ξ
ξ00
√3τ2 − (C∗
1 )2 (−τ8/3 + 3C1τ2 − 3
)τ4(−τ8/3 + 3C1τ2 − 3
) dτ + c∗1,
for any ξ ∈ (ξ01, ξ02), where c∗1 is a real constant.
Next, we consider ψC1,C∗1= ψ Θ dened by
ψC1,C∗1(ξ, θ) =
((χ ξ
)(ξ) cos
(θ(θ)
),(χ ξ
)(ξ) sin
(θ(θ)
),(ν ξ
)(ξ))
=
(χ(ξ) cos
θ
C∗1
, χ(ξ) sinθ
C∗1
, ν(ξ)
), (ξ, θ) ∈ DC1 ,
3.2. Complete biconservative surfaces in S3 83
where C1 > 4/(33/2
)is a positive constant, C∗
1 ∈(0,√(
33/2)/(33/2C1 − 4
) ), χ(ξ) =
C∗1/ξ and
ν(ξ) = ±∫ ξ
ξ00
√3τ2 − (C∗
1 )2 (−τ8/3 + 3C1τ2 − 3
)τ4(−τ8/3 + 3C1τ2 − 3
) dτ + c∗1,
for any ξ ∈ (ξ01, ξ02), with c∗1 a real constant.
Remark 3.39. The mean curvature function of ψC1,C∗1is given by
fC1,C∗1=
9ξ2 − (C∗1 )
2 (−2ξ8/3 + 9C1ξ2 − 18
)6C∗
1
√9ξ2 − 3 (C∗
1 )2 (−ξ8/3 + 3C1ξ2 − 3
)and we can see that it depends on both C1 and C
∗1 .
Remark 3.40. From now on, we will take ξ00 = (9C1/4)3/2 ∈ (ξ01, ξ02) and C∗
1 ∈(0,√(
33/2)/(33/2C1 − 4
) ).
The function ν has the following properties which follows easily.
Lemma 3.41. Let
ν0(ξ) =
∫ ξ
ξ00
√3τ2 − (C∗
1 )2 (−τ8/3 + 3C1τ2 − 3
)τ4(−τ8/3 + 3C1τ2 − 3
) dτ, ξ ∈ (ξ01, ξ02) ,
i.e., we x the sign in (3.23) and we choose c∗1 = c∗1,0 = 0. Then
(i) limξξ01 ν0(ξ) = ν0,−1 > −∞ and limξξ02 ν0(ξ) = ν0,1 <∞;
(ii) ν0 is strictly increasing and
limξξ01
ν ′0(ξ) = limξξ02
ν ′0(ξ) = ∞;
(iii) limξξ01 ν′′0 (ξ) = −∞ and limξξ02 ν
′′0 (ξ) = ∞.
Proof. We prove (i) in a similar way to the proof of Lemma 3.35. More precisely, we
consider the functions
H(ξ) =√3ξ2 − (C∗
1 )2 (−ξ8/3 + 3C1ξ2 − 3
)and G(ξ) =
1
ξ2,
with ξ ∈ [ξ01, ξ02]. It easy to see that these functions have a maximum and a min-
imum, which we denote by MH and mH , respectively, for H, and with MG and mG,
respectively, for G. Of course, mH , mG, MH , and MG are positive constants.
Let us assume that ξ < ξ00. Then ν0(ξ) < 0 for any ξ ∈ (ξ01, ξ00) and it is easy to
see that
ν0(ξ) ≥MHMG
∫ ξ
ξ00
1√−τ8/3 + 3C1τ2 − 3
dτ.
84 Chapter 3. Complete biconservative surfaces in R3 and S3
Thus, to show that limξξ01 ν0(ξ) is nite it is enough to prove that
limξξ01
∫ ξ
ξ00
1√−τ8/3 + 3C1τ2 − 3
dτ > −∞.
In order to prove this, we consider again the smooth function T (ξ) = −ξ8/3+3C1ξ2−3,
for any ξ ∈ (0,∞). We have
T (ξ) = T (ξ01) + T ′ (ξ01) (ξ − ξ01) + (ξ − ξ01)µ1(ξ), ξ > 0,
where µ1 is a continuous function on (0,∞) and limξ→ξ01 µ1(ξ) = 0.
Since T (ξ01) = 0 and T ′ (ξ01) = −8ξ5/301 /3 + 6C1ξ01 we obtain
T (ξ) =
(−8
3ξ5/301 + 6C1ξ01
)(ξ − ξ01) + (ξ − ξ01)µ1(ξ)
= (ξ − ξ01)
(−8
3ξ5/301 + 6C1ξ01 + µ1(ξ)
), ξ > 0.
We note that(−8ξ
5/301 /3 + 6C1ξ01
)(ξ − ξ01) > 0 for ξ ∈ (ξ01, ξ00). This is, indeed,
equivalent to −8ξ5/301 /3 + 6C1ξ01 < 0, i.e., ξ01 > (9C1/4)
3/2, which is true.
Now, we consider the functions β(ξ) = 1/
(√(−8ξ
5/301 /3 + 6C1ξ01
)(ξ − ξ01)
)and
γ(ξ) = 1/√
−ξ8/3 + 3C1ξ2 − 3, for any ξ ∈ (ξ01, ξ00). Since limξξ01 (γ(ξ)/β(ξ)) = 1 ∈(0,∞) and
limξξ01
∫ ξ
ξ00
β(τ)dτ = limε→0
∫ ξ01−ε
ξ00
β(τ)dτ = −2
√ξ00 − ξ01
−83ξ
5/301 + 6C1ξ01
∈ R∗−,
we get that limξξ01
∫ ξξ00γ(τ) dτ > −∞, which means that limξξ01 ν0(ξ) = ν0,−1 >
−∞.
In the same way, we can prove that limξξ02
∫ ξξ00
1/√−τ8/3 + 3C1τ2 − 3 dτ < ∞,
and then
limξξ02
ν0(ξ) = ν0,1 <∞.
In order to prove (ii), we rst note that
ν ′0(ξ) =
√3ξ2 − (C∗
1 )2 (−ξ8/3 + 3C1ξ2 − 3
)ξ4(−ξ8/3 + 3C1ξ2 − 3
) > 0, ξ ∈ (ξ01, ξ02) ,
and, therefore, ν0 is a strictly increasing function and
limξξ01
ν ′0(ξ) = limξξ02
ν ′0(ξ) = ∞.
3.2. Complete biconservative surfaces in S3 85
To prove (iii), we can rewrite the derivative of ν0 as
ν ′0(ξ) =
√3
ξ2(−ξ8/3 + 3C1ξ2 − 3
) − (C∗1 )
2
ξ4,
and by a straightforward computation we obtain that
ν ′′0 (ξ) =1
2√3ξ2 − (C∗
1 )2 (−ξ8/3 + 3C1ξ2 − 3
)(
1(−ξ8/3 + 3C1ξ2 − 3
)3/2 ·(−6(−ξ8/3 + 3C1ξ
2 − 3)
ξ− 3
(−8
3ξ5/3 + 6C1ξ
))+
+4 (C∗
1 )2√
−ξ8/3 + 3C1ξ2 − 3
ξ3
).
We have −8ξ5/301 /3+ 6C1ξ01 > 0 and −8ξ
5/302 /3+ 6C1ξ02 < 0 since these inequalities are
equivalent to relations (3.16). Therefore,
limξξ01
ν ′′0 (ξ) = −∞ and limξξ02
ν ′′0 (ξ) = ∞.
Remark 3.42. The immersion ψC1,C∗1depends on the sign ± and on the constant c∗1
in the expression of ν. We denote by S±C1,C∗
1 ,c∗1the image of ψC1,C∗
1.
We note that the boundary of S±C1,C∗
1 ,c∗1is given by the curves(
C∗1
ξ01cos
θ
C∗1
,C∗1
ξ01sin
θ
C∗1
, ν (ξ01)
)and (
C∗1
ξ02cos
θ
C∗1
,C∗1
ξ02sin
θ
C∗1
, ν (ξ02)
)These curves are circles in ane planes in R3 parallel to the Oxy plane and their radii
are C∗1/ξ01 and C
∗1/ξ02, respectively.
At a boundary point, using the coordinates (ν, θ), we get that the tangent plane to
the closure of S±C1,C∗
1 ,c∗1is spanned by a vector which is tangent to the corresponding
circle and by the vector (0, 0, 1). Thus, the tangent plane is parallel to the rotational
axis Oz.
Geometrically, we start with a piece of type S±C1,C∗
1 ,c∗1and by symmetry to the planes
where the boundary lie, we get our complete surface SC1,C∗1; the process is periodic and
we perform it along the whole Oz axis.
86 Chapter 3. Complete biconservative surfaces in R3 and S3
Analytically, we x C1 and C∗1 , and alternating the sign and with appropriate choices
of the constant c∗1, we can construct a complete surface of revolution SC1,C∗1in R3 which
on an open subset is locally isometric to (DC1 , gC1). In fact, these choices of + and −,and of the constants c∗1 are uniquely determined by the rst choice of +, or of −, andof the constant c∗1. We start with + and c∗1 = c∗1,0.
Further, we will give more details about this construction. Let us consider the
prole curve σ0(ξ) = (χ(ξ), ν0(ξ)), for any ξ ∈ (ξ01, ξ02). Obviously, ν0 : (ξ01, ξ02) →(ν0,−1, ν0,1) is a dieomorphism and we can consider ν−1
0 : (ν0,−1, ν0,1) → (ξ01, ξ02), with
ν−10 : ξ0 = ξ0(ν), ν ∈ (ν0,−1, ν0,1). One can reparametrize σ0 such that it becomes the
graph of a function depending on the variable ν, ν ∈ (ν0,−1, ν0,1).
In order to extend our surface to the upper part, we ask the line ν = ν0,1 to be a
symmetry axis. Therefore 2ν0,1 = ν0(ξ) + ν1(ξ), where ν1 : (ξ01, ξ02) → R, and then we
get ν1(ξ) = 2ν0,1 − ν0(ξ); thus c∗1 = c∗1,1 = 2ν0,1. It is easy to see that
limξξ01
ν1(ξ) = 2ν0,1 − ν0,−1, limξξ02
ν1(ξ) = ν0,1,
and, since ν ′1(ξ) = −ν ′0(ξ) < 0, for any ξ ∈ (ξ01, ξ02), it follows that ν1 is strictly
decreasing and ν1 (ξ01, ξ02) = (ν0,1, 2ν0,1 − ν0,−1). Since ν1 is a dieomorphism on its
image, we can consider ν−11 : (ν0,1, 2ν0,1 − ν0,−1) → (ξ01, ξ02), with ν−1
1 : ξ1 = ξ1(ν),
ν ∈ (ν0,1, 2ν0,1 − ν0,−1).
It is easy to see that
limνν0,1
ξ1(ν) = ξ02, limν2ν0,1−ν0,−1
ξ1(ν) = ξ01,
and, since(ν−11
)′(ν) = 1/ (ν ′1(ξ1(ν))) < 0, for any ν ∈ (ν0,1, 2ν0,1 − ν0,−1), it follows
that ν−11 is strictly decreasing.
Next, we dene a function F1 : (ν0,−1, 2ν0,1 − ν0,−1) → R by
F1(ν) =
ξ1(ν), ν ∈ (ν0,1, 2ν0,1 − ν0,−1)
ξ02, ν = ν0,1
ξ0(ν), ν ∈ (ν0,−1, ν0,1)
,
and we will prove that F1 is at least of class C3.
Obviously, F1 is continuous.
In order to prove that F1 is of class C1, rst we consider ν ∈ (ν0,−1, ν0,1). In this
case, we have
F ′1(ν) = ξ′0(ν) =
1
ν ′0(ξ0(ν))
and
limνν0,1
F ′1(ν) = lim
νν0,1ξ′0(ν) = lim
νν0,1
1
ν ′0(ξ0(ν))= lim
ξξ02
1
ν ′0(ξ)= 0.
3.2. Complete biconservative surfaces in S3 87
Then, if we consider ν ∈ (ν0,1, 2ν0,1 − ν0,−1), one gets
F ′1(ν) = ξ′1(ν) =
1
ν ′1(ξ1(ν))
and
limνν0,1
F ′1(ν) = lim
νν0,1ξ′1(ν) = lim
νν0,1
1
ν ′1(ξ1(ν))= lim
ξξ02
1
ν ′1(ξ)= lim
ξξ02
1
−ν ′0(ξ)= 0.
Therefore, limνν0,1 F′1(ν) = limνν0,1 F
′1(ν) = 0 ∈ R, which means that there exists
F ′1(ν0,1) = 0 and F1 is of class C
1.
To prove that F1 is of class C2 we consider the same two situations. First assume
that ν ∈ (ν0,−1, ν0,1). In this case, one has
F ′′1 (ν) = −ν
′′0 (ξ0(ν)) · ξ′0(ν)(ν ′0(ξ0(ν)))
2= − ν ′′0 (ξ0(ν))
(ν ′0(ξ0(ν)))3
and
limνν0,1
F ′′1 (ν) = − lim
ξξ02
ν ′′0 (ξ)
(ν ′0(ξ))3.
From the denition of ν0, by a straightforward computation, we obtain the expressions
for the rst and second derivatives of ν0, and replacing them in the above relation we
get
limνν0,1
F ′′1 (ν) =
1
6ξ302
(−8
3ξ2/302 + 6C1
)∈ R.
Now, assume that ν ∈ (ν0,1, 2ν0,1 − ν0,−1). In this case
F ′′1 (ν) = ξ′′1 (ν) = −ν
′′1 (ξ1(ν)) · ξ′1(ν)(ν ′1(ξ1(ν)))
2= − ν ′′1 (ξ1(ν))
(ν ′1(ξ1(ν)))3
and
limνν0,1
F ′′1 (ν) = − lim
ξξ02
ν ′′1 (ξ)
(ν ′1(ξ))3= − lim
ξξ02
ν ′′0 (ξ)
(ν ′0(ξ))3.
Therefore,
limνν0,1
F ′′1 (ν) = lim
νν0,1F ′′1 (ν) =
1
6ξ302
(−8
3ξ2/302 + 6C1
)∈ R,
which means that there exists F ′′1 (ν0,1) =
16ξ
302
(−8
3ξ2/302 + 6C1
), and F1 is of class C
2.
Now, we prove that F1 is of class C3. First, assume ν ∈ (ν0,−1, ν0,1). In this case
F ′′′1 (ν) =
−ν ′′′0 (ξ0(ν)) · ν ′0(ξ0(ν)) + 3(ν ′′0 (ξ0(ν)))2
(ν ′0(ξ0(ν)))5
and
limνν0,1
F ′′′1 (ν) = lim
ξξ02
−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2
(ν ′0(ξ))5
.
88 Chapter 3. Complete biconservative surfaces in R3 and S3
From the denition of ν0, the rst and second derivatives of ν0, we can compute the
third derivative of ν0, and substituting them in the above relation, one obtains
limνν0,1
F ′′′1 (ν) = 0.
The second case is when ν ∈ (ν0,1, 2ν0,1 − ν0,−1). In this case, we have
F ′′′1 (ν) =
−ν ′′′1 (ξ1(ν)) · ν ′1(ξ1(ν)) + 3(ν ′′1 (ξ1(ν)))2
(ν ′1(ξ1(ν)))5
and
limνν0,1
F ′′′1 (ν) = lim
ξξ02
−ν ′′′1 (ξ) · ν ′1(ξ) + 3(ν ′′1 (ξ))2
(ν ′1(ξ))5
= limξξ02
−(−ν ′′′0 (ξ)) · (−ν ′0(ξ)) + 3(−ν ′′0 (ξ))2
−(ν ′0(ξ))5
= − limξξ02
−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2
(ν ′0(ξ))5
.
Therefore,
limνν0,1
F ′′′1 (ν) = lim
νν0,1F ′′′1 (ν) = 0 ∈ R,
which shows that there exists F ′′′1 (ν0,1) = 0 and F1 is of class C
3.
In order to extend our surface to the lower part, we ask the line ν = ν0,−1 to be a
symmetry axis. Therefore, 2ν0,−1 = ν0(ξ) + ν−1(ξ), where ν−1 : (ξ01, ξ02) → R, and we
get ν−1(ξ) = 2ν0,−1 − ν0(ξ); thus c∗1 = c∗1,−1 = 2ν0,−1. It is easy to see that
limξξ02
ν−1(ξ) = 2ν0,−1 − ν0,1, limξξ01
ν−1(ξ) = ν0,−1,
and, since ν ′−1(ξ) = −ν ′0(ξ) < 0, for any ξ ∈ (ξ01, ξ02), it follows that ν−1 is strictly
decreasing and ν−1 (ξ01, ξ02) = (2ν0,−1 − ν0,1, ν0,−1). Since ν−1 is a dieomorphism
on its image, one can consider ν−1−1 : (2ν0,−1 − ν0,1, ν0,−1) → (ξ01, ξ02), with ν−1
−1 :
ξ−1 = ξ−1(ν), ν ∈ (2ν0,−1 − ν0,1, ν0,−1).
It is easy to see that
limν2ν0,−1−ν0,1
ξ−1(ν) = ξ02, limνν0,−1
ξ−1(ν) = ξ01,
and, since(ν−1−1
)′(ν) = 1/
(ν ′−1(ξ−1(ν))
)< 0, for any ν ∈ (2ν0,−1 − ν0,1, ν0,−1), we get
that ν−1−1 is strictly decreasing.
Further, we dene the function F−1 : (2ν0,−1 − ν0,1, ν0,1) → R by
F−1(ν) =
ξ0(ν), ν ∈ (ν0,−1, ν0,1)
ξ01, ν = ν0,−1
ξ−1(ν), ν ∈ (2ν0,−1 − ν0,1, ν0,−1)
.
3.2. Complete biconservative surfaces in S3 89
In a similar way to F1 case, we prove that F−1 is at least of class C3.
Obviously, F−1 is continuous.
To prove that F−1 is of class C1, we rst consider ν ∈ (ν0,−1, ν0,1). In this case
F ′−1(ν) = ξ′0(ν) =
1
ν ′0(ξ0(ν))
and
limνν0,−1
F ′−1(ν) = lim
νν0,−1
ξ′0(ν) = limνν0,−1
1
ν ′0(ξ0(ν))
= limξξ01
1
ν ′0(ξ)= 0.
Then, we consider ν ∈ (2ν0,−1 − ν0,1, ν0,−1). In this case
F ′−1(ν) = ξ′−1(ν) =
1
ν ′−1(ξ−1(ν))
and
limνν0,−1
F ′−1(ν) = lim
νν0,−1
ξ′−1(ν) = limνν0,−1
1
ν ′−1(ξ−1(ν))
= limξξ01
1
ν ′−1(ξ)= lim
ξξ01
1
−ν ′0(ξ)= 0.
Therefore, limνν0,−1 F′−1(ν) = limνν0,−1 F
′−1(ν) = 0 ∈ R, which means that there
exists F ′−1(ν0,−1) = 0 and F−1 is of class C
1.
In order to prove that F−1 is of class C2, we consider the same two situations. First,
let ν ∈ (ν0,−1, ν0,1). In this case, one has
F ′′−1(ν) = −ν
′′0 (ξ0(ν)) · ξ′0(ν)(ν ′0(ξ0(ν)))
2= − ν ′′0 (ξ0(ν))
(ν ′0(ξ0(ν)))3
and
limνν0,−1
F ′′−1(ν) = − lim
ξξ01
ν ′′0 (ξ)
(ν ′0(ξ))3.
By a straightforward computation, we can see that
limνν0,−1
F ′′−1(ν) = −1
6ξ301
(−8
3ξ2/301 + 6C1
)∈ R.
Second, assume that ν ∈ (2ν0,−1 − ν0,1, ν0,−1). Then
F ′′−1(ν) = ξ′−1(ν) = −
ν ′′−1(ξ−1(ν)) · ξ′−1(ν)
(ν ′−1(ξ−1(ν)))2= −
ν ′′−1(ξ−1(ν))
(ν ′−1(ξ−1(ν)))3
and
limνν0,−1
F ′′−1(ν) = − lim
ξξ01
ν ′′−1(ξ)
(ν ′−1(ξ))3= − lim
ξξ01
ν ′′0 (ξ)
(ν ′0(ξ))3.
90 Chapter 3. Complete biconservative surfaces in R3 and S3
Therefore,
limνν0,−1
F ′′−1(ν) = lim
νν0,−1
F ′′−1(ν) = −1
6ξ301
(−8
3ξ2/301 + 6C1
)∈ R,
which means that there exists F ′′−1(ν0,−1) = −(1/6)ξ301
(−8ξ
2/301 /3 + 6C1
)and F−1 is of
class C2.
Further, we prove that F−1 is of class C3. First, consider ν ∈ (ν0,−1, ν0,1). In this
case
F ′′′−1(ν) =
−ν ′′′0 (ξ0(ν)) · ν ′0(ξ0(ν)) + 3(ν ′′0 (ξ0(ν)))2
(ν ′0(ξ0(ν)))5
and
limνν0,−1
F ′′′−1(ν) = lim
ξξ01
−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2
(ν ′0(ξ))5
= 0.
If ν ∈ (2ν0,−1 − ν0,1, ν0,−1), we have
F ′′′−1(ν) =
−ν ′′′−1(ξ−1(ν)) · ν ′−1(ξ−1(ν)) + 3(ν ′′−1(ξ−1(ν)))2
(ν ′−1(ξ−1(ν)))5
and
limνν0,−1
F ′′′−1(ν) = lim
ξξ01
−ν ′′′−1(ξ) · ν ′−1(ξ) + 3(ν ′′−1(ξ))2
(ν ′−1(ξ))5
= − limξξ01
−ν ′′′0 (ξ) · ν ′0(ξ) + 3(ν ′′0 (ξ))2
(ν ′0(ξ))5
.
Therefore,
limνν0,−1
F ′′′−1(ν) = lim
νν0,−1
F ′′′−1(ν) = 0 ∈ R,
which means that there exists F ′′′−1(ν0,−1) = 0 and F−1 is of class C
3.
Now, we extend the functions F1 and F−1 to the whole line R. This construction
will be done by symmetry to the lines ν = ν0,k, k ∈ Z∗.
We dene ν0,2 = 2ν0,1 − ν0,−1, ν0,3 = 2ν0,2 − ν0,1 = 3ν0,1 − 2ν0,−1, etc.; then
ν0,−2 = 2ν0,−1−ν0,1, ν0,−3 = 2ν0,−2−ν0,−1 = 3ν0,−1−2ν0,1, etc.. In this way we obtain
ν0,k =
k ν0,1 − (k − 1)ν0,−1, k ≥ 1
−k ν0,−1 + (k + 1)ν0,1, k ≤ −1.
The functions νk are obtained in the same way. For example, ν1(ξ) = 2ν0,1 − ν0(ξ),
ν2(ξ) = 2ν0,2 − ν1(ξ) = 2ν0,1 − 2ν0,−1 + ν0(ξ), etc.; then ν−1(ξ) = 2ν0,−1 − ν0(ξ),
ν−2(ξ) = 2ν0,−2 − ν−1(ξ) = 2ν0,−1 − 2ν0,1 + ν0(ξ), etc.. In general, we have
νk(ξ) =
2ν0,k − νk−1(ξ), k ≥ 1
2ν0,k − νk+1(ξ), k ≤ −1.
3.2. Complete biconservative surfaces in S3 91
We note that for νk we have the following formulas
νk(ξ) =
k (ν0,1 − ν0,−1) + ν0(ξ), k = 2p, p ∈ Z(k + 1)ν0,1 − (k − 1)ν0,−1 − ν0(ξ), k = 2p+ 1, p ∈ Z
.
Denoting the inverse of the function νk by ξk, we dene the function
F (ν) =
ξ01, ν = ν0,k, k = 2p, p ≥ 1
ξ02, ν = ν0,k, k = 2p+ 1, p ≥ 0
ξk(ν), ν ∈ (ν0,k, ν0,k+1) , k ≥ 1
ξ02, ν = ν0,1
ξ0(ν), ν ∈ (ν0,−1, ν0,1)
ξ01, ν = ν0,−1
ξk(ν), ν ∈ (ν0,k−1, ν0,k) , k ≤ −1
ξ01, ν = ν0,k, k = 2p− 1, p ≤ 0
ξ02, ν = ν0,k, k = 2p, p ≤ −1
,
which is at least of class C3.
Remark 3.43. When C1 = C∗1 = 1, c∗1 = 0 and ξ00 = (9/4)3/2, the plots of
ν0(ξ) =
∫ ξ
ξ00
√3τ2 −
(−τ8/3 + 3C1τ2 − 3
)τ4(−τ8/3 + 3C1τ2 − 3
) dτ,
ν1(ξ) = 2ν02 − ν(ξ), h−1(ξ) = 2ν01 − ν(ξ), and of corresponding prole curves σ0(ξ) =
(1/ξ, ν0(ξ)), σ1(ξ) = (1/ξ, ν1(ξ)), and σ−1(ξ) = (1/ξ, ν−1(ξ)), for ξ ∈ (ξ01, ξ02), are
represented in Figures 3.7, 3.8, 3.9, 3.10.
Remark 3.44. The function F is periodic and its main period is 2 (ν0,1 − ν0,−1).
Remark 3.45. The function F depends on C1 and C∗1 .
We dene σk(ξ) = (χ(ξ), νk(ξ)), ξ ∈ (ξ01, ξ02), where k ∈ Z. From Theorem 3.38,
we know that (DC1 , gC1) is isometric to the surface of revolution given by
ΨC1,C∗1(ξ, θ) =
(χ(ξ) cos
θ
C∗1
, χ(ξ) sinθ
C∗1
, νk(ξ)
), (ξ, θ) ∈ DC1 .
We can reparameterize σk and one obtains
σk(ν) =
σ (ξk(ν)) = ((χ ξk)(ν), ν) = ((χ F )(ν), ν) , ν ∈ (ν0,k, ν0,k+1) , k ≥ 1
σ (ξ0(ν)) = ((χ ξ0)(ν), ν) = ((χ F )(ν), ν) , ν ∈ (ν0,−1, ν0,1) , k = 0
σ (ξk(ν)) = ((χ ξk)(ν), ν) = ((χ F )(ν), ν) , ν ∈ (ν0,k−1, ν0,k) , k ≤ −1
.
Now, let us consider the periodic curve
σ(ν) = ((χ F )(ν), ν) , ν ∈ R.
92 Chapter 3. Complete biconservative surfaces in R3 and S3
Figure 3.7: Plot of ν0. Figure 3.8: Plot of ν0, ν1 and ν−1.
Figure 3.9: Plot of σ0. Figure 3.10: Plot of σ0, σ1 and σ−1.
The curve σ is the prole curve of SC1,C∗1and it is the graph of the function χ F
depending on ν and dened on the whole Oz (or Oν). We note that σ is at least of
class C3.
Theorem 3.46. The surface of revolution given by
ΨC1,C∗1(ν, θ) =
((χ F )(ν) cos θ
C∗1
, (χ F )(ν) sin θ
C∗1
, ν
), (ν, θ) ∈ R2,
is complete and, on an open dense subset, it is locally isometric to (DC1 , gC1). The
induced metric is given by
gC1,C∗1(ν, θ) =
3F 2(ν)
3F 2(ν)− (C∗1 )
2 (−F 8/3(ν) + 3C1F 2(ν)− 3)dν2 +
1
F 2(ν)dθ2,
(ν, θ) ∈ R2. Moreover, gradK = 0 at any point of that open dense subset, and 1−K > 0
everywhere.
3.2. Complete biconservative surfaces in S3 93
From Theorem 3.46 we easily get the following result.
Proposition 3.47. The universal cover of the surface of revolution given by ΨC1,C∗1
is R2 endowed with the metric gC1,C∗1. It is complete, 1 − K > 0 on R2 and, on an
open dense subset, it is locally isometric to (DC1 , gC1) and gradK = 0 at any point.
Moreover any two surfaces(R2, gC1,C∗
1
)and
(R2, gC1,C∗′
1
)are isometric.
Proof. We only have to prove the last statement. We construct the isometry between(R2, gC1,C∗
1
)and
(R2, gC1,C∗′
1
)in a natural way, in the sense that, for example, it maps
the interval (ν0,−1, ν0,1) corresponding to C∗1 onto the interval (ν0,−1, ν0,1) corresponding
to C∗′1 . Repeating this process, we obtain an (at least) C3 dieomorphism of R2. It is
easy to see that such dieomorphism is a global isometry.
The second step is to construct eectively the biconservative immersion from the
surface(R2, gC1,C∗
1
)in S3, or from SC1,C∗
1in S3. The geometric ideea of the construction
is the following: from each piece S±C1,C∗
1 ,c∗1of SC1,C∗
1we go back to (DC1 , gC1) and then,
using ϕC1 and a specic choice of+ or− and of the constant c1, we get our biconservative
immersion ΦC1,C∗1. Again, the choices of + and −, and of the constant c1 are uniquely
determined (modulo 2π, for c1) by the rst choice of +, or of −, and of the constant
c1.
Further, we will give more details about this construction. We recall that, from
Theorem 3.31 and Lemma 3.35, we have that ΦC1 : (DC1 , gC1) → S3,
ϕC1(ξ, θ) =
(√1− 1
C1ξ2cos ζ,
√1− 1
C1ξ2sin ζ,
cos(√C1θ)√
C1ξ,sin(
√C1θ)√C1ξ
),
with ζ(ξ) = ± (ζ0(ξ) + c1), is a biconservative immersion in S3 and
limξξ01
ζ0(ξ) = ζ0,−1 > −∞, limξξ02
ζ0(ξ) = ζ0,1 <∞.
In order to construct a biconservative immersion from(R2, gC1,C∗
1
)in S3, starting
with the rst component of the parametrization, we consider the following continuous
functions dened on [ξ01, ξ02]:
Φ1k(ξ) =
√
1− 1C1ξ2
cos (ζ0(ξ) + c1,k) , ξ ∈ (ξ01, ξ02)√1− 1
C1ξ201cos (ζ0,−1 + c1,k) , ξ = ξ01√
1− 1C1ξ202
cos (ζ0,1 + c1,k) , ξ = ξ02
,
where c1,k ∈ R for any k ∈ Z.Next, consider the function Φ1 : R → R dened by
Φ1(ν) =
(Φ1k F
)(ν), ν ∈ [ν0,k, ν0,k+1] , k ≥ 1(
Φ10 F
)(ν), ν ∈ [ν0,−1, ν0,1](
Φ1k F
)(ν), ν ∈ [ν0,k−1, ν0,k] , k ≤ −1
. (3.28)
94 Chapter 3. Complete biconservative surfaces in R3 and S3
We will prove that Φ1 is of class C3. Since F is a periodic function, with the main period
2 (ν0,1 − ν0,−1), it is enough to ask Φ1 to be a C3 function on the interval (ν0,−2, ν0,2) =
(2ν0,−1 − ν0,1, 2ν0,1 − ν0,−1). This means that it is enough to study the behaviour of F
at ν0,−1 and ν0,1.
First, we ask Φ1 to be continuous at ν0,−1 and ν0,1, i.e.,
limνν0,1
Φ1(ν) = limνν0,1
Φ1(ν) ∈ R, limνν0,−1
Φ1(ν) = limνν0,−1
Φ1(ν) ∈ R.
Since
limνν0,1
Φ1(ν) = limνν0,1
Φ10(F (ν)) = lim
νν0,1Φ10(ξ0(ν))
= limξξ02
Φ10(ξ) =
√1− 1
C1ξ202cos (ζ0,1 + c1,0) ∈ R
and
limνν0,1
Φ1(ν) = limνν0,1
Φ11(F (ν)) = lim
νν0,1Φ11(ξ1(ν))
= limξξ02
Φ11(ξ) =
√1− 1
C1ξ202cos (ζ0,1 + c1,1) ∈ R,
we get that cos (ζ0,1 + c1,0) = cos (ζ0,1 + c1,1). Therefore, we have two cases, as c1,1 =
c1,0 + 2s1π or c1,1 = −2ζ0,1 − c1,0 + 2s1π, where s1 ∈ Z, i.e.,
c1,1 ≡ c1,0 (mod 2π) or c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π) .
In a similar way, for ν0,−1, we have
limνν0,−1
Φ1(ν) = limνν0,−1
Φ10(F (ν)) = lim
νν0,−1
Φ10(ξ0(ν))
= limξξ01
Φ10(ξ) =
√1− 1
C1ξ201cos (ζ0,−1 + c1,0) ∈ R
and
limνν0,−1
Φ1(ν) = limνν0,−1
Φ11(F (ν)) = lim
νν0,−1
Φ11(ξ−1(ν))
= limξξ01
Φ11(ξ) =
√1− 1
C1ξ201cos (ζ0,−1 + c1,−1) ∈ R.
Hence, we must have cos (ζ0,−1 + c1,0) = cos (ζ0,−1 + c1,−1). Therefore we have again
two cases as c1,−1 = c1,0 +2s−1π or c1,−1 = −2ζ0,−1 − c1,0 +2s−1π, where s−1 ∈ Z, i.e.,c1,−1 ≡ c1,0 (mod 2π) or c1,−1 ≡ (−2ζ0,−1 − c1,0) (mod 2π).
3.2. Complete biconservative surfaces in S3 95
Further, we ask that Φ1 to be a C1 function on the interval (ν0,−2, ν0,2). Thus we
must see what happens at ν0,−1 and ν0,1.
When ν ∈ (ν0,−1, ν0,1), we have
(Φ1)′(ν) =
(Φ1
0
)′(ξ0(ν))ξ
′0(ν) =
(Φ1
0
)′(ξ0(ν))
1
ν′0(ξ0(ν))
=
1
C1ξ30(ν)√1− 1
C1ξ20(ν)
cos (ζ0(ξ0(ν)) + c1,0)−
√1− 1
C1ξ20(ν)·
·√νξ
4/30 (ν)
(−1 + C1ξ20(ν))
√−ξ8/30 (ν) + 3C1ξ20(ν)− 3
sin (ζ0(ξ0(ν)) + c1,0)
·
·ξ20(ν)
√−ξ8/30 (ν) + 3C1ξ20(ν)− 3√
3ξ20(ν)− (C∗1 )
2(−ξ8/30 (ν) + 3C1ξ20(ν)− 3
) .
Thus
limνν0,1
(Φ1)′(ν) = lim
ξξ02
1
C1ξ3√1− 1
C1ξ2
cos (ζ0(ξ) + c1,0)−√
1− 1
C1ξ2·
·√C1ξ
4/3
(−1 + C1ξ2)√−ξ8/3 + 3C1ξ2 − 3
sin (ζ0(ξ) + c1,0)
]·
· ξ2√−ξ8/3 + 3C1ξ2 − 3√
3ξ2 − (C∗1 )
2 (−ξ8/3 + 3C1ξ2 − 3)
= −
√C1
3
(1− 1
C1ξ202
)ξ7/302
−1 + C1ξ202sin (ζ0,1 + c1,0) ∈ R
and
limνν0,−1
(Φ1)′(ν) = lim
ξξ01
1
C1ξ3√
1− 1C1ξ2
cos (ζ0(ξ) + c1,0)−√
1− 1
C1ξ2·
·√C1ξ
4/3
(−1 + C1ξ2)√
−ξ8/3 + 3C1ξ2 − 3sin (ζ0(ξ) + c1,0)
]·
· ξ2√−ξ8/3 + 3C1ξ2 − 3√
3ξ2 − (C∗1 )
2 (−ξ8/3 + 3C1ξ2 − 3)
= −
√C1
3
(1− 1
C1ξ201
)ξ7/301
−1 + C1ξ201sin (ζ0,−1 + c1,0) ∈ R.
96 Chapter 3. Complete biconservative surfaces in R3 and S3
When ν ∈ (ν0,1, ν0,2), we have
(Φ1)′(ν) =
(Φ1
1
)′(ξ1(ν))ξ
′1(ν) =
(Φ1
1
)′(ξ1(ν))
1
ν′1(ξ1(ν))
=
1
C1ξ31(ν)√1− 1
C1ξ21(ν)
cos (ζ0 (ξ1(ν)) + c1,1)−
√1− 1
C1ξ21(ν)·
·√C1ξ
4/31 (ν)
(−1 + C1ξ21(ν))
√−ξ8/31 (ν) + 3C1ξ21(ν)− 3
sin (ζ0 (ξ1(ν)) + c1,1)
·
·
−ξ21(ν)
√−ξ8/31 (ν) + 3C1ξ21(ν)− 3√
3ξ21(ν)− (C∗1 )
2(−ξ8/31 (ν) + 3C1ξ21(ν)− 3
) .
Thus
limνν0,1
(Φ1)′(ν) = − lim
ξξ02
1
C1ξ3√
1− 1C1ξ2
cos (ζ0(ξ) + c1,1)−√1− 1
C1ξ2·
·√C1ξ
4/3
(−1 + C1ξ2)√
−ξ8/3 + 3C1ξ2 − 3sin (ζ0(ξ) + c1,1)
]·
· ξ2√−ξ8/3 + 3C1ξ2 − 3√
3ξ2 − (C∗1 )
2 (−ξ8/3 + 3C1ξ2 − 3)
=
√C1
3
(1− 1
C1ξ202
)ξ7/302
−1 + C1ξ202sin (ζ0,1 + c1,1) ∈ R.
When ν ∈ (ν0,−2, ν0,−1), we have
(Φ1)′(ν) =
(Φ1
−1
)′(ξ−1(ν))ξ
′−1(ν) =
(Φ1
−1
)′(ξ−1(ν))
1
ν′−1(ξ−1(ν))
=
1
C1ξ3−1(ν)√1− 1
C1ξ2−1(ν)
(ζ0 (ξ−1(ν)) + c1,−1)−√
1− 1
C1ξ2−1(ν)·
·√C1ξ
4/3−1 (ν)(
−1 + C1ξ2−1(ν))√
−ξ8/3−1 (ν) + 3C1ξ2−1(ν)− 3sin (ζ0 (ξ−1(ν)) + c1,−1)
·
·
−ξ2−1(ν)
√−ξ8/3−1 (ν) + 3C1ξ2−1(ν)− 3√
3ξ2−1(ν)− (C∗1 )
2(−ξ8/3−1 (ν) + 3C1ξ2−1(ν)− 3
) .
3.2. Complete biconservative surfaces in S3 97
Hence
limνν0,−1
(Φ1)′(ν) = − lim
ξξ01
1
C1ξ3√
1− 1C1ξ2
cos (ζ0(ξ) + c1,−1)−√
1− 1
C1ξ2·
·√C1ξ
4/3
(−1 + C1ξ2)√
−ξ8/3 + 3C1ξ2 − 3sin (ζ0(ξ) + c1,−1)
]·
· ξ2√−ξ8/3 + 3C1ξ2 − 3√
3ξ2 − (C∗1 )
2 (−ξ8/3 + 3C1ξ2 − 3)
=
√C1
3
(1− 1
C1ξ201
)ξ7/301
−1 + C1ξ201sin (ζ0,−1 + c1,−1) ∈ R.
The function Φ1 is of class C1 on (ν0,−2, ν0,2) if and only if
limνν0,1
(Φ1)′(ν) = lim
νν0,1
(Φ1)′(ν) ∈ R and lim
νν0,−1
(Φ1)′(ν) = lim
νν0,−1
(Φ1)′(ν) ∈ R.
The above equalities are equivalent to
sin (ζ0,1 + c1,0) = − sin (ζ0,1 + c1,1) and sin (ζ0,−1 + c1,0) = − sin (ζ0,−1 + c1,−1) .
We recall that, from the continuity of Φ1, there are two possibilities for each c1,1 and
c1,−1 as follows
• If c1,1 = c1,0 + 2s1π, i.e., c1,1 ≡ c1,0 (mod 2π), we get
sin (ζ0,1 + c1,1) = sin (ζ0,1 + c1,0 + 2s1π) = sin (ζ0,1 + c1,0) ;
• If c1,1 = −2ζ0,1 − c1,0 + 2s1π, i.e., c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π), we get
sin (ζ0,1 + c1,1) = sin (ζ0,1 − 2ζ0,1 − c1,0 + 2s1π) = − sin (ζ0,1 + c1,0) ;
• If c1,−1 = c1,0 + 2s−1π, i.e., c1,−1 ≡ c1,0 (mod 2π), we get
sin (ζ0,−1 + c1,−1) = sin (ζ0,−1 + c1,0 + 2s−1π) = sin (ζ0,−1 + c1,0) ;
• If c1,−1 = −2ζ0,−1 − c1,0 + 2s−1π, i.e., c1,−1 ≡ (−2ζ0,−1 − c1,0) (mod 2π), we get
sin (ζ0,−1 + c1,−1) = sin (ζ0,−1 − 2ζ0,−1 − c1,0 + 2s−1π) = − sin (ζ0,−1 + c1,0) .
Then we can choose
c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π) and c1,−1 ≡ (−2ζ0,−1 − c1,0) (mod 2π) . (3.29)
With this choice, we show that Φ1 is of class C2 on (ν0,−2, ν0,2).
98 Chapter 3. Complete biconservative surfaces in R3 and S3
The reasoning is similar to that from the proof of C1 smoothness of Φ1.
When ν ∈ (ν0,−1, ν0,1), we have(Φ1)′′
(ν) =(Φ10
)′′(ξ0(ν))
1
(ν ′0(ξ0(ν)))2 −
(Φ10
)′(ξ0(ν))
ν ′′0 (ξ0(ν))
(ν ′0(ξ0(ν)))3 .
From the expressions of Φ10 and of ν0, one can compute (Φ1)′′(ν) and we obtain
limνν0,1
(Φ1)′′
(ν) =
[− C1ξ
14/302
3 (−1 + C1ξ202)2
(1− 1
C1ξ202
)1/2
+
+− 4
3ξ2/302 + 3C1
3C1
(1− 1
C1ξ202
)−1/2]cos (ζ0,1 + c1,0) ∈ R
and
limνν0,−1
(Φ1)′′
(ν) =
[− C1ξ
14/301
3 (−1 + C1ξ201)2
(1− 1
C1ξ201
)1/2
+
+− 4
3ξ2/301 + 3C1
3C1
(1− 1
C1ξ201
)−1/2]cos (ζ0,−1 + c1,0) ∈ R.
When ν ∈ (ν0,1, ν0,2), we have(Φ1)′′
(ν) =(Φ11
)′′(ξ1(ν))
1
(ν ′1(ξ1(ν)))2 −
(Φ11
)′(ξ1(ν))
ν ′′1 (ξ1(ν))
(ν ′1(ξ1(ν)))3 ,
and, as above, one obtains
limνν0,1
(Φ1)′′
(ν) =
[− C1ξ
14/302
3 (−1 + C1ξ202)2
(1− 1
C1ξ202
)1/2
+
+− 4
3ξ2/302 + 3C1
3C1
(1− 1
C1ξ202
)−1/2]cos (ζ0,1 + c1,1) ∈ R
For ν ∈ (ν0,−2, ν0,−1), we have(Φ1)′′
(ν) =(Φ1−1
)′′(ξ−1(ν))
1(ν ′−1(ξ−1(ν))
)2 −(Φ1−1
)′(ξ−1(ν)) ·
ν ′′−1(ξ−1(ν))(ν ′−1(ξ−1(ν))
)3and one gets
limνν0,−1
(Φ1)′′
(ν) =
[− C1ξ
14/301
3 (−1 + C1ξ201)2
(1− 1
C1ξ201
)1/2
+
+− 4
3ξ2/301 + 3C1
3C1
(1− 1
C1ξ201
)−1/2]cos (ζ0,−1 + c1,−1) ∈ R
From (3.29) one obtains
limνν0,1
(Φ1)′′
(ν) = limνν0,1
(Φ1)′′
(ν) ∈ R, limνν0,−1
(Φ1)′′
(ν) = limνν0,−1
(Φ1)′′
(ν) ∈ R,
3.2. Complete biconservative surfaces in S3 99
and consequently, for this choice of c1,1 and c1,−1, we get that Φ1 is of class C2 on
(h0,−2, h0,2).
Using the same method, one can prove that Φ1 is of class C3 on (ν0,−2, ν0,2). More
precisely, one sees that
limνν0,1
(Φ1)′′′
(ν) = limνν0,1
(Φ1)′′′
(ν) =
=
√C1
(1− 1
C1ξ202
)ξ13/302
(6− 6C1 (C
∗1 )
2+ 8
3 (C∗1 )
2ξ2/302
)4 · 35/2(−1 + C1ξ202)
·
·(6C1 −
8
3ξ2/302
)+C
3/21 ξ702
√1− 1
C1ξ202
33/2 (−1 + C1ξ202)3 −
ξ7/302
(6C1 − 8
3ξ2/302
)2√3C1
√1− 1
C1ξ202
−
−5√C1ξ
13/302
(6C1 − 8
3ξ2/302
)√1− 1
C1ξ202
35/2 (−1 + C1ξ202)+
+C
3/21 ξ
19/302
(6C1 − 8
3ξ2/302
)√1− 1
C1ξ202
33/2 (−1 + C1ξ202)2
sin (ζ0,1 + c1,0) ∈ R
and
limνν0,−1
(Φ1)′′′
(ν) = limνν0,−1
(Φ1)′′′
(ν) =
=
√C1
(1− 1
C1ξ201
)ξ13/301
(6− 6C1 (C
∗1 )
2+ 8
3 (C∗1 )
2ξ2/301
)4 · 35/2(−1 + C1ξ201)
·
·(6C1 −
8
3ξ2/301
)+C
3/21 ξ701
√1− 1
C1ξ201
33/2 (−1 + C1ξ201)3 −
ξ7/301
(6C1 − 8
3ξ2/301
)2√3C1
√1− 1
C1ξ201
−
−5√C1ξ
13/301
(6C1 − 8
3ξ2/301
)√1− 1
C1ξ201
35/2 (−1 + C1ξ201)+
+C
3/21 ξ
19/301
(6C1 − 8
3ξ2/301
)√1− 1
C1ξ201
33/2 (−1 + C1ξ201)2
sin (ζ0,−1 + c1,0) ∈ R.
In general, if we ask Φ1 to be of class C3 on R, since F is a periodic function, it can be
shown that we have the following relations between two consecutive c1,k, where k ∈ Z:
c1,k ≡
(−2ζ0,1 − c1,k−1) (mod 2π) , k = 2p+ 1, p ∈ N(−2ζ0,−1 − c1,k−1) (mod 2π) , k = 2p, p ∈ N(−2ζ0,−1 − c1,k+1) (mod 2π) , k = 2p− 1, p ∈ Z−
(−2ζ0,1 − c1,k+1) (mod 2π) , k = 2p, p ∈ Z−
, (3.30)
100 Chapter 3. Complete biconservative surfaces in R3 and S3
or, equivalently,
c1,k ≡
(−2ζ0,1 − c1,k−1) (mod 2π) k = 2p+ 1, p ∈ Z(−2ζ0,−1 − c1,k−1) (mod 2π) k = 2p, p ∈ Z
.
We note that for c1,k, we also have the following formulas
c1,k ≡
(k (ζ0,1 − ζ0,−1) + c1,0) (mod 2π) , k = 2p, p ∈ Z((k − 1)ζ0,−1 − (k + 1)ζ0,1 − c1,0) (mod 2π) , k = 2p+ 1, p ∈ Z
. (3.31)
To study the second component of the parametrization ϕC1 , we work in a similar
way as for the rst one. We consider the following continuous functions dened on
[ξ01, ξ02]:
Φ2k(ξ) =
(−1)k
√1− 1
C1ξ2sin (ζ0(ξ) + c1,k) , ξ ∈ (ξ01, ξ02)
(−1)k√1− 1
C1ξ201sin (ζ0,−1 + c1,k) , ξ = ξ01
(−1)k√
1− 1C1ξ202
sin (ζ0,1 + c1,k) , ξ = ξ02
,
where c1,k ∈ R, for any k ∈ Z, are given by (3.30).
Then, we consider the function Φ2 : R → R dened by
Φ2(ν) =
(Φ2k F
)(ν), ν ∈ [ν0,k, ν0,k+1] , k ≥ 1(
Φ20 F
)(ν), ν ∈ [ν0,−1, ν0,1](
Φ2k F
)(ν), ν ∈ [ν0,k−1, ν0,k] , k ≤ −1
. (3.32)
We show that, with these choices of the constants c1,k, Φ2 is of class C3. The proof
is similar to the proof of C3 smoothness of Φ1, as we will see further.
Since F is a periodic function, with main period 2 (ν0,1 − ν0,−1), it is enough to
prove that Φ2 is of class C3 on (ν0,−2, ν0,2). This means that it is enough to study what
happens at ν0,−1 and ν0,1.
Since sin (ζ0,−1 + c1,−1) = − sin (ζ0,−1 + c1,0) and sin (ζ0,1 + c1,1) = − sin (ζ0,1 + c1,0),
it is easy to see that Φ2 is continuous at ν0,−1 and ν0,1, as
limνν0,1
Φ2(ν) = limνν0,1
Φ2(ν) =
√1− 1
C1ξ202sin (ζ0,1 + c1,0) ∈ R
and
limνν0,−1
Φ2(ν) = limνν0,−1
Φ2(ν) =
√1− 1
C1ξ201sin (ζ0,−1 + c1,0) ∈ R.
Therefore, we get that Φ2 is continuous on (ν0,−2, ν0,2).
Since cos (ζ0,−1 + c1,−1) = cos (ζ0,−1 + c1,0) and cos (ζ0,1 + c1,1) = cos (ζ0,1 + c1,0),
we obtain that
limνν0,1
(Φ2)′(ν) = lim
νν0,1
(Φ2)′(ν) =
√C1
3
(1− 1
C1ξ202
)ξ7/302
−1 + C1ξ202cos (ζ0,1 + c1,0) ∈ R
3.2. Complete biconservative surfaces in S3 101
and
limνν0,−1
(Φ2)′(ν) = lim
νν0,−1
(Φ2)′(ν) =
√C1
3
(1− 1
C1ξ201
)ξ7/301
−1 + C1ξ201cos (ζ0,−1 + c1,0) ∈ R.
Therefore, we get that Φ2 is of class C1 on (ν0,−2, ν0,2).
Next, from sin (ζ0,−1 + c1,−1) = − sin (ζ0,−1 + c1,0) and sin (ζ0,1 + c1,1) = − sin (ζ0,1 + c1,0),
we get that
limνν0,1
(Φ2)′′
(ν) = limνν0,1
(Φ2)′′
(ν) =
=
[− C1ξ
14/302
3 (−1 + C1ξ202)2
(1− 1
C1ξ202
)1/2
+
+− 4
3ξ2/302 + 3C1
3C1
(1− 1
C1ξ202
)−1/2]sin (ζ0,1 + c1,0) ∈ R
and
limνν0,−1
(Φ2)′′
(ν) = limνν0,−1
(Φ2)′′
(ν) =[− C1ξ
14/301
3 (−1 + C1ξ201)2
(1− 1
C1ξ201
)1/2
+
+− 4
3ξ2/301 + 3C1
3C1
(1− 1
C1ξ201
)−1/2]sin (ζ0,−1 + c1,0) ∈ R.
Therefore, we have that Φ2 is of class C2 on (ν0,−2, ν0,2).
Similarly, for the third derivative of Φ2, using the relations cos (ζ0,−1 + c1,−1) =
cos (ζ0,−1 + c1,0) and cos (ζ0,1 + c1,1) = cos (ζ0,1 + c1,0), one obtains
limνν0,1
(Φ2)′′′
(ν) = limνν0,1
(Φ2)′′′
(ν) =
= −
√C1
(1− 1
C1ξ202
)ξ13/302
(6− 6C1 (C
∗1 )
2+ 8
3 (C∗1 )
2ξ2/302
)4 · 35/2(−1 + C1ξ202)
·
·(6C1 −
8
3ξ2/302
)+C
3/21 ξ702
√1− 1
C1ξ202
33/2 (−1 + C1ξ202)3 −
ξ7/302
(6C1 − 8
3ξ2/302
)2√3C1
√1− 1
C1ξ202
−
−5√C1ξ
13/302
(6C1 − 8
3ξ2/302
)√1− 1
C1ξ202
35/2 (−1 + C1ξ202)+
+C
3/21 ξ
19/302
(6C1 − 8
3ξ2/302
)√1− 1
C1ξ202
33/2 (−1 + C1ξ202)2
cos (ζ0,1 + c0) ∈ R
102 Chapter 3. Complete biconservative surfaces in R3 and S3
and
limνν0,−1
(Φ2)′′′
(ν) = limνν0,−1
(Φ2)′′′
(ν) =
= −
√C1
(1− 1
C1ξ201
)ξ13/301
(6− 6C1 (C
∗1 )
2+ 8
3 (C∗1 )
2ξ2/301
)4 · 35/2(−1 + C1ξ201)
·
·(6C1 −
8
3ξ2/301
)+C
3/21 ξ701
√1− 1
C1ξ201
33/2 (−1 + C1ξ201)3 −
ξ7/301
(6C1 − 8
3ξ2/301
)2√3C1
√1− 1
C1ξ201
−
−5√C1ξ
13/301
(6C1 − 8
3ξ2/301
)√1− 1
C1ξ201
35/2 (−1 + C1ξ201)+
+C
3/21 ξ
19/301
(6C1 − 8
3ξ2/301
)√1− 1
C1ξ201
33/2 (−1 + C1ξ201)2
cos (ζ0,−1 + c1,0) ∈ R.
Therefore, we see that Φ2 is of class C3 on (ν0,−2, ν0,2).
Since F is periodic, one can prove that Φ2 is of class C3 on whole R.For the third component of the parametrization ϕC1 , we consider the following func-
tion
Φ30(ξ) =
1√C1ξ
, ξ ∈ [ξ01, ξ02] ,
It is obvious that Φ30 is a smooth function on [ξ01, ξ02].
Let us consider a new function Φ3 : R → R dened by
Φ3(ν) = (Φ30 F )(ν), ν ∈ R. (3.33)
Since F is at least of class C3 on R and Φ30 is smooth on [ξ01, ξ02], it follows that Φ
3 is
at least of class C3 on R.For the fourth component of the parametrization ϕC1 , we dene Φ4 as Φ3, i.e.,
Φ4(ν) = (Φ40 F )(ν), ν ∈ R, (3.34)
where Φ40(ξ) = 1/
(√C1ξ
), for any ξ ∈ [ξ01, ξ02].
Now, we can conclude with the following theorem.
Theorem 3.48. The map ΦC1,C∗1:(R2, gC1,C∗
1
)→ S3, dened by
ΦC1,C∗1(ν, θ) = ϕC1(F (ν), θ) =
(Φ1(ν),Φ2(ν),Φ3(ν) cos(
√C1θ),Φ
4(ν) sin(√C1θ)
),
(ν, θ) ∈ R2, where Φ1, Φ2, Φ3 and Φ4 are given by (3.28), (3.32), (3.33) and (3.34),
respectively, and the constants c1,k are given by (3.31), is a biconservative immersion.
3.2. Complete biconservative surfaces in S3 103
Proof. Obviously, for ν ∈ (ν0,k, ν0,k+1), when k ≥ 1, or ν ∈ (ν0,−1, ν0,1), or ν ∈(ν0,k−1, ν0,k), when k ≤ −1, ΦC1,C∗
1is an isometric immersion and it is biconservat-
ive. As ΦC1,C∗1is a map of class C3 and the biconservative equation is a third-degree
equation, by continuity, we get that ΦC1,C∗1is biconservative on R2.
Remark 3.49. We note that ΦC1,C∗1has self-intersections (along circles).
We end this chapter by verifying (partially) the correctness of the above construc-
tion.
More precisely, if we denote by S+C1,c1,0
the image of ΦC1,C∗1((ν0,−1, ν0,1)× R), we
can see that its boundary in R4 is given by the curves:(√1− 1
C1ξ201cos (ζ0,−1 + c1,0) ,
√1− 1
C1ξ201sin (ζ0,−1 + c1,0) ,
cos(√C1θ
)√C1ξ01
,sin(√C1θ
)√C1ξ01
)
and(√1− 1
C1ξ202cos (ζ0,1 + c1,0) ,
√1− 1
C1ξ202sin (ζ0,1 + c1,0) ,
cos(√C1θ
)√C1ξ02
,sin(√C1θ
)√C1ξ02
).
These curves are two circles in the ane planes(√1− 1
C1ξ201cos (ζ0,−1 + c1,0) ,
√1− 1
C1ξ201sin (ζ0,−1 + c1,0) , 0, 0
)+ span e3, e4
and(√1− 1
C1ξ202cos (ζ0,1 + c1,0) ,
√1− 1
C1ξ202sin (ζ0,1 + c1,0) , 0, 0
)+ span e3, e4 ,
respectively. The radii of the lower circle is 1/(√C1ξ01
)and of the upper one
1/(√C1ξ02
), respectively.
Now if we consider S−C1,c1,1
the image of ΦC1,C∗1((ν0,1, ν0,2)× R), we can see that its
boundary in R4 is given by the curves:(√1− 1
C1ξ202cos (ζ0,1 + c1,1) ,−
√1− 1
C1ξ202sin (ζ0,1 + c1,1) ,
cos(√C1θ
)√C1ξ02
,sin(√C1θ
)√C1ξ02
)
and(√1− 1
C1ξ201cos (ζ0,−1 + c1,1) ,−
√1− 1
C1ξ201sin (ζ0,−1 + c1,1) ,
cos(√C1θ
)√C1ξ01
,sin(√C1θ
)√C1ξ01
).
These curves are two circles and the radii of them are 1/(√C1ξ02
)(for the lower
circle) and 1/(√C1ξ01
)(for the upper circle).
Since c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π), it is easy to see that the lower circle from
the boundary of S−C1,c1,1
coincide with the upper circle from the boundary of S+C1,c1,0
.
104 Chapter 3. Complete biconservative surfaces in R3 and S3
Then, at a common boundary point, we get that the tangent plane to the closure in R4,
S+C1,c1,0
, of S+C1,c1,0
is spanned by a vector tangent to the boundary and the vector− ξ4/302√
3(C1ξ202 − 1
) sin (ζ0,1 + c1,0) ,ξ4/302√
3(C1ξ202 − 1
) cos (ζ0,1 + c1,0) , 0, 0
.
At the same boundary point, the tangent plane to S−C1,c1,1
is spanned by a vector tangent
to the boundary and the vector ξ4/302√
3(C1ξ202 − 1
) sin (ζ0,1 + c1,1) ,ξ4/302√
3(C1ξ202 − 1
) cos (ζ0,1 + c1,1) , 0, 0
.
As c1,1 ≡ (−2ζ0,1 − c1,0) (mod 2π), the two tangent planes coincide.
However, we must then check that we have a C3 smooth gluing.
Further, we consider C1 = C∗1 = 1, c∗1 = 0 and we start with + in the expression of
ν.
The construction of a complete biconservative surface in S3 can be summarized in
the diagram from Figure 3.11.
Some numerical experiments suggest that ΦC1,C∗1is not periodic and it has self-
intersections along circles parallel to Ox3x4.
The projection of ΦC1,C∗1on the Ox1x2 plane is a curve which lies in the annulus
of radii√1− 1/
(C1ξ201
)and
√1− 1/
(C1ξ202
). It has self-intersections and is dense in
the annulus. This is illustrated in Figure 3.12.
In Figure 3.13, we represent the signed curvature of the prole curve of SC1,C∗1, and
it suggests the fact that we have a smooth gluing.
The signed curvature of the curve obtained projecting ΦC1,C∗1on the Ox1x2 plane
is represented in Figure 3.14 and it suggests the fact that ΦC1,C∗1is at least of class C3.
However, we can state the following open problem.
Open problem. Does there exist a non-CMC biconservative immersion ΦC1,C∗1that
is double periodic, thus providing a compact non-CMC biconservative surface in S3?
Concerning the biharmonic surfaces in S3 we have the following classication result.
Theorem 3.50 ([13]). Let φ : M2 → S3 be a proper-biharmonic surface. Then φ(M)
is an open part of the small hypersphere S2(1/√2).
Figure 3.11: The idea of the construction of complete biconservative surfaces in S3.
106 Chapter 3. Complete biconservative surfaces in R3 and S3
Figure 3.12: The projection of ΦC1,C∗1on the Ox1x2 plane.
Figure 3.13: The signed curvature of the prole curve of SC1,C∗1.
Figure 3.14: The signed curvature of the curve obtained projecting ΦC1,C∗1on the Ox1x2
plane.
Chapter 4Biconservative surfaces in
arbitrary Riemannian manifolds
In this chapter we present general properties of biconservative surfaces in arbitrary
Riemannian manifolds. We nd the link between biconservativity, the property of the
shape operator AH to be a Codazzi tensor eld, the holomorphicity of a generalized Hopf
function and the quality of the surface to have constant mean curvature. Then, we give
a description of the metric and of the shape operator AH for a CMC biconservative
surface. Finally, we nd a Simons type formula for biconservative surfaces and then use
it to study their geometry.
The major part of the results from this chapter is original and it can be also found
in [55]. Some of them are known results, but obtained in a dierent way.
4.1 More characterizations of biconservative sub-
manifolds
In this section we characterize biconservative submanifolds satisfying some additional
geometric hypotheses.
We begin with a study on the basic properties of submanifolds with AH parallel,
as they are the simplest biconservative submanifolds. First, we dene the principal
curvatures of a submanifold Mm of Nn as being the eigenvalue functions of AH .
Proposition 4.1. Let φ :Mm → Nn be a submanifold and λ1 ≥ · · · ≥ λm the principal
curvatures of M . If ∇AH = 0, then
(i) M is biconservative;
(ii) λi are constant functions on M (in particular M is CMC);
107
108 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
(iii) A∇⊥XH
(Y )−A∇⊥YH
(X) =(RN (X,Y )H
)⊤, for any X,Y ∈ C(TM);
(iv) traceA∇⊥· H
(·) = − trace(RN (·,H)·
)⊤.
Proof. For the sake of completeness, we give the proof of the second item. More pre-
cisely, we show that λi are constant functions on M . Let us consider an arbitrary
point p ∈ M . Since AH(p) is symmetric, then AH(p) is diagonalizable. We denote by
λ1,p ≥ · · · ≥ λm,p the eigenvalues of AH(p) and then dene the continuous functions
λi :M → R, λi(p) = λi,p, for any p ∈M and any i = 1,m.
Further, we consider eii=1,m an orthonormal basis in TpM which diagonalizes
AH(p), i.e., (AH(p)) (ei) = λi(p)ei, for any i = 1,m.
Consider q ∈ M , q = p, and γ : [a, b] → M a smooth curve such that γ(a) = p and
γ(b) = q. We dene the vector elds Xi = Xi(t), along γ, such that DXidt (t) = 0, for
any t and Xi(a) = ei. It is easy to see that W (t) = (AH(γ(t))) (Xi(t)) is also a vector
eld along γ and
DW
dt(t) =
(∇γ′(t)AH
)(Xi(t)) +AH
(DXi
dt(t)
)= 0.
Now, since D(λi(p)Xi)dt (t) = 0, we get that W (t) and λi(p)Xi are parallel along γ. Since
for t = a they are equal, it follows that they coincide for any t, and in particular, for
t = b. Therefore, λi(p) are eigenvalues of AH(q). As q was chosen in an arbitrary way,
we get that λi are constant functions on M , for any i = 1,m.
We will nd, later in this chapter, some converse results of this proposition. More
precisely in the case when m = 2, we will show that, under some standard hypotheses,
a biconservative surface satises ∇AH = 0.
Corollary 4.2. Let φ : Mm → Nn(c) be a submanifold with ∇AH = 0 and c ∈ R.Then
(i) A∇⊥XH
(Y ) = A∇⊥YH
(X), for any X,Y ∈ C(TM);
(ii) traceA∇⊥· H
(·) = 0.
In the particular case of surfaces, we have a stronger result.
Proposition 4.3. Let φ : M2 → Nn be a surface. If ∇AH = 0, then M is pseudoum-
bilical or at.
Proof. Let λ1 ≥ λ2 be the principal curvatures ofM . Since ∇AH = 0, from Proposition
4.1, we have that λ1 and λ2 are constant functions on M .
4.1. More characterizations of biconservative submanifolds 109
If λ1 = λ2, obviously, M is pseudoumbilical.
If λ1 > λ2, around any point of M we can consider a local orthonormal frame eld
X1, X2 which diagonalizes AH , i.e, AH (Xi) = λiXi, for i ∈ 1, 2.Using the Ricci formula we get
R(X,Y )AH(Z) = AH(R(X,Y )Z),
for any X,Y, Z ∈ C(TM) and then
λiR (X1, X2)Xi =AH (R (X1, X2)Xi)
=AH (R(X1, X2, X1, Xi)X1 +R(X1, X2, X2, Xi)X2)
=λ1R(X1, X2, X1, Xi)X1 + λ2R(X1, X2, X2, Xi)X2,
for i ∈ 1, 2.For both i's, we obtain K = 0 on U , where K is the Gaussian curvature of M given
by K = R (X1, X2, X1, X2).
If (Mm, g) is a Riemannian manifold and T is a parallel symmetric tensor eld of
type (1, 1), then its eigenvalue functions λ1 ≥ · · · ≥ λm are constant functions on M
and, obviously, T is a Codazzi tensor eld. If m = 2, the converse result also holds.
Proposition 4.4. Let(M2, g
)be a surface and consider T a symmetric tensor eld of
type (1, 1). Let λ1 ≥ λ2 be the eigenvalue functions of T . If λ1 and λ2 are constant
functions on M and T is a Codazzi tensor eld, then ∇T = 0. Moreover, if λ1 > λ2,
then(M2, g
)is at.
Proof. If λ1 = λ2 = λ, it follows that T = λI and ∇T = 0.
If λ1 > λ2, around any point of M we can consider a local orthonormal frame eld
X1, X2 which diagonalizes T . By a direct computation, one obtains(∇XjT
)(Xi) + T
(∇XjXi
)= λi∇XjXi, (4.1)
for i, j ∈ 1, 2.Since T is a Codazzi tensor eld, for appropriate choices of i and j in (4.1), we get
∇XiXj = 0, for any i, j ∈ 1, 2. It follows that K vanishes at any point of M and
∇T = 0 on M .
When T = AH we have the next result.
Corollary 4.5. Let φ :M2 → Nn be a surface and λ1 ≥ λ2 be the principal curvatures
of M . If λ1 and λ2 are constant functions on M and AH is a Codazzi tensor eld, then
∇AH = 0.
110 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
Remark 4.6. Let φ :Mm → Nn be a submanifold with the principal curvatures being
constant functions. Then M is a CMC submanifold.
We note that, in general, AH is not a Codazzi tensor eld. In the following we study
the properties of submanifolds with AH a Codazzi tensor eld, and their connection with
biconservativity, as this is the next natural step after the case when AH is parallel.
We begin with a result which follows easily from the Codazzi equation, equation
(1.9) and Proposition 1.32.
Proposition 4.7. Let φ :Mm → Nn be a submanifold. If AH is a Codazzi tensor eld,
then
(i) A∇⊥XH
(Y )−A∇⊥YH
(X) =(RN (X,Y )H
)⊤, for any X,Y ∈ C(TM);
(ii) trace∇AH = m grad(|H|2
);
(iii) traceA∇⊥· H
(·) = m2 grad
(|H|2
)− trace
(RN (·,H)·
)⊤;
(iv) M is biconservative if and only |H| is constant.
If N is an n-dimensional space form, Proposition 4.7 gives the following corollary.
Corollary 4.8. Let φ : Mm → Nn(c) be a submanifold. If AH is a Codazzi tensor
eld, then
(i) A∇⊥XH
(Y )−A∇⊥YH
(X) = 0, for any X,Y ∈ C(TM);
(ii) trace∇AH = m grad(|H|2
);
(iii) traceA∇⊥· H
(·) = m2 grad
(|H|2
);
(iv) M is biconservative if and only if |H| is constant.
Next, we have a similar result to Proposition 4.7, where, rather than assuming AH
to be a Codazzi tensor eld, we assume that |H| = 0 at any point and Aη is a Codazzi
tensor eld, where η = H/|H| ∈ C(NM).
Proposition 4.9. Let φ : Mm → Nn be a submanifold. Assume |H| = 0 on M and
denote η = H/|H|. If Aη is a Codazzi tensor eld, then
(i) A∇⊥XH
(Y )−A∇⊥YH
(X) = X(log |H|)AH(Y )−Y (log |H|)AH(X)+(RN (X,Y )H
)⊤,
for any X,Y ∈ C(TM);
(ii) trace∇AH = AH(grad(log |H|)) +m|H| grad |H|;
(iii) traceA∇⊥· H
(·) = AH (grad(log |H|))− trace(RN (·,H)·
)⊤;
4.2. Properties of biconservative surfaces 111
(iv) M is biconservative if and only if
AH(grad |H|) = −m2|H|2 grad |H|.
Corollary 4.10. Let φ : Mm → Nn(c) be a submanifold. Assume that |H| = 0 on M
and denote η = H/|H|. If Aη is a Codazzi tensor eld, then
(i) A∇⊥XH
(Y )− A∇⊥YH
(X) = X(log |H|)AH(Y )− Y (log |H|)AH(X), for any X,Y ∈C(TM);
(ii) trace∇AH = AH(grad(log |H|)) +m|H| grad |H|;
(iii) traceA∇⊥· H
(·) = AH (grad(log |H|));
(iv) M is biconservative if and only if
AH(grad |H|) = −m2|H|2 grad |H|.
To end the section consider those submanifolds in space forms having H parallel
in the normal bundle (PMC submanifolds). Using the Codazzi equation, we get the
following result.
Proposition 4.11. Let φ :Mm → Nn(c) be a PMC submanifold. Then
(i) M is biconservative;
(ii) AH is a Codazzi tensor eld;
(iii) ⟨(∇AH) (·, ·), ·⟩ is totally symmetric;
(iv) trace∇AH = 0.
4.2 Properties of biconservative surfaces
In this section we nd some of the main properties of biconservative surfaces.
One of the main results is Theorem 4.19. Before stating it, we have a lemma
which holds for an arbitrary symmetric tensor eld T of type (1, 1), then present some
properties satised when div T = 0, and nally bring all these together in Theorem
4.18.
Lemma 4.12. Let(M2, g
)be a surface and let T be a symmetric tensor eld of type
(1, 1). We have
112 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
(i)
div T = grad t− ⟨Z12, X2⟩X1 + ⟨Z12, X1⟩X2,
where t = traceT , X1, X2 is a local orthonormal frame eld on M and
Z12 = (∇X1T ) (X2)− (∇X2T ) (X1) ;
(ii) ⟨T (∂z), ∂z⟩ is holomorphic if and only if grad t = 2div T ;
(iii) ⟨T (∂z), ∂z⟩ is holomorphic if and only if
grad t = 2⟨Z12, X2⟩X1 − 2⟨Z12, X1⟩X2.
Proof. Since the rst and the third statements of the theorem follow by standard com-
putation, we will only give the proof of the second item.
SinceM is oriented, the metric g can be written locally as g = e2ρ(du2 + dv2
), where
(u, v) are positively oriented local coordinates and ρ = ρ(u, v) is a smooth function. As
usual, we denote
∂z =1
2(∂u − i∂v) and ∂z =
1
2(∂u + i∂v) .
Therefore, ⟨T (∂z), ∂z⟩ is holomorphic if and only if ∂z ⟨T (∂z), ∂z⟩ = 0. We can see that
the Christoel symbols are given by
Γ212 = Γ1
11 = −Γ122 = ρu,
where ρu = ∂ρ∂u and
Γ112 = Γ2
22 = −Γ211 = ρv.
Thus, we obtain
∇∂u∂v =∇∂v∂u = Γ112∂u + Γ2
12∂v = ρv∂u + ρu∂v,
∇∂u∂u =Γ111∂u + Γ2
11∂v = ρu∂u − ρv∂v,
∇∂v∂v =Γ122∂u + Γ2
22∂v = −ρu∂u + ρv∂v.
After some straightforward computations, we get
∂z ⟨T (∂z), ∂z⟩ =e2ρ
8(−tu + 2⟨div T, ∂u⟩+ i (tv − 2⟨div T, ∂v⟩)) ,
where t = traceT . Now, it is easy to see that ⟨T (∂z), ∂z⟩ is holomorphic if and only if
grad t = 2div T .
4.2. Properties of biconservative surfaces 113
Remark 4.13. We note that ⟨Z12, X2⟩X1 − ⟨Z12, X1⟩X2 does not depend on the local
orthonormal frame eld X1, X2. Thus, there exists a unique global vector eld Z
such that for any local orthonormal frame eld X1, X2, on its domain of denition,
we have
Z = ⟨Z12, X2⟩X1 − ⟨Z12, X1⟩X2.
Therefore, we obtain the global formula
div T = grad t− Z.
Lemma 4.12 is the key ingredient to prove the following four propositions. First,
let(M2, g
)be a surface and consider T a symmetric tensor eld of type (1, 1) with
t = traceT .
Proposition 4.14. If t is constant, then the following relations are equivalent
(i) T is a Codazzi tensor eld;
(ii) ⟨T (∂z) , ∂z⟩ is holomorphic;
(iii) div T = 0.
Proposition 4.15. If div T = 0, then the following relations are equivalent
(i) T is a Codazzi tensor eld;
(ii) ⟨T (∂z) , ∂z⟩ is holomorphic;
(iii) t is constant.
Proposition 4.16. If ⟨T (∂z) , ∂z⟩ is holomorphic, then the following relations are equi-
valent
(i) T is a Codazzi tensor eld;
(ii) t is constant;
(iii) div T = 0.
Proposition 4.17. If T is a Codazzi tensor eld, then the following relations are equi-
valent
(i) t is constant;
(ii) ⟨T (∂z) , ∂z⟩ is holomorphic;
(iii) div T = 0.
114 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
Summarizing, we can now state the next theorem.
Theorem 4.18. Let(M2, g
)be a surface and consider T be a symmetric tensor eld
of type (1, 1). Then any two of the following relations imply each of the others
(i) div T = 0;
(ii) t is constant;
(iii) ⟨T (∂z) , ∂z⟩ is holomorphic;
(iv) T is a Codazzi tensor eld.
Considering T = S2 or T = AH , we get the following result.
Theorem 4.19. Let φ :M2 → Nn be a surface. Then any two of the following relations
imply each of the others
(i) M is biconservative;
(ii) |H| is constant;
(iii) ⟨AH (∂z) , ∂z⟩ is holomorphic;
(iv) AH is a Codazzi tensor eld.
Proof. For the sake of completeness, we present a sketch the proof.
First, we recall that S2 = −2|H|2I +4AH , traceS2 = 4|H|2, traceAH = 2|H|2, and
divS2 = −2 grad(|H|2
)+ 4divAH . (4.2)
It is then easy to see that
⟨S2 (∂z) , ∂z⟩ = 4⟨AH (∂z) , ∂z⟩,
and, therefore, ⟨AH (∂z) , ∂z⟩ is holomorphic if and only if ⟨S2 (∂z) , ∂z⟩ is holomorphic.
The idea of the proof is to choose a condition and then prove the equivalence between
each two other conditions using, in principal, Theorem 4.18 with T = AH or T = S2.
For example, assume that (i) holds. To prove that (ii) implies (iii) and (iv) we note
that since divS2 = 0 and |H| is constant, from (4.2), we get divAH = 0. Now, from
Theorem 4.18 with T = AH , we get (iii) and (iv). Conversely, from Proposition 4.7, we
have that hypotheses (i) and (iv) imply (ii). Since ⟨AH (∂z) , ∂z⟩ is holomorphic if and
only if ⟨S2 (∂z) , ∂z⟩ is holomorphic, from Theorem 4.18 with T = S2, it follows that (i)
and (iii) imply (ii). Using the above equivalences that we have already proved, it is
easy to see that (iii) is equivalent to (iv).
The other cases follow easily in a similar way.
4.2. Properties of biconservative surfaces 115
Remark 4.20. We note that some of the implications in Theorem 4.19 was also ob-
tained in [50], and the fact that ⟨AH (∂z) , ∂z⟩ is holomorphic if and only if |H| isconstant was also proved in [43] under the hypothesis of biharmonicity.
Remark 4.21. If φ :M2 → Nn is a non-pseudoumbilical CMC biconservative surface,
then the set of pseudoumbilical points has no accumulation points. Also, we deduce
that if M2 is a CMC biconservative surface and is a topological sphere, then it is
pseudoumbilical (see [50]); this should be compared with the classical result: a PMC
surface M2 of genius 0 in a space form is pseudoumbilical (see [35]).
Using Corollary 4.5, Remark 4.6 and Theorem 4.19, one obtains the next result.
Theorem 4.22. Let φ : M2 → Nn be a biconservative surface. We denote by λ1 and
λ2 the principal curvatures of M . If λ1 and λ2 are constant functions on M , then
∇AH = 0.
Remark 4.23. If we replace the hypothesis M is a biconservative surface in Theorem
4.22 by ⟨AH (∂z) , ∂z⟩ is a holomorphic function the conclusion still holds.
Since any CMC surface in a 3-dimensional space form is biconservative, using The-
orem 4.19, we easily get the following well-known properties of CMC surfaces.
Corollary 4.24. Let φ :M2 → N3(c), c ∈ R, be a CMC surface. Then
(i) M is biconservative;
(ii) AH is a Codazzi tensor eld;
(iii) ⟨AH (∂z) , ∂z⟩ is holomorphic.
For a PMC surface in an arbitrary manifold, AH is not necessarily a Codazzi tensor
eld, but when the surface is biconservative, this does happen.
Corollary 4.25. Let φ : M2 → Nn be a PMC biconservative surface. Then AH is a
Codazzi tensor eld and(RN (X,Y )H
)⊤= 0 for any X,Y ∈ C(TM).
Proof. First, we note that ∇⊥H = 0 implies |H| constant, and, if M is biconservative,
from Theorem 4.19, we have that AH is a Codazzi tensor eld.
Now, to show that(RN (X,Y )H
)⊤= 0 we only have to replace ∇⊥H = 0 in the
Codazzi equation and use the fact that AH is a Codazzi tensor eld.
The next theorem gives a description in terms of |H| and the principal curvatures
of M of the metric and of the shape operator in the direction of H for a CMC bicon-
servative surface in an arbitrary manifold.
116 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
Theorem 4.26. Let φ :M2 → Nn be a CMC biconservative surface. Denote by λ1 and
λ2 the principal curvatures of M and by µ = λ1−λ2 their dierence. Then, around any
non-pseudoumbilical point p there exists a local chart (U ;u, v) which is both isothermal
and a line of curvature coordinate system for AH . Moreover, on U , we have
⟨·, ·⟩ = 1
µ⟨·, ·⟩0,
and AH is given by
⟨AH(·), ·⟩ =(|H|2
µ+
1
2
)du2 +
(|H|2
µ− 1
2
)dv2,
or, equivalently, by
AH =
(|H|2
µ+
1
2
)du⊗ ∂u +
(|H|2
µ− 1
2
)dv ⊗ ∂v,
where ⟨·, ·⟩0 is the Euclidean metric on R2.
In the particular case when n = 3, we obtain that µ satises
µ0
∆ µ+
∣∣∣∣ 0
grad µ
∣∣∣∣20
+ 2µ
(KN + |H|2 − µ2
4|H|2
)= 0, (4.3)
where KN is the sectional curvature of N3 along M2.
Proof. Let λ1 and λ2 be the principal curvatures of M and p a non-pseudoumbilical
point in M . Then λ1, λ2 are continuous, and there exists an open neighborhood U
around p such that they are smooth functions on U , and µ = λ1 − λ2 is a positive
smooth function on U .
Let X1, X2 be a local orthonormal frame eld on U such that AH (Xi) = λiXi,
for any i ∈ 1, 2. Further, consider the connection forms on U , dened by
∇X1 = ω21X2 and ∇X2 = ω1
2X1.
Obviously, ω21 = −ω1
2. From Theorem 4.19 one obtains that AH is a Codazzi tensor
eld, i.e., on U , we have
(∇AH) (X1, X2) = (∇AH) (X2, X1) .
Using the denition of ωji , we get, on U ,
ω21 (X1) =
1
µX2 (λ1) and ω2
1 (X2) =1
µX1 (λ2) .
Now, since | traceB| = 2|H| is a constant, it is easy to see that X2 (λ1) = (X2(µ)) /2
and X1 (λ2) = − (X1(µ)) /2. If we denote byω1, ω2
the local orthornormal coframe
eld on U dual to X1, X2, one gets
ω21 =
1
2
((X2(logµ))ω
1 − (X1(logµ))ω2).
4.2. Properties of biconservative surfaces 117
After some straightforward computations, we obtain[X1õ,X2õ
]= 0,
and, therefore, on U there exist coordinate functions u and v such that ∂u = X1/√µ
and ∂v = X2/√µ. Moreover the expression of the metric in isothermal coordinates on
U is
⟨·, ·⟩ = 1
µ
(du2 + dv2
)=
1
µ⟨·, ·⟩0.
Since λ1 and λ2 are principal curvatures of M , it is easy to see that
⟨AH(·), ·⟩ =1
µ
(λ1du
2 + λ2dv2).
Using λ1 + λ2 = 2|H|2 and λ1 − λ2 = µ, we conclude that
λ1 = |H|2 + µ
2and λ2 = |H|2 − µ
2. (4.4)
In the n = 3 case, from the Gauss equation, it follows that
K = KN (X1, X2) + |H|2 − µ2
4|H|2, (4.5)
where KN is the sectional curvature of N along M .
From (1.6) with ρ = −(logµ)/2 we get
K = −µ2
0
∆ (logµ)
= − 1
2µ
∣∣∣∣ 0
grad µ
∣∣∣∣20
− 1
2
0
∆ µ,
where ⟨·, ·⟩0 is the Euclidean metric on R2,0
∆ and0
grad are the Laplacian and the
gradient, respectively, with respect to ⟨·, ·⟩0.Therefore, replacing K in (4.5), we have that µ is a solution of (4.3).
Remark 4.27. Biconservative surfaces in BianchiCartanVranceanu spaces, which are
3-dimensional spaces with non-constant sectional curvature, were studied in [51].
Remark 4.28. If N is a 3-dimensional space form, the same result holds without the
hypothesis of biconservativity, as a CMC surface is automatically biconservative.
We note that, since K = −(∆(logµ))/2, the next result is obvious.
Corollary 4.29. Let φ :M2 → Nn be a CMC biconservative surface. Assume that M
is compact and does not have pseudoumbilical points. Then M is a topological torus.
118 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
Using Theorem 4.19 and Corollary 4.29 we obtain the following corollary.
Corollary 4.30. Let φ : M2 → Nn be a CMC biconservative surface. Assume that
M is compact and does not have pseudoumbilical points. If K ≥ 0 or K ≤ 0, then
∇AH = 0 and K = 0.
We end this section with two results which basically say that a CMC biconservative
surface in Nn can be also immersed in N3(c) having either AH or S2, as shape operator.
Theorem 4.31. Let φ :M2 → Nn be a biconservative surface. We denote by λ1 and λ2
the principal curvatures of M corresponding to φ. Assume that λ1 and λ2 are constants
and λ1 > λ2. We have:
(i) there exists locally ψ : M2 → N3(c) an isoparametric surface such that AφHφ is
the shape operator of ψ in the direction of the unit normal vector eld, where
c =µ2
4− |Hφ|4 ;
moreover,∣∣Hψ
∣∣ = |Hφ|2.
(ii) there exists locally ψ : M2 → N3(c) an isoparametric surface such that Sφ2 is the
shape operator of ψ in the direction of the unit normal vector eld, where
c = 4(µ2 − |Hφ|4
);
moreover,∣∣Hψ
∣∣ = 2 |Hφ|2.
Proof. First, we consider a symmetric tensor eld Aψ of type (1, 1) on M given by
Aψ(X) = AφHφ(X), X ∈ C(TM).
As AφHφ is a Codazzi tensor eld, Aψ satises (formally), the Codazzi equation for a
surface in a 3-dimensional space form.
Since the principal curvatures of M corresponding to φ, λ1 and λ2, are constants
and λ1 > λ2, from Proposition 4.3 it follows that K = 0. Now (formally) from the
Gauss equation for a surface in a 3-dimensional space form N3(c), and from (4.4), we
obtain
c =− detAψ
=− detAφHφ =µ2
4− |Hφ|4 .
Therefore, there exists locally an immersion ψ : M2 → N3(c) such that its shape
operator in the direction of the unit normal vector eld is Aψ. Moreover, the surface is
isoparametric as λ1 and λ2 are constants.
4.3. A Simons type formula for S2 119
We have |τ(ψ)| = 2∣∣Hψ
∣∣ = traceAψ and in the same time
traceAφHφ = λ1 + λ2 = 2 |Hφ|2 = |τ(φ)|2
2.
From the denition of Aψ one easily sees∣∣Hψ
∣∣ = |Hφ|2.Secondly, we dene the shape operator associated to the surface ψ : M2 → N3(c)
as
Aψ(X) = Sφ2 (X), X ∈ C(TM),
where c ∈ R.Since Sφ2 = −2 |Hφ|2 I + 4AφHφ , using the same argument as in the previous case,
one obtains
c =− detAψ
=− detSφ2 = 4(µ2 − |Hφ|4
)and |τ(ψ)| = |τ(φ)|2, i.e., ∣∣∣Hψ
∣∣∣ = 2 |Hφ|2 .
Theorem 4.32. Let φ : M2 → Nn be a biconservative surface. Denote by λ1 and λ2
the principal curvatures of M corresponding to φ. Assume that λ1 and λ2 are constants
and λ1 = λ2. If K = 0, then we have:
(i) there exists locally an umbilical surface ψ : M2 → N3(c) such that AφHφ is the
shape operator of ψ in the direction of the unit normal vector eld, where
c = − |Hφ|4 ;
moreover,∣∣Hψ
∣∣ = |Hφ|2.
(ii) there exists locally an umbilical surface ψ :M2 → N3(c) such that Sφ2 is the shape
operator of ψ in the direction of the unit normal vector eld, where
c = −4 |Hφ|4 ;
moreover,∣∣Hψ
∣∣ = 2 |Hφ|2.
4.3 A Simons type formula for S2
As we have already mentioned we present here some converse results of Proposition
4.1 (see Theorem 4.22, Theorem 4.39 for the compact case, and Theorem 4.41 for the
complete non-compact case).
120 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
As in the previous section, we will rst compute the rough Laplacian ∆RT for an
arbitrary symmetric tensor eld T of type (1, 1) on M with div T = 0, and then ∆RS2.
Proposition 4.33. Let(M2, g
)be a surface and T a symmetric tensor eld of type
(1, 1). Assume that div T = 0. Then
trace(∇2T
)= 2KT − tKI − (∆t)I −∇ grad t. (4.6)
Proof. Let p ∈M be an arbitrary point and X1, X2 a local orthornormal frame eld,
geodesic around p. Clearly, at p we have
(trace
(∇2T
))(Xj) =
2∑i=1
(∇2T
)(Xi, Xi, Xj) .
The right hand term can be rewritten as
2∑i=1
(∇2T
)(Xi, Xi, Xj) =
2∑i=1
((∇2T
)(Xi, Xi, Xj)−
(∇2T
)(Xi, Xj , Xi)
)+
2∑i=1
(∇2T
)(Xi, Xj , Xi) .
After some straightforward computations, at p, one obtains
2∑i=1
(∇2T
)(Xi, Xi, Xj) =
2∑i=1
(Xi⟨div T,Xj⟩Xi −Xi⟨div T,Xi⟩Xj − (Xi (Xjt))Xi +
+(Xi (Xit))Xj +(∇2T
)(Xi, Xj , Xi)
),
where t = traceT .
Further, applying the Ricci formula, since div T = 0 and
2∑i=1
(∇2T
)(Xj , Xi, Xi) = 0
at p, it follows that,(trace
(∇2T
))(Xj) = (2KT − (∆t) I −∇ grad t−KtI) (Xj) .
Therefore, at p, one obtains
trace(∇2T
)= 2KT − tKI − (∆t)I −∇ grad t.
Since p was arbitrary chosen, we get that the above result holds on M .
Using (4.6), we can compute the Laplacian of the squared norm of S2 and obtain
a Simons type formula (here, instead of the second fundamental form we have the
stress-bienergy tensor).
4.3. A Simons type formula for S2 121
Proposition 4.34. Let φ :M2 → Nn be a biconservative surface. Then,
12∆ |S2|2 = −2K |S2|2 + div
((⟨S2, grad
(|τ(φ)|2
)⟩)♯)
+K|τ(φ)|4
+12∆(|τ(φ)|4
)+∣∣grad (|τ(φ)|2)∣∣2 − |∇S2|2
. (4.7)
Proof. First, using the fact that traceS2 = |τ(φ)|2 and applying (4.6) with T = S2 one
obtains
∆RS2 = −2KS2 +∇ grad(|τ(φ)|2
)+(K|τ(φ)|2 +∆
(|τ(φ)|2
))I, (4.8)
where ∆RS2 = − trace(∇2S2
).
Then, from (1.2) and since M is biconservative, one gets
div(S2(grad
(|τ(φ)|2
)))= ⟨S2,Hess
(|τ(φ)|2
)⟩. (4.9)
From (1.1), with T = S = S2, it is easy to see that
1
2∆ |S2|2 =
⟨∆RS2, S2
⟩− |∇S2|2 . (4.10)
Further, since ⟨I, S2⟩ = traceS2 = |τ(φ)|2 and
∆(|τ(φ)|2
)|τ(φ)|2 = 1
2∆(|τ(φ)|4
)+∣∣grad (|τ(φ)|2)∣∣2 ,
from (4.8), (4.9), and (4.10), it follows (4.7).
Remark 4.35. If φ :M2 → Nn is a CMC biconservative surface, then S2 is a Codazzi
tensor eld and (4.8) follows from a well-known formula in [21].
Remark 4.36. Equation (4.8) was obtained, in a dierent way, in [43] but for bihar-
monic maps (a stronger hypothesis) from surfaces.
Integrating (4.7) we get the following integral formula.
Proposition 4.37. Let φ :M2 → Nn be a compact biconservative surface. Then∫M
(|∇S2|2 + 2K
(|S2|2 −
|τ(φ)|4
2
))vg =
∫M
∣∣grad (|τ(φ)|2)∣∣2 vg, (4.11)
or, equivalently,∫M
(|∇AH |2 + 2K
(|AH |2 − 2|H|4
))vg =
5
2
∫M
∣∣grad (|H|2)∣∣2 vg.
Proof. Since M is compact, by integrating (4.7) it is easy to get (4.11). To obtain the
second equation, i.e., an equivalent expression to (4.11), in terms of AH and |H|, werst recall that
|S2|2 = 16 |AH |2 − 24|H|4 (4.12)
122 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
and
∇XS2 = −2(X(|H|2
))I + 4∇XAH ,
for any X ∈ C(TM).
Then, by some standard computations, we obtain
|∇S2|2 = 16 |∇AH |2 − 24∣∣grad (|H|2
)∣∣2 .Finally, we can rewrite (4.11) as∫
M
(16 |∇AH |2 − 24
∣∣grad (|H|2)∣∣2+ (2K (16 |AH |2 − 24|H|4 − 8|H|4
))vg =
= 16
∫M
∣∣grad (|H|2)∣∣2 vg
and by a straightforward computation we get∫M
(|∇AH |2 + 2K
(|AH |2 − 2|H|4
))vg =
5
2
∫M
∣∣grad (|H|2)∣∣2 vg. (4.13)
Remark 4.38. It is easy to see that 2 |S2|2−|τ(φ)|4 = 32(|AH |2 − 2|H|4
)≥ 0, and the
equality occurs if and only if S2 =(|τ(φ)|2
/2)I, or, equivalently, M is pseudoumbilical.
From (4.13) we easily get the following result.
Theorem 4.39. Let φ : M2 → Nn be a compact CMC biconservative surface. If
K ≥ 0, then ∇AH = 0 and M is at or pseudoumbilical.
Proof. Since |H| is constant, from (4.13), one obtains
|∇AH |2 + 2K(|AH |2 − 2|H|4
)= 0.
Therefore ∇AH = 0 and K(|AH |2 − 2|H|4
)= 0. From the last equality, it follows that
K = 0, i.e., M is at, or |AH |2 − 2|H|4 = 0, i.e., M is pseudoumbilical.
Remark 4.40. Theorem 4.39 was obtained in [43] under the hypothesis of biharmon-
icity.
In the following, we study complete non-compact biconservative surfaces.
Theorem 4.41. Let φ : M2 → Nn be a complete non-compact CMC biconservative
surface with K ≥ 0. IfN
Riem ≤ k0, where k0 is a non-negative constant, then ∇AH = 0.
4.3. A Simons type formula for S2 123
Proof. As |H| is constant, τ(φ) = 2H and |S2|2 = 16 |AH |2 − 24|H|4, from (4.7) we get
− 1
2∆ |S2|2 = 32K
(|AH |2 − 2|H|4
)+ |∇S2|2 . (4.14)
Since K and |AH |2 − 2|H|4 are always non-negative (see the hypothesis and Remark
4.38, respectively), we get that ∆ |S2|2 ≤ 0, i.e., |S2|2 is a subharmonic function.
Next, we show that |S2|2 is bounded from above. From (4.12) it is easy to see that
|S2|2 is bounded from above if and only if |AH |2 is bounded from above.
Let us consider X1, X2 a local orthonormal frame eld onM and η, η1, · · · , ηn−3a local orthonormal coframe eld on M such that H = |H|η. From the Gauss equation
we have
K−N
Riem (X1, X2) =⟨B (X1, X1) , B (X2, X2)⟩ − |B (X1, X2)|2
=⟨Aη (X1) , X1⟩⟨Aη (X2) , X2⟩ − (⟨Aη (X1) , X2⟩)2
+
n−3∑α=1
(⟨Aηα (X1) , X1⟩⟨Aηα (X2) , X2⟩ − (⟨Aηα (X1) , X2⟩)2
)=detAη +
n−3∑α=1
detAηα ,
whereN
Riem (X1, X2) = RN (X1, X2, X1, X2) .
It is clear that ⟨H, ηα⟩ = 0, and then traceAηα = 2⟨H, ηα⟩ = 0, for any α ∈ 1, 2, · · ·n− 3.As Aηα is symmetric, we note that detAηα ≤ 0 for any α and
∑n−3α=1 detAηα ≤ 0. Then,
we get
K−N
Riem (X1, X2) ≤ detAη. (4.15)
Let us consider µ1 and µ2 the principal curvatures of Aη. Then
detAη =µ1µ2 =(µ1 + µ2)
2 −(µ21 + µ22
)2
=(traceAη)
2 − |Aη|2
2
=4|H|2 − |Aη|2
2.
From (4.15) one obtains
|Aη|2 ≤ 4|H|2 − 2K + 2N
Riem (X1, X2) .
Since K ≥ 0 andN
Riem ≤ k0, it follows that
|Aη|2 ≤ 4|H|2 + 2k0.
124 Chapter 4. Biconservative surfaces in arbitrary Riemannian manifolds
Therefore, we have just proved that |AH |2 is bounded from above.
It is well known that a complete surface with K ≥ 0 is parabolic ([36]), i.e., any
subharmonic function bounded from above is constant. Thus, as |S2|2 is bounded from
above and subharmonic, it follows that |S2|2 is a constant. Using |∇S2|2 = 16 |∇AH |2
and (4.14) one obtains that ∇AH = 0, and thereforeM is either at or pseudoumbilical.
4.3.1 Exemples of submanifolds with ∇AH = 0
As we have seen, a PMC surface in a space formNn(c), n ≥ 4, is trivially biconservative.
But, if the surface is only CMC, then it is not necessarily biconservative. In [50] it
was proved that if a surface is biconservative and CMC in N4(c), with c = 0, then the
surface has to be PMC, i.e., the trivial case for our problem. We just recall here that,
if c = 1, then a PMC surface in S4 is either a minimal surface of a small hypersphere
of radius a, a ∈ (0, 1), in S4, or a CMC surface in a small or great hypersphere in S4
(see [74,75]). Of course, if we consider a CMC biconservative surface M2 of genus 0 in
R4, it is pseudoumbilical and therefore it is PMC, i.e., M2 is a 2-sphere (see [35]). In
R4, there were obtained all CMC biconservative surfaces which are not PMC. They
are given by the isometric immersion φ : R2 → R4 dened by
φ(u, v) = γ(u) + (v + a)e4,
where γ : R → R3 is a smooth curve parametrized by arc-length with positive constant
curvature κ and torsion τ = 0. By a straightforward computation we obtain that the
second fundamental form of the surface is given by
B (∂u, ∂u) = κ(u)N(u), B (∂u, ∂v) = 0, B (∂v, ∂v) = 0,
where T (u), N(u), B(u) is the Frenet frame eld of the curve γ. Then, one obtains
the expression of the mean curvature vector eld
H(u, v) =κ
2N(u),
the shape operator in the direction of H
AH (∂u) =κ2
2∂u, AH (∂v) = 0
and
∇⊥∂uH =
κ
2τ(u)N(u), ∇⊥
∂vH = 0.
It is easy to see that φ is a biconservative immersion, i.e., it satises
grad(|H|2
)+ 2 traceA∇⊥
· H(·) + 2 trace
(RR4
(·,H)·)⊤
= 0.
Therefore, φ satises all hypotheses of Theorem 4.41 which implies that AH is parallel,
a fact which can be also checked by a direct computation.
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