core.ac.uk · Declaration I hereby declare that this thesis was composed in its entirety by myself and that the work contained therein is my own, except where explicitly stated otherwise

ON THE LOCALLY CONFORMALLY FLAT

HYPER-SURFACES WITH NON-NEGATIVE SCALAR

CURVATURE IN R5

ZHOU JIURU

NATIONAL UNIVERSITY OF SINGAPORE

2013

ON THE LOCALLY CONFORMALLY FLAT

HYPER-SURFACES WITH NON-NEGATIVE SCALAR

CURVATURE IN R5

ZHOU JIURU

(M. Sc., Nanjing University)

A THESIS SUBMITTED

FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2013

Copyright c© 2013 by Zhou Jiuru.

All rights reserved.

Declaration

I hereby declare that this thesis was composed in its entirety by myself and that thework contained therein is my own, except where explicitly stated otherwise in the text.

Besides, I understand that I have duly acknowledged all the sources of informationwhich have been used in the thesis.

Finally, this thesis has also not been submitted for any degree in any universitypreviously.

Zhou Jiuru

ZHOU JIURU

Acknowledgements

It is my honor to give my sincerest gratitude to my supervisor, Professor Xu Xing-wang, who is always kind and full of humor. In the past few years, he has taught me theprofessional knowledge and shown me the virtue and principle of a researcher. His var-ious knowledge and charming personality will benefit me for my whole life. He alwaysshares his own ideas and experience in research without reservation, and is patient toexplain any question, easy or tough. I also appreciate Prof. Xu for his unselfish help ofthis thesis, and without his guidance, it would have not been finished.

During the time when I am studying in Department of Mathematics, National Uni-versity of Singapore, I have learnt a lot, and here I would like to thank Dr. Han Fei forsharing his own research experience and many interesting mathematical stories. I feeldeeply grateful to my friends and classmates, especially Ngo Quoc Anh, Zhang Hong,Cai Ruilun, Ye Shengkui. I would also like to thank the University and the departmentfor support.

Special thanks should also be given to my master degree’s supervisor, Dr. Mei Ji-aqiang for building up my mathematical background. I thank Prof. Wang Hongyu forhis help at the beginning of my postgraduate life and his support during the year whenI stayed in Yangzhou University. I also thank my friends Xu Haifeng, Zhu Peng, GuPeng for their help.

Finally, for all the people who have ever cared me, helped me, I would like to offermy undying gratitude.

Dedication

To my parents and my wife...

Contents

Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Studies on open surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Studies on locally conformally flat open four dimensional manifolds . . . . . 2

1.3 Results of this paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Backgrounds and Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1 Riemann connection, Curvatures and second fundamental form . . . . . . . . 7

2.2 Qn curvature and Qn curvature equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Chern-Gauss-Bonnet formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 High Dimensional Chern-Gauss-Bonnet formula . . . . . . . . . . . . . . . . . . . 15

3.1 Chern-Gauss-Bonnet formula for Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Geometric conditions for metric to be normal . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Chern-Gauss-Bonnet formula in Local version . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Chern-Gauss-Bonnet formula for conformally flat manifolds . . . . . . . . . . . . 33

4 Controlling the number of ends by the mean curvature . . . . . . . . . . . . 37

5 Mean curvature and embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 Decomposition of the conformal factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Contents Contents

5.2 Immersion and Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

6 Conclusions and further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

viii

Summary

This thesis studies the geometry and topology of manifolds from an extrinsic pointof view. Suppose M4 is a complete noncompact locally conformally flat hyper-surfacewith nonnegative scalar curvature immersed in R5. Given some conditions on the secondfundamental form and the mean curvature, we should show that if the L4 norm of themean curvature of M , i.e. (

∫M |H|

4 dvM )14 is bounded by some constant which does

not depend on the manifold M , then M is embedded in R5. This result should be ageneralization of S. Muller and V. Sverak’s result on two dimensional manifolds whichimmersed in Rn.

List of Notations and Conventions

∇ Riemann connectionRm Riemann curvature tensorRic Ricci curvature tensorR Scalar curvature

Br(x) A ball centered at point x with radius rB Unit ball centered at the origin of Euclidean spaceωn Volume of the unit sphere Sn in the Euclidean space (Rn+1, |dx|2)

−∫B f dx Average integral of f given by −

∫B f dx = 1

|B|∫B f dx

1

Introduction

A central problem in global differential geometry is connections between the geom-etry and the topology of a manifold. One of the most important results of such style isthe Chern-Gauss-Bonnet formula for closed Riemannian manifolds (Chern, 1944, [8]).Actually, before Chern’s work, some mathematicians thought that differential geometrywas a dead end. Although there were plenty of satisfying results in classical differentialgeometry, all were based on local analysis, which blocked the depth development ofdifferential geometry. Afterwards, it is S.S. Chern who brought a new life to differentialgeometry and had used analysis, topology, algebra to study differential geometry froma global point of view, and also established global differential geometry.

For modern differential geometry, a powerful branch is geometric analysis leadingby S.T. Yau. It mainly uses differential equations to study differential geometry, andhas created excellent work for many mathematical problems such as Yang-Mills fields,Calabi-Yau manifolds, Ricci flow. The Poincare conjecture, which is one of the sevenMillennium Prize Problems, was solved in 2002 by using Ricci flow. Therefore, moreattention should be paid to consider using differential equations in studying differentialgeometry.

In the rest of the chapter, we will introduce a problem in differential geometry, andalso a brief review of several results done by various mathematicians, which providesthe background and motivation of this thesis.

1.1 Studies on open surfaces

As mentioned previously, the Chern-Gauss-Bonnet formula is a very important resultin differential geometry. Actually, after Chern’s work, many mathematicians tried togeneralize this formula to open manifolds, and studied the total curvature to openmanifolds.

For an open surfaceM , the total Gaussian curvature is bounded by the Euler numberof the surface up to a multiplication of a constant, as long as the Gaussian curvatureis absolutely integrable, i.e. ∫

MK dvM ≤ 2πχ(M), (1.1)

and such an open surface can be conformally compactified by attaching finite points(Huber, 1957, [17]). Furthermore, the deficit between the Euler number of the surface

2 1 Introduction

and the total Gaussian curvature is just the total isoperimetric number (Finn, 1965,[13]), i.e.,

χ(M)− 1

2π

∫MK dvM =

l∑i

µi, (1.2)

where l is the number of ends of M , and µi is the isoperimetric ratio of each end.Therefore, the Chern-Gauss-Bonnet formula is excellently generalized on open surfaces.

However, these are all intrinsic properties. For extrinsic properties, suppose an opensurface M with finite number of ends is immersed into an n dimensional Euclideanspace Rn, we consider the total integral of the second fundamental form.

Suppose ∫M|A|2 dvM < +∞,

then the Chern-Gauss-Bonnet formula (1.2) also holds, and in this case, the totalisoperimetric number is equal to the total number of ends of the open surface (countedwith multiplicity) (Muller and Sverak, 1995, [25]), i.e.,

χ(M)− 1

2π

∫MK dvM =

l∑i

mi,

where l is still the number of ends and mi is the multiplicity of each end. Furthermore,if∫M |A|

2 dvM < 8π for n = 3, or∫M |A|

2 dvM ≤ 4π for n ≥ 4, then M is embed-ded (Muller and Sverak, 1995, [25]). These conclusions indicate that some geometricassumptions can deduce topologic results, but they are just on surfaces, i.e. two dimen-sional manifolds. Since we have no idea if they also hold on general cases, our researchshould be concentrated on general manifolds.

1.2 Studies on locally conformally flat open four dimensionalmanifolds

For four dimensional manifolds, Huber’s result on the upper bound of the totalGaussian curvature can be generalized to complete four dimensional manifolds of pos-itive sectional curvature outside a compact set (R. Greene and H. Wu, 1976, [16]), butthe deficit between the Euler number and the total Gauss curvature of the manifoldsis hard to deduce. This gap is filled only after Branson creating Qn curvature.

As we all know, isothermal coordinates always exist on a surface, which means asurface is always locally conformally flat. Therefore, it is very natural to study the lo-cally conformally flat manifolds. Actually, for a locally conformally flat four dimensionalclose manifold M , the total Gaussian curvature is equal to the total Q4 curvature up toa multiplication of a constant. For locally conformally flat open four dimensional mani-folds with finitely many simple ends whose scalar curvature is nonnegative at each endand the Q4 curvature is integrable, we can use the Q4 curvature equation to determinethe deficit between the Euler number of M and the total Q4 curvature (Chang, Qingand Yang, 2000, [7]). With appropriate conditions on scalar curvature, Ricci curvaturetensor and Q4 curvature, they obtain that such manifold can be compactified by ad-joining a finite number of points (Chang, Qing and Yang, 2000, [7]). This work providesa method to study the locally conformally flat manifolds. However, for general openmanifolds, we still do not have enough knowledge about the total Gaussian curvature.

1.3 Results of this paper 3

1.3 Results of this paper

Based on the analysis of previous work, applications of the Chern-Gauss-Bonnetformula need to be studied for locally conformally flat four dimensional manifolds.Therefore, in this thesis, we try to generalize Muller and Sverak’s work ([25]) on surfacesto locally conformally flat four dimensional manifolds. In [25], they showed the followingresults,

Theorem 1.1 Let M → Rn be a complete, connected, non-compact surface immersedinto Rn. Assume that either ∫

M|A|2 dvM < 8π, n = 3,

or ∫M|A|2 dvM ≤ 4π, n ≥ 4.

Then M is embedded.

In this thesis, we study the topology and geometry of locally conformally flat openfour dimensional manifolds immersed into the Euclidean space R5. More specifically,we will show the following theorem,

Theorem 1 Let M → R5 be a complete, simply connected, noncompact, locally con-formally flat hyper-surface immersed into R5 with 16H2 − |A|2 to be non-negative and∆(16H2 − |A|2) ∈ H1(M). Assume that

(

∫M|H|4dv)

14 < C2, (1.3)

where

C2 =1

13C1.

Then M is embedded.

For the constant C1 in Theorem 1, please refer to Chapter 4.

This theorem shows that some weak conditions on the mean curvature and sec-ond fundamental form, which are extrinsic quantities, can control the topology of themanifold.

First let us give some standard notations. Let M be a complete, connected, non-compact, oriented, locally conformally flat four dimensional manifold immersed intoR5, i.e. M is a locally conformally flat hyper-surface of R5. We denote the secondfundamental form of M by A, and Q4 curvature of M by Q4. Choose a conformalparametrization

f : Ω∗ → Σ ⊂M → R5,

whereΩ∗ = x ∈ R4, |x| > 1,

and Σ is a neighborhood of an end of M . Under this conformal parametrization, wedenote the conformal metric to be g = e2ug0, where g0 is the standard flat metric.

With the conditions in Theorem 1, we obtain,

4 1 Introduction

Theorem 2 Suppose H4 and ∆(16H2 − |A|2) ∈ H1(M), R = 16H2 − |A|2 ≥ 0. Thenin local coordinates (Ω∗, e2ug0), the metric e2ug0 is normal. Furthermore, the conformalfactor has the following decomposition,

u(x) = u0(x) + α log |x|+ h(x),

wherelimx→∞

u0(x) = 0,

and h is a biharmonic function on Ω∗ ∪ ∞.

For the definition of normal metric, please refer to Chapter 5. By the decompositionof the conformal factor, we can prove that the parametrization function f behaves likexm in some sense.

Theorem 3 Choose a conformal parametrization

f : Ω∗ → Σ ⊂M → R5.

Then we have the following limit,

limx→∞

|f(x)||x|α+1

=eλ

α+ 1,

whereλ = lim

x→∞h(x).

Finally, we can prove Theorem 1.

As shown in Theorem 2, given weak assumptions on the mean curvature and thesecond fundamental form, we have a nice decomposition of the conformal factor u. Theresults show that the conformal factor u behaves like a log function at infinity. Hence itbecomes possible to study the conformal factor. Furthermore, with this decomposition,the deficit between the Euler number of M and the total Q4 curvature is equal to thetotal number of ends of M . This is similar to the result in [25], where they claimed inCor 4.2.5 that if a surface is immersed into a Euclidean space Rn with

∫M |A|

2 dv <∞,the deficit between the Euler number and the total Gaussian curvature is the totalnumber of ends of the given surface. This decomposition is also used to study theconformal parametrization function f , which can be seen in Theorem 3.

Theorem 1 is the key result of this thesis, which says that under suitable conditionson the mean curvature and the second fundamental form, the immersion of the manifoldbecomes an embedding. The main step to obtain this result is to use conditions onthe total integral of the mean curvature to find the decomposition of the conformalfactor. After that, we find the behavior of the conformal parametrization function, i.e.the conformal parametrization function behaves like xm. Then we use the integrabilitycondition on the mean curvature to control the number of ends of the manifold. Finally,the immersion is actually an embedding.

We work on this result, because it shows some connections between the topologyand the geometry of a manifold, which is one of the central problems in differentialgeometry.

The following section shows the structure of this thesis.

1.3 Results of this paper 5

Chapter 2 provides some necessary material and conventions in differential geometry.Chapter 3 discusses the high dimensional Chern-Gauss-Bonnet theorem which is dueto X. W. Xu [33]. Chapter 4 shows the way to use the mean curvature to control thenumber of ends of a manifold with nonnegative scalar curvature. Chapter 5 presentsthe main result of this thesis, i.e. using the mean curvature to control the topology oflocally conformally flat four dimensional hyper-surfaces.

2

Backgrounds and Preparation

In this chapter, we are going to introduce necessary materials and clarify conventionswhich we need for the rest of the thesis. Most of these can be found in standardtextbooks on differential geometry or Riemannian geometry.

2.1 Riemann connection, Curvatures and second fundamental form

Following [9], given a complete Riemannian manifold (M, g,∇) with Riemann metricg and the induced Riemann connection ∇, the Riemann curvature tensor is defined bythe following formulas,

R(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z, (2.1)

Rm(X,Y, Z,W ) = 〈R(X,Y )Z,W 〉. (2.2)

In local coordinates, (2.1) and (2.2) is equivalent to the following equations,

R(∂i, ∂j)∂k = Rlijk∂l,

Rijkl = 〈R(∂i, ∂j)∂k, ∂l〉 = gmlRmijk. (2.3)

According to (2.3), by taking the trace of Riemann tensor, we get Ricci curvaturetensor Ric. Furthermore, if we take the trace of Ricci curvature tensor, we obtain scalarcurvature R. Hence, in local coordinates,

Ricij = gkmRkijm,

R = gijRicij .

Now we use moving frames to deal with the Riemann connection and curvaturetensors. Suppose (M, g,∇) is a Riemannian manifold, ei is a local frame on someopen subset U ⊂M , and let ωi be the dual coframe. Then there exist one forms ωji such that

∇ei = ωji ej . (2.4)

Then

Ωji = dωji − ω

ki ∧ ω

jk =

1

2Rjiklω

k ∧ ωl (2.5)

is the curvature form.

8 2 Backgrounds and Preparation

By lowering down the index, i.e.,

Ωij = gkjΩki ,

we get

Ωij =1

2Rijklω

k ∧ ωl. (2.6)

Next, let us consider differential geometry from an extrinsic point of view.

Suppose Mn is a differentiable manifold. Mn is immersed into the Euclidean space(Rn+1, |dx|2), i.e. Mn is a hyper-surface of Rn+1. We will denote by ∇ the Riemannianconnection on Rn+1 induced by the canonical metric |dx|2, and denote by ∇ the inducedRiemann connection on Mn.

Suppose X,Y are tangent vector fields on M , the second fundamental form A isdefined to be the normal projection of the Riemann connection,

A(X,Y ) = (∇XY )⊥.

For any point p ∈ M and the unit normal vector field N on M , the second fun-damental form A determines a self-adjoint linear map AN : TpM → TpM , i.e. theWeingarten operator defined as follows:

〈AN (X), Y 〉 = 〈A(X,Y ), N〉.

For any point p ∈M , we denote by λ1, λ2, . . . , λn the eigenvalues of the symmetrictransformation AN of the tangent space TpM . These eigenvalues λ1, λ2, . . . , λn arecalled the principal curvatures at p. The Gaussian curvature K is defined to be thedeterminant of the second fundamental form,

Kn = λ1 · λ2 · · ·λn−1 · λn. (2.7)

For the hyper-surface M → Rn+1, suppose Sn ⊂ Rn+1 is the unit sphere centeredat the original point of Rn+1, the Gauss map

G : M → Sn ⊂ Rn+1

maps every point of the hyper-surface to the normal vector at that point,

∀ p ∈M, G(p) = N(p),

where N(p) is the unit normal vector at p. With this definition, we have the followingWeingarten equation,

〈dGp(X), Y 〉 = −〈G(p), A(X,Y )〉,

for tangent vector fields X,Y of M .

Hence, by taking an orthonormal frame e1, e2, · · · , en of the tangent bundle of M ,we have the norm

|dG| = |A|.

The following equation, i.e. Gauss equation, describes the difference between theintrinsic and extrinsic connections,

Rm(X,Y, Z,W ) = 〈A(X,W ), A(Y, Z)〉 − 〈A(X,Z), A(Y,W )〉.

2.2 Qn curvature and Qn curvature equation 9

The scalar second fundamental form h is the symmetric 2-tensor on M defined by

h(X,Y ) = 〈A(X,Y ), N〉.

In local coordinates, we have the local form of the Gauss equation,

Rijkl = hilhjk − hikhjl. (2.8)

Hence, after contracting (2.8) once and twice respectively, the Ricci curvature tensorand scalar curvature can be represented in terms of the second fundamental form,

Rjk = nH · hjk − hlkhjl,R = n2H2 − |A|2,

where

H =1

n

n∑i=1

hii =1

n

n∑i=1

λi

is defined to be the mean curvature.

2.2 Qn curvature and Qn curvature equation

Next, let us consider the high order curvature which is a generalization of the scalarcurvature in two dimensional case, i.e. the Qn curvature.

Suppose (Mn, g0) is a complete Riemannian manifold. For conformal geometry, con-sider the conformal metric g = e2ug0 on M ; then an operator A on the manifold M issaid to be conformally covariant of bidegree (a, b) if

Ag(φ) = e−buAg0(eauφ), (2.9)

for φ ∈ C∞(M).

The simplest example is the Laplace-Beltrami operator ∆ for two dimensional man-ifold, where

∆ = div ∇ =1√|g|

∂

∂xi

(gij√|g| ∂∂xj

).

It is a conformally covariant operator of bidegree (0, 2). More precisely, it satisfies

∆g = e−2u∆g0 .

Furthermore, for this operator, we have the following Gaussian curvature equation

∆g0u+Kge2u = Kg0 , (2.10)

where K is the Gauss curvature.

It is quite natural to ask whether there exists a high order conformally covariantoperator Pn, and whether there exists a local curvature invariant Qn satisfying thefollowing conformal transformation formula,

enuQn,g = Qn,g0 + Pn,g0u, (2.11)

for n > 2.

Thanks to C. R. Graham, R. Jenne, L. Mason and G. Sparling, they show thefollowing result in [15],


Theorem 2.1 For Riemannian manifold (Mn, g), let m be a positive even integer sat-isfying

m =

any positive even integer, n odd

m ≤ n, n even.

Then there exists a conformally covariant operator Pn on C∞(M) of degree (m−n2 , m+n2 ),

such that the leading symbol of Pn is the same as the symbol of ∆n2 . In particular, on

(Rn, |dx|2), Pn = ∆n2 .

For a special case when n is even, and m = n, the conformally covariant operator Pnis of bidegree (0, n). In [2], T. Branson proved that there exists Qn curvature satisfying(2.11), whereQn can be represented by the metric g, the connection∇ and the curvaturetensor Rm. Then (2.11) is said to be the Qn curvature equation.

Suppose (M, g) is a complete Riemannian manifold, if for every point p ∈M , thereexists a local coordinate system (xi, Up) such that the metric g is conformal to the flatmetric, we write

gij = e2uδij , (2.12)

where u is some function defined on the neighborhood Up of p. Then the manifold M issaid to be locally conformally flat. Since for a flat metric, the curvature tensor vanishes,the Qn curvature for g0 also vanishes. Hence, for a locally conformally flat manifold(M, g), we have the Qn curvature equation

Qn,g = e−nu∆n2 u. (2.13)

In the following, we say Qn = Qn,g is the Qn curvature for locally conformally flatmanifold Mn, if there is no confusion.

Although we know the existence of Qn curvature due to [2], it is hard to write theexplicit formula of Qn curvature for arbitrary n. For some lower dimensional case, weknow the expressions of Qn.

For n = 2, it is obvious thatP2 = ∆,

and

Q2 = K =R

2.

For n = 4, we have the following forth order differential operator, i.e. the Paneitzoperator

P4 = ∆2 + δ(2

3Rg − 2 Ric)d,

where δ is the divergence operator, R is the scalar curvature, Ric is the Ricci curvaturetensor, and d is the differential. Q4 curvature of a four dimensional manifold is definedby

Q4 =1

6(−∆R+

1

4R2 − 3|E |2) (2.14)

= −1

6∆R+

1

6R2 − 1

2|Ric |2, (2.15)

where E in (2.14) is the traceless Ricci curvature

E = Ric−1

4Rg.

2.3 Chern-Gauss-Bonnet formula 11

We can calculate that

|E |2 = |Ric |2 − R2

4.

If we consider the conformal metric g = e2ug0 on M , then we have the conformalchange formula of the Q4 curvature Q4,g0 of (M, g0) and Q4,g of (M, g) as follows,

P4,g0u+Q4,g0 = Q4,ge4u. (2.16)

2.3 Chern-Gauss-Bonnet formula

It is well-known that for an orientable, closed even dimensional Riemannian manifoldMn with n = 2m, Chern-Gauss-Bonnet theorem states∫

MΩ = χ(M), (2.17)

where

Ω =1

2nπmm!

∑i1,··· ,in

εi1,··· ,inΩi1i2 ∧ · · · ∧Ωin−1in (2.18)

and χ(M) is the Euler characteristic number of M . Here, εi1,··· ,in is defined as follows,

εi1,··· ,in =

1, i1, · · · , in is an even permutation of 1, · · · , n−1, i1, · · · , in is an odd permutation of 1, · · · , n0, i1, · · · , in are not all distinct.

In the following, we are going to recall some results of Chern-Gauss-Bonnet theoremon open manifolds.

For an n dimensional differentiable manifold M , consider the Kulkarni-Nomizu prod-uct

: S2M × S2M → CM,

where S2M = T ∗M⊗ST ∗M is the bundle of symmetric 2-tensor, and CM is the bundleof curvature tensors, defined by:

(α β)ijkl = αilβjk + αjkβil − αikβjl − αjlβik.

The Riemann curvature tensor can be decomposed into three orthogonal part,

Rm = W +R

2n(n− 1)g g +

1

n− 2E g,

= W + S g,

where W is the Wely tensor and S = 1n−2 [Ric− R

2(n−1)g] is the Weyl-Schouten tensor.

With the help of Weyl-Schouten tensor, we have the characterization of a manifold,whose dimension is not less than 4, to be locally conformally flat,

Property 2.2 A Riemannian manifold (Mn, g) with n ≥ 4 is locally conformally flat⇐⇒ the Weyl tensor W = 0.


Notice that we have defined the Gaussian curvature Kn in (2.7). Actually, there isan intrinsic expression of Kn,

Kn dvM =1

n!

∑i1,··· ,in

εi1,··· ,inΩi1i2 ∧ · · · ∧Ωin−1in . (2.19)

Now we determine the explicit formula of Gaussian curvature for a locally confor-mally flat four dimensional Riemannian manifold in terms of various curvatures.

Consider the curvature form and note that for a locally conformally flat manifold,the Weyl tensor W vanishes. Hence, take an orthonormal frame such that it diagonalizes(Sij) to be diag(µ1, µ2, . . . , µ4)

Ωij

=1

2Rijklω

k ∧ ωl

=1

2(S g)ijklω

k ∧ ωl

=1

2(Silω

j ∧ ωl − Sjlωi ∧ ωl)

= −(µi + µj)ωi ∧ ωj . (2.20)

Therefore, by (2.20), we obtain that∑i1,··· ,i4

εi1,··· ,i4Ωi1i2 ∧Ωi3i4

= 8[Ω12 ∧Ω34 −Ω13 ∧Ω24 +Ω23 ∧Ω14]

= 8[(µ1 + µ2)(µ3 + µ4)ω1 ∧ ω2 ∧ ω3 ∧ ω4 − (µ1 + µ3)(µ2 + µ4)ω1 ∧ ω3 ∧ ω2 ∧ ω4

+(µ2 + µ3)(µ1 + µ4)ω2 ∧ ω3 ∧ ω1 ∧ ω4]

= 16σ2(S)ω1 ∧ ω2 ∧ ω3 ∧ ω4. (2.21)

By (2.19) and (2.21), we get

K4 =2

3σ2(S). (2.22)

Now we calculate that

2σ2(S)

= (trace S)2 − trace(S2)

=1

4(R− 2

3R)2 − 1

4

4∑i,j

(Ricij −1

6Rgij)

2

= −1

4|Ric |2 +

R2

12. (2.23)

Therefore, combining (2.22) and (2.23), we obtain that

K4 =1

3(−1

4|Ric |2 +

R2

12). (2.24)

Therefore, by (2.15)

2.3 Chern-Gauss-Bonnet formula 13

Q4 = 6K4 −1

6∆R.

In this case, by (2.18),

Q4 dvM = 8π2Ω − 1

6∆RdvM . (2.25)

Hence, for a locally conformally flat four dimensional closed Riemannian manifold,by divergence theorem, the Chern-Gauss-Bonnet formula says

1

8π2

∫MQ4 dvM = χ(M). (2.26)

A. Chang, J. Qing and P. Yang generalized Chern-Gauss-Bonnet theorem to openmanifolds in [7],

Theorem 2.3 Suppose that (M, g) is a complete four dimensional manifold with finitenumber of conformally flat simple ends. And suppose that

(a). The scalar curvature is non-negative at infinity at each end.

(b). The Q4 curvature is integrable.

Then

χ(M)− 1

8π2

∫M

(|W |2 +Q4) dvM =

l∑i=1

µi, (2.27)

where

µi = limr→∞

(∫∂Br(0) e

3ui dσ(x)) 4

3

4(2π2)13

∫Br(0)\B e

4ui dx,

and l is the number of the ends.

3

High Dimensional Chern-Gauss-Bonnet formula

In this chapter, we want to introduce Chern-Gauss-Bonnet theorem for high evendimensional locally conformally flat open manifolds. For more general case, please see[33]. Since this result and the method of the proof will be applied to prove the maintheorem of this thesis, we will provide the argument for completeness of this thesis. Inorder to state the results, let us present some definitions and notations.

Definition 3.1 Suppose (M, g) is a complete open locally conformally flat n dimen-sional manifold such that

M = N ∪ l⋃

i=1

Ei,

where (N, g) is a compact locally conformally flat Riemannian manifold with boundary

∂N =

l⋃i=1

∂Ei,

and each end Ei is simple in the sense that,

(Ei, gi) = (Rn\B, e2uig0),

for some function ui. Such manifolds will be called complete locally conformally flat ndimensional manifolds with simple ends.

Actually, there are many examples of such manifolds. For example, complete metricswith constant positive scalar curvature on Sn\piki=1, i.e. Sn with k points deletedconstructed by R. Schoen [28] and R. Mazzeo and F. Pacard [24].

The high dimension Chern-Gauss-Bonnet theorem can be stated in the following,

Theorem 3.2 Suppose (M, g) is an open complete locally conformally flat n = 2mdimensional manifold with simple ends. Let l be the number of ends. Assume that

(a) the scalar curvature is non-negative at infinity on each end;

(b) the Qn curvature is absolutely integrable.

Then, we have the following formula,

χ(M)− 2

(n− 1)!ωn

∫MQn dvg =

l∑i=1

µi,

16 3 High Dimensional Chern-Gauss-Bonnet formula

where

µi = limr→∞

[∫∂Br(0) e

(n−1)ui dσ]nn−1

n(ωn)1

n−1∫Br(0)\B e

nui dx.

3.1 Chern-Gauss-Bonnet formula for Rn

In the following, we consider the even dimensional Euclidean space Rn. Similar to[7], we define normal metric for high dimensional manifolds,

Definition 3.3 A conformal metric e2ug0 on the Euclidean space Rn with the Qn cur-vature to be absolutely integrable is said to be normal if the function u has the followingdecomposition,

u(x) =2

(n− 1)!ωn

∫Rn

[log

|y||x− y|

]Qn(y)enu(y) dy + C0,

whereQn(y) = e−nu(y)[(−∆)

n2 u](y),

C0 is a constant and ωn is the volume of the unit sphere Sn in the Euclidean spaceRn+1.

Following Chang, Qing, and Yang [6] and N. Trudinger [32], we can define the mixedvolumes and corresponding isoperimetric ratios as follows,

Vn(r) =

∫Br(0)

enu dx,

and

Vk(r) =1

n

∫∂Br(0)

rk−n+1(1 + r∂u

∂r)n−k−1eku dσ,

for k = 1, 2, . . . , n− 1.

By the above definition, for all 1 ≤ j ≤ n − 1 and 1 ≤ k ≤ n − j, we define theisoperimetric ratio to be,

Ck,k+j(r) =V

k+jj(n−1)

k

(nωn)1

n−1Vk

j(n−1)

k+j

.

We want to prove the following theorem,

Theorem 3.4 Suppose e2ug0 is a smooth complete normal metric on Rn with absolutelyintegrable Qn curvature. Then

α = limr→∞

Cn−1,n(r) = 1− 2

(n− 1)!ωn

∫RnQne

nu dy ≥ 0.

Furthermore, if α > 0, thenlimr→∞

Ck,k+j(r) = α,

for all 1 ≤ j ≤ n− 1 and 1 ≤ k ≤ n− j.

3.1 Chern-Gauss-Bonnet formula for Rn 17

Notice that the above theorem was obtained both by X.W Xu [33] and C.B. Ndiaye,J. Xiao [26] using different method. In order to prove Theorem 3.4, we need severallemmas.

Lemma 3.5 Suppose e2ug0 is a complete normal metric. Then the following identityholds,

limr→∞

(1 + r∂u

∂r) = 1− 2

(n− 1)!ωn

∫RnQn(y)enu(y) dy, (3.1)

where

u(r) = −∫∂Br(0)

u(x) dσ.

Proof. Since the metric is normal, by direct calculation, we have

1 + r∂u

∂r

= 1− 2

(n− 1)!ωn

∫Rn

x · (x− y)

|x− y|2Qn(y)enu(y) dy

= 1− 2

(n− 1)!ωn

∫RnQn(y)enu(y) dy

+2

(n− 1)!ωn

∫Rn

[1− x · (x− y)

|x− y|2

]Qn(y)enu(y) dy

= 1− 2

(n− 1)!ωn

∫RnQn(y)enu(y) dy

+2

(n− 1)!ωn

∫Rn

[y · (x− y)

|x− y|2

]Qn(y)enu(y) dy. (3.2)

By the formula that

d

dr−∫∂Br(x0)

udσ = −∫∂Br(x0)

du

drdσ,

which is proved in the Appendices, we can see in order to finish the proof of Lem 3.5,we only need to show that the spherical average integral of the last term in (3.2) tendsto zero as r →∞. Set

α =1

nωn

∫Rn|Qn(y)|enu(y) dy.

Since Qn is absolutely integrable, ∀ ε > 0, ∃ a sufficiently large R0 > 0 such that∫|y|≥R0

|Qn(y)|enu(y) dy ≤ ε

4. (3.3)

For such fixed R0, there exists a sufficiently large R1 > R0 > 0 such that if r > R1, forany positive integer k ≤ n− 2, we have[

R0

r −R0

]k≤ ε

4α. (3.4)

We also have, for |y| ≥ R0, by Holder inequality,


−∫∂Br(0)

[|y||x− y|

]kdσx

≤ |y|k[−∫∂Br(0)

1

|x− y|n−2dσx

] kn−2

= 1, (3.5)

since 1|x−y|n−2 is a harmonic function.

By (3.3), (3.4) and (3.5), for r = |x| ≥ R1, we have,

|−∫∂Br(0)

∫Rn

[y · (y − x)

|x− y|2

]kQn(y)enu(y) dy

dσx|

≤∫Rn

[−∫∂Br(0)

(|y||x− y|

)k dσx

]|Qn(y)|enu(y) dy

=

∫|y|≤R0

[−∫∂Br(0)

(|y||x− y|

)k dσx

]|Qn(y)|enu(y) dy

+

∫|y|≥R0

[−∫∂Br(0)

(|y||x− y|

)k dσx

]|Qn(y)|enu(y) dy

≤ ε

4+ε

4≤ ε. (3.6)

With k = 1 in (3.6), take spherical average of (3.2) and let r →∞, we get (3.1). 2

Lemma 3.6 Suppose the metric e2ug0 on Rn is a normal metric. Then for any k > 0,we have

−∫∂Br(0)

eku dσ = ekueo(1), (3.7)

where o(1)→ 0 as r →∞.

Proof. Since e2ug0 is a normal metric, we decompose u as follows,

u(x)

=2

(n− 1)!ωn

∫B |x|

2

(0)

[log

(|y||x− y|

)]Qn(y)enu(y) dy + C0

+2

(n− 1)!ωn

∫Rn\B |x|

2

(0)

[log

(|y||x− y|

)]Qn(y)enu(y) dy

=2

(n− 1)!ωn

∫B |x|

2

(0)

[log

(|y||x|

)]Qn(y)enu(y) dy + C0

+2

(n− 1)!ωn

∫B |x|

2

(0)

[log

(|x||x− y|

)]Qn(y)enu(y) dy

+2

(n− 1)!ωn

∫Rn\B |x|

2

(0)

[log

(|y||x− y|

)]Qn(y)enu(y) dy

= f(|x|) + u1(x) + u2(x). (3.8)


We can see that f(|x|) is a radial symmetric function which is the major part of u

when |x| → ∞. For sufficiently large |x|, if |x|12 ≤ |y| ≤ 1

2 |x|, then

1

2|x| ≤ |x| − |y| ≤ |x− y| ≤ |x|+ |y| ≤ 3

2|x|,

Hence,

|u1(x)|

= | 2

(n− 1)!ωn

∫B |x|

2

(0)

[log

(|x||x− y|

)]Qn(y)enu(y) dy|

≤ 2

(n− 1)!ωn

∫B|x|

12

(0)

∣∣∣∣log

(|x||x− y|

)∣∣∣∣ |Qn(y)|enu(y) dy

+2

(n− 1)!ωn

∫B |x|

2

(0)\B|x|

12

(0)

∣∣∣∣log

(|x||x− y|

)∣∣∣∣ |Qn(y)|enu(y) dy

≤ C · log|x|

12

|x|12 − 1

+ C1

∫B |x|

2

(0)\B|x|

12

(0)|Qn(y)|enu(y) dy, (3.9)

where C,C1 do not depend on x.

Hence,|u1(x)| = o(1) as |x| = r →∞. (3.10)

Next, we want to show that the average integral of u2 has the estimate that u2(r) =o(1) as r →∞. In order to prove it, we rewrite u2 by Fubini theorem,

−∫∂Br(0)

u2(x) dσ

=2

(n− 1)!ωn

∫Rn\B |x|

2

(0)

−∫∂Br(0)

[log

(|y||x− y|

)]dσ

Qn(y)enu(y) dy. (3.11)

From (3.11), we can show that

L =

∣∣∣∣∣−∫∂Br(0)

log

(|y||x− y|

)dσ

∣∣∣∣∣is bounded for |y| ≥ 1

2 |x|.Notice that we have proved in Lemma 3.5 the inequality (3.5), i.e., if |y| ≥ 1

2 |x|,

−∫∂Br(0)

|y||x− y|

≤ 1. (3.12)

Hence, by (3.12) and Jenson’s inequality,

exp

[−∫∂Br(0)

log|y||x− y|

dσ

]

≤ −∫∂Br(0)

|y||x− y|

dσ

≤ 1. (3.13)


On the other hand, it also follows by Jenson’s inequality and the fact that |x− y| ≤|x|+ |y| and |y| ≥ |x|2 ,

exp

[−−∫∂Br(0)

log|y||x− y|

dσ

]

≤ −∫∂Br(0)

|x− y||y|

dσ

≤ −∫∂Br(0)

1 +|x||y|

dσ

≤ 3. (3.14)

Combining (3.13) and (3.14), we obtain that L is bounded from above by a constantwhich does not depend on x and y. Hence,

−∫∂Br(0)

u2(x) dσ = o(1) as r →∞. (3.15)

By the estimate (3.10) and the fact that f(|x|) is a radial symmetric function, wehave

k−∫∂Br(0)

(u(x)− u2(x)) dσ

= log

[−∫∂Br(0)

ek(u(x)−u2(x)) dσ

]+ o(1)

= log−∫∂Br(0)

eku(x) dσ + o(1)

+ log

[1 +

−∫∂Br(0) e

k(u(x)−u2(x))(1− eku2(x))dσ

−∫∂Br(0) e

ku(x)dσ

]. (3.16)

In order to prove (3.7), by (3.16), we only need to show that the last term on theright hand side of (3.16) is o(1) as r →∞, i.e.

−∫∂Br(0) e

k(u(x)−u2(x))(1− eku2(x))dσ

−∫∂Br(0) e

ku(x)dσ= o(1), as r →∞. (3.17)

Since f(|x|) is a radial symmetric function, then by Jenson’s inequality, (3.10) and(3.15) we obtain that, as r →∞

|−∫∂Br(0) e

k(u(x)−u2(x))(1− eku2(x)) dσ

−∫∂Br(0) e

ku(x) dσ|

≤ekf(|x|)eo(1)|−

∫∂Br(0)(1− e

ku2(x)) dσ|exp[−

∫∂Br(0) u(x) dσ]

≤expk−

∫∂Br(0)[u(x)− u2(x)] dσeo(1)|−

∫∂Br(0)(1− e

ku2(x))σ|exp[−

∫∂Br(0) u(x) dσ]

= eo(1)

∣∣∣∣∣−∫∂Br(0)

(1− eku2(x)) dσ

∣∣∣∣∣ . (3.18)


In order to prove the last term in (3.18) is o(1) as r → ∞, we follow [13] and [6].By changing variable we get

|−∫∂Br(0)

(1− eku2(x)) dσ| = |−∫∂B1(0)

(eku2(rσ) − 1) dσ|. (3.19)

Next, we estimate the measure of the set EM = σ ∈ Sn−1| |u2(rσ)| > M for anygiven real number M > 0. If we denote the measure of EM by |EM |, then we have thefollowing estimate,

M |EM |

≤∫EM

|u2(rσ)| dσ

=1

nωn

∫Rn\B |x|

2

(0)

[∫EM

| log

(|y||x− y|

)| dσ]|Qn(y)|enu(y) dy. (3.20)

First we claim that for |y| ≥ 12 |x|,∫

EM

| log|y||x− y|

| dσ ≤ (C2 + C3 log1

|EM |)|EM |, (3.21)

for some constants C2 and C3 depending only on the dimension n.

In order to see this, by Jenson’s inequality,

exp

[−∫EM

| log|y|

|rσ − y|| dσ]

≤ −∫EM

exp

[| log

|y||rσ − y|

|]dσ

= −∫EM

|y||rσ − y|

dσ +−∫EM

|rσ − y||y|

dσ. (3.22)

The second integration of the last term of (3.22) is bounded by log 3.

Now we estimate the first integral of the last term of (3.22). If |y| 6= r, then by theestimate (3.5) in Lem 3.5, we have

−∫EM

|y||rσ − y|

dσ

≤ 1

|EM |

∫Sn

|y||rσ − y|

dσ

≤ ωn|EM |

. (3.23)

On the other hand, if |y| = r, we can estimate it as follows:

1

|EM |

∫EM

|y||rσ − y|

=1

|EM |

∫EM

1

|σ − y|y| |

dσ

≤ n− 1

n− 2ω

1n−1

n−1 |EM |− 1n−1 . (3.24)


Notice that the last step in (3.24) is the potential estimate (7.32) in [14] on page 159by choosing a point P which is not belong to EM , and project the domain EM into Rnby stereographic projection. Observe that the set EM is not equal to the whole sphereis r is sufficient large as we have shown that u2(r) tends to zero as r →∞.

Hence, it is clear that our claim (3.21) follows from (3.22), (3.23) and (3.24).

Therefore, by (3.20) and (3.21), we get

M ≤ o(1)(C2 + C3 log1

|EM |), (3.25)

where o(1)→ 0 as r →∞. Solve (3.25) for |EM | in terms of M , we get

|EM | ≤ C4e− Mo(1) . (3.26)

Hence, we obtain

|−∫∂Br(0)

(1− eku2(x)) dσ|

= k|∫ ∞

0(ek(±M) − 1)|EM | dM |

= o(1). (3.27)

Finally, by (3.15), (3.16), (3.17), (3.18) and (3.27), we obtain that

k−∫∂Br(0)

u(x) dσ = log

[−∫∂Br(0)

eku(x) dσ

]+ o(1). (3.28)

Lemma 3.6 follows by taking exponential on both sides of equation (3.28). 2

Lemma 3.7 For any positive real number k < n− 1, we have

−∫∂Br(x)

|r∂u∂r− r∂u

∂r|k dσ = o(1) as r →∞. (3.29)

Proof. It follows from the equation (3.2) in the proof of Lemma 3.5 that

|r∂u∂r− r∂u

∂r|

= | − 2

(n− 1)!ω

∫Rn

[x · (x− y)

|x− y|2−−∫∂Br(0)

x · (x− y)

|x− y|2dσ

]Qn(y)enu(y)dy|

≤ 2

(n− 1)!ωn

∫Rn

∣∣∣∣∣x · (x− y)

|x− y|2−−∫∂Br(0)

x · (x− y)

|x− y|2dσ

∣∣∣∣∣ |Qn(y)|enu(y)dy

=2

(n− 1)!ωn

∫Rn

∣∣∣∣∣[1− x · (x− y)

|x− y|2

]−−∫∂Br(0)

[1− x · (x− y)

|x− y|2

]dσ

∣∣∣∣∣ |Qn(y)|enu(y)dy

≤ 2

(n− 1)!ωn

∫Rn

[|y||x− y|

+−∫∂Br(0)

|y||x− y|

dσ

]|Qn(y)|enu(y) dy. (3.30)


Hence, by taking the kth−power of (3.30) and integrating over the sphere ∂B1(0),then using Holder inequality, we obtain that

−∫∂Br(0)

|r∂u∂r− r∂u

∂r|k dσ

≤[

2

(n− 1)!ωn

]k−∫∂Br(0)

∫Rn

(|y||x− y|

+−∫∂Br(0)

y

|x− y|dσ

)k|Qn(y)|enu(y) dy

dσ·[∫

Rn|Qn(y)|enu(y) dy

]k−1

=

[2

(n− 1)!ωn

]k ∫Rn

−∫∂Br(0)

(|y||x− y|

+−∫∂Br(0)

y

|x− y|dσ

)kdσ

|Qn(y)|enu(y) dy

·[∫

Rn|Qn(y)|enu(y) dy

]k−1

= o(1), as r →∞, (3.31)

by the estimate (3.6) in the proof of Lemma 3.5 if k ≤ n − 2. If n − 2 < k < n − 1,(3.31) follows from (3.6) by the observation that

−∫∂Br(0)

(|y||x− y|

)kdσ ≤ C(n, k),

where C(n, k) is a constant only depends on n and k − 1. 2

Lemma 3.8 For any positive real number k < n−1, there exists a constant C depend-ing only on n and k such that,

max−∫∂Br(0)

|r∂u∂r|k dσ,−

∫∂Br(0)

|r∂u∂r|k dσ ≤ C. (3.32)

Proof. The proof follows from the estimate (3.6) in the proof of Lemma 3.5. 2

Proof of Theorem 3.4: Denote

β = limr→∞

(1 + r∂u

∂r).

Notice that the completeness of the metric g = e2ug0 implies that the correspondingradial symmetric metric g = e2ug0 is also a complete metric. Hence we can concludefrom this completeness that β ≥ 0. Next, we consider two cases,

Case 1. β = 0.

Since β = 0, there exists a sufficiently large R0, such that if r ≥ R0, we have

|1 + r∂u

∂r| ≤ 1

n. (3.33)

Integrating (3.33), we get

rn−1enu(r) ≤ Rn−10 enu(R0), (3.34)


for all r ≥ R0, which in turn implies that

| ddr

[rnenu(r)]| ≤ C1,

for some constant C1 depending only on R0 and n. Therefore, rnenu(r) is uniformlycontinuous. Notice that if Vn(r) is bounded for all r, then by Jenson’s inequality, wecan see that

∫ r0 t

nenu(t) dt is also bounded for all r, hence we obtain that

limr→∞

rnenu(r) = 0. (3.35)

It follows from (3.35) and the definition of Cn−1,n that

limr→∞

Cn−1,n(r) = 0. (3.36)

If Vn(r) is unbounded, i.e.,limr→∞

V (r) =∞,

then if Vn−1(r) is bounded, it follows from the definition of Cn−1,n that the equation(3.36) holds in this case. If

limr→∞

Vn−1(r) =∞,

then by the L’Hospital’s Law,

limr→∞

Cn−1,n(r)

= limr→

[Vn−1(r)]nn−1

(nωn)1

n−1Vn(r)

= limr→∞

[−∫∂Br(0) e

(n−1)u dσ]1

n−1−∫∂Br(0) e

(n−1)u(1 + r ∂u∂r ) dσ

−∫∂Br(0) e

nu dσ

= 0

= β. (3.37)

Here in the second equality we have used Lemma 3.7, Lemma 3.8 and Holder in-equality.

Case 2. β > 0.

If β > 0, then there exists a sufficiently large R0 > 0 such that for all r ≥ R0, thefollowing inequality holds,

1 + r∂u

∂r≥ β

2. (3.38)

Integrate (3.38), we get

u(r) ≥ u(R0) + (β

2− 1)(log

r

R0). (3.39)

It follows from (3.39) that

rn−1e(n−1)u(r) ≥ e(n−1)u(R0)R(β2−1)(n−1)

0 rβ2

(n−1), (3.40)

which clearly implies that Vn−1(r)→∞, and so is for Vn(r).

3.2 Geometric conditions for metric to be normal 25

Similarly we can show that for all 1 ≤ k ≤ n− 2, Vk(r) also tends to infinity.

Then by Lemma 3.7, Lemma 3.8 and Holder inequality, we obtain that

Vk(r) = Vk(r) + o(1), (3.41)

as r →∞ for all 1 ≤ k ≤ n−1, where Vk(r) is the mixed volume with respect to metrice2u(r)g0. To be precise, for k = n − 1, (3.41) holds by Lemma 3.5. For k ≤ n − 2, wecan prove by the following equality,

Vk(r)

=1

n

∫∂Br(0)

rk−n+1(1 + r∂u

∂r)n−k−1eku dσ

=1

n

∫∂Br(0)

rk−n+1

[(1 + r

∂u

∂r)n−k−1 − (1 + r

∂u

∂r)n−k−1

]eku dσ

+1

n

∫∂Br(0)

rk−n+1(1 + r∂u

∂r)n−k−1eku dσ

=1

n

∫∂Br(0)

rk−n+1(r∂u

∂r− r∂u

∂r) ·

n−k−2∑l=0

[(1 + r

∂u

∂r)l(1 + r

∂u

∂r)n−k−2−l

]eku dσ

+rk−n+1(1 + r∂u

∂r)n−k−1

[1

n|∂Br(0)|

∫∂Br(0)

(eku − eku) dσ

]+ Vk(r). (3.42)

By Holder inequality, Lemma 3.7 and Lemma 3.8, we can see that every term in(3.42) is o(1) as r →∞ except Vk(r).

Similar argument gives us the following identity

d

drVn(r) =

d

drVn(r) + o(1),

as r →∞.

For radial symmetric metric eug0, we can see that Ck,k+j(r) = 1 + r ∂u∂r , except forthe case k = n − 1 and k = 1. For k = n − 1, we again use L’Hospital’s rule sincethe ratio is the type of infinity over infinity similar to the equation (3.37). Hence ourtheorem 3.4 follows.

3.2 Geometric conditions for metric to be normal

In this section, we will show that there is a large class of metrics which are normal.The main result of this section is the following theorem.

Theorem 3.9 Suppose e2ug0 is a complete C∞ metric on Rn with absolutely integrableQn curvature and its scalar curvature is non-negative at infinity. Then the metric isnormal, hence the conclusion of theorem 3.4 holds.

Proof. Let

v(x) =1

(n− 1)!ωn

∫Rn

(log

|y||x− y|

)Qn(y)eu(y) dy,

Since Qn is integrable, v(x) is well-defined, and (−∆)n2 v = Qne

4u. If we set w = u− v,(−∆)

n2w = 0.


We want to show that under the assumption in this theorem, (−∆)(u − v) = 0.Since (−∆)

n2w = 0, (−∆)

n2−1w is harmonic. Let p = n

2 . Hence, by mean value property,∀x0 ∈ Rn, and ∀ r > 0, we have,

(∆p−1w)(x0)

= −∫Br(x0)

(∆p−1w)(y)dy

=n

ωn−1rn

∫∂Br(x0)

∂

∂r(∆p−2w)(y)dy

=n

r

∂

∂r−∫∂Br(x0)

(∆p−2w)(y)dy. (3.43)

Multiple (3.43) by rn on both sides and integral on r to get,

r2

2n(∆p−1w)(x0) = −

∫∂Br(x0)

(∆p−2w)(y)dy − limr→0−∫∂Br(x0)

(∆p−2w)(y)dy

= −∫∂Br(x0)

(∆p−2w)(y)dy − (∆p−2w)(x0),

i.er2

2n(∆p−1w)(x0) + (∆p−2w)(x0) = −

∫∂Br(x0)

(∆p−2w)(y)dy. (3.44)

Multiple (3.44) by nrn−1 on both sides and integral on r, we get

rn+2

2(n+ 2)(∆p−1w)(x0) + rn(∆p−2w)(x0) =

n

ωn

∫Br(x0)

(∆p−2w)(y)dy. (3.45)

Divide both sides of (3.45) by rn, we have

r2

2(n+ 2)(∆p−1w)(x0) + (∆p−2w)(x0) = −

∫Br(x0)

(∆p−2w)(y)dy.

Now repeat the above argument p− 1 times to get

P (r)

= C1(n, p)r2(p−1)(∆p−1w)(x0) + C2(n, p)r2(p−2)(∆p−2w)(x0)

+ · · ·+ Cp−1(n, p)r2(∆w)(x0)

= −∫Br(x0)

w(y) dy − w(x0), (3.46)

where all Ci(n, p) are positive constants depending only on p and n.

By calculation and Fubini’s theorem,

−∫∂Br(x0)

∆v(x)dσ

= − 2

(n− 1)!ωn

∫Rn

(−∫∂Br(x0)

1

|x− y|2dσx

)Qne

nu(y) dy, (3.47)

and

3.2 Geometric conditions for metric to be normal 27

d

dr−∫∂Br(x0)

v(x)dσ

= − 2

(n− 1)!ωn

∫Rn

(−∫∂Br(x0)

(x− x0) · (x− y)

r|x− y|2dσx

)Qne

nu(y) dy. (3.48)

By (3.47) and (3.48), and since Qn is integrable,

|n− 2

2(d

dr−∫∂Br(x0)

v(x) dσ)2 +−∫∂Br(x0)

∆v(x) dσ|

≤ |n− 2

2(d

dr−∫∂Br(x0)

v(x) dσ)|2 + |−∫∂Br(x0)

∆v(x) dσ|

≤ C 1

r2, (3.49)

where C is a constant only depends on the absolute integral of the Qn curvature.

By the definition of P (r) and w,

−∫∂Br(x0)

u(x) dσ

= u(x0) +−∫∂Br(x0)

v(x) dσ − v(x0) + P (r).

Therefore,

n− 2

2[d

dr−∫∂Br(x0)

u(x) dσ]2 +−∫∂Br(x0)

∆u(x) dσ

=n− 2

2[d

dr−∫∂Br(x0)

v(x) dσ]2 +−∫∂Br(x0)

∆v(x) dσ

+∆P (r) +n− 2

2[d

drP (r)]2

+(n− 2)d

drP (r) · [ d

dr−∫∂Br(x0)

v(x) dσ]. (3.50)

Let us consider the scalar curvature equation under the conformal change g = e2ug0

on Rn,

∆u+n− 2

2|∇u|2 = − R

2(n− 1)e2u.

Since R ≥ 0 for sufficiently large r, then by Cauchy inequality, we have

n− 2

2[d

dr−∫∂Br(x0)

u(x) dσ]2 +−∫∂Br(x0)

∆u(x) dσ

= −∫∂Br(x0)

∆u(x) dσ +n− 2

2[−∫∂Br(x0)

d

dru(x) dσ]2

≤ −∫∂Br(x0)

∆u(x) +n− 2

2|∇u|2 dσ

= −−∫∂Br(x0)

R

2(n− 1)e2u dσ

≤ 0. (3.51)


Hence, in the right side of the equation (3.50), the leading term is C1(n, p)24(p −1)2r4p−6[(−∆)p−1w]2(x0). Therefore, if we divide the factor r4p−6 throughout the equa-tion (3.50) and let r goes to infinity and combine with the equation (3.51), we get[(−∆)p−1w](x0) = 0 if 2p > 3. Once [(−∆)p−1w](x0) = 0, the next leading term will beC2(n, p)24(p − 2)2r4p−10[(−∆)p−2w(x0)]2, and then repeat this argument to concludethat [(−∆)p−2w](x0) = 0. Hence, we get P (r) = 0. Since x0 is arbitrary in Rn, weconclude that w is a harmonic function in the whole space Rn.

Next, we show that u is a constant function. In order to see this, we first note that∇iw is also a harmonic function for all 1 ≤ i ≤ n. Hence, by mean value formula forharmonic functions, we have, for any x0 ∈ Rn,

|wi(x0)|2 = |−∫∂Br(0)

wi(x) dσ|2 ≤ −∫∂Br(0)

|∇w|2 dσ. (3.52)

On the other hand, by the definition of u,

n− 2

4|∇w|2

≤ n− 2

2(|∇u|2 + |∇v|2)

≤ − R

2(n− 1)e2u +

n− 2

2|∇v|2 −∆v. (3.53)

Taking the average integral over the sphere ∂Br(0) and using the estimate (3.49) forv, we conclude that |wi(x0)| = 0. Since x0 is arbitrarily chosen, we have wi = 0. Thusw is a constant function. Hence, u is a normal metric. Therefore, Theorem 3.9 follows.

3.3 Chern-Gauss-Bonnet formula in Local version

In this section, we generalize the result in previous section to local version. Themain result is the following theorem.

Theorem 3.10 Suppose (Rn\B1(0), e2ug0) is complete a conformal metric with abso-lutely integrable Qn curvature and non-negative scalar curvature at infinity. Then wehave the following formula,

limr→∞

[Vn−1(r)]nn−1

nω1

n−1n Vn(r)

=

2

(n− 1)!ωn

∫∂B1(0)

∂[(−∆)n2−1u]

∂rdσ + 1

− 2

(n− 1)!ωn

∫Rn\B1(0)

Qn(y)enu(y) dy, (3.54)

whereQn(y) = e−nu(y)[(−∆)

n2 ](y),

Vn(r) =

∫Br(0)\B1(0)

enu(y) dy,

and

Vn−1(r) =

∫∂Br(0)

e(n−1)u(y) dy.

3.3 Chern-Gauss-Bonnet formula in Local version 29

In order to proof this theorem, we need the definition of normal metric in the localversion.

Definition 3.11 A conformal metric e2ug0 with absolutely integrable Qn curvature issaid to be normal on Rn\B1(0) if

u(x) =2

(n− 1)!ωn

∫Rn\B1(0)

[log

|y||x− y|

]Qn(y)enu(y) dy

+β0 log |x|+ (n+ 1

2− p)

p−2∑j=1

βj|x|2j

+ h(x), (3.55)

where p = n2 , and h(x) is some p−harmonic function in Rn\B1(0) with the following

properties,

1.

rn−1 ∂

∂r

[−∫∂Br(0)

(−∆)n2−1h dσ

]= o(1), (3.56)

as r →∞.

2. For all 2 ≤ k ≤ p,

r2(k−1)−∫∂Br(0)

|(−∆)k−1h(x)| dσ (3.57)

is bounded for r large if n is odd and is o(1) as r →∞ if n is even.

Similarly to Lemma 3.5, for a normal metric which is complete at infinity, we havethe following identity,

Lemma 3.12 If e2ug0 is a normal metric, complete at infinity, then

limr→∞

(1 + r∂[−∫∂Br(0) u dσ]

∂r)

= 1 + β0 −2

(n− 1)!ωn

∫Rn\B1(0)

Qn(y)enu(y) dy ≥ 0.

Proof. Similar to the proof of Lemma 3.5. 2

With all the preparation, now we want to prove Theorem 3.9.

Proof of Theorem 3.10. We first show that Theorem 3.9 holds true for normalmetric. For normal metric, we have

(−∆)n2−1u =

1

(n− 2)ωn

∫Rn\B1(0)

1

|x− y|n−2Qn(y)enu(y) dy

−β0b(n)

|x|n−2+ (−∆)

n2−1h. (3.58)

By the definition of Qn curvature and integration by parts, it follows that

30 3 High Dimensional Chern-Gauss-Bonnet formula∫Br(0)\B1(0)

Qn(x)enu(x) dx

=

∫Br(0)\B1(0)

(−∆)n2 u dx

=

∫∂B1(0)

∂[(−∆)n2−1u]

∂rdσ −

∫∂Br(0)

∂[(−∆)n2−1u]

∂rdσ

=1

(n− 2)ωn−1

∫Rn\B1(0)

[∫∂Br(0)

∂

∂r

(1

|x− y|n−2

)dσ

]Qn(y)enu(y) dy

−β0b(n)(n− 2)ωn−1 +

∫∂B1(0)

∂

∂r

[(−∆)

n2−1u

]dσ −

∫∂Br(0)

∂

∂r

[(−∆)

n2−1h

]dσ.

(3.59)

Notice that here we have used the fact that the rest of integrals are zero and also

b(n)(n− 2)ωn−1 =(n− 1)!

2ωn.

Therefore, divide equation (3.59) by the factor (n−1)!2 ωn to obtain

2

(n− 1)!ωn

∫Br(0)\B1(0)

Qn(x)enu(x) dx

=2

(n− 1)!ωn

∫Rn\B1(0)

[−∫∂Br(0)

rn−1 ∂

∂r

(1

|x− y|n−2

)dσ

]Qn(y)enu(y) dy

−β0 +2

(n− 1)!ωn

∫∂B1(0)

[(−∆)n2−1u]

∂rdσ

− 2

(n− 1)!ωn

∫∂Br(0)

[(−∆)n2−1h]

∂rdσ. (3.60)

By (3.56), let r →∞, we get

2

(n− 1)!ωn

∫∂B1(0)

∂[(−∆)n2−1u]

∂rdσ = β0. (3.61)

Hence, Theorem 3.10 for normal metric follows from Lemma 3.8 and the observationthat

limr→∞

(1 + r∂[−∫∂Br(0) u dσ]

∂r) = lim

r→∞

[Vn−1(r)]nn−1

(nωn)1nVn(r)

, (3.62)

which follows from the L’Hospital rule just same as in whole space situation.

In the following, we are going to show that our assumption in Theorem 3.10 impliesthat e2ug0 is a normal metric.

Let

v(x) =2

(n− 1)!ωn

∫Rn\B1(0)

(log

|y||x− y|

)Qn(y)eu(y) dy,

Since Qn is integrable, v(x) is well-defined, and (−∆)n2 v = Qne

nu. If we set w = u− v,then w is a p−harmonic function, where p = n

2 . We can see that for any x0 ∈ Rn\B1(0),

3.3 Chern-Gauss-Bonnet formula in Local version 31

as long as r ≤ |x0| − 1, the equation (3.46) holds. If we choose R0 sufficiently large sothat the scalar curvature R ≥ 0 for any point x with |x| ≥ R0, then for any x0 such that|x0| ≥ 2R0, we have R ≥ 0 on the ball B |x0|

2

(x0). Hence, for such an x0, the equation

(3.51) holds too for any r ≤ |x0|2 .

Hence, by (3.49), (3.50) and (3.51), we get

r2(∆P (r) +n− 2

2[d

drP (r)]2) ≤ C, (3.63)

where the constant C is independent of x0 and r. Here, we have used the fact that,with the definition of v, we have

r2

∣∣∣∣∣−∫∂Br(0)

∆v dσ

∣∣∣∣∣ ≤ C1,

and

r

∣∣∣∣∣ ddr(−∫∂Br(0)

v dσ

)∣∣∣∣∣ ≤ C2,

for some constants C1 and C2 which depend only on n and the absolute integral of Qncurvature.

By the equation (3.46), and choosing r = |x0|2 , it follows that

|x0|2k∣∣∣[(−∆)kw](x0)

∣∣∣ ≤ C, (3.64)

where C only depends on n and∫Rn\B1(0) |Q(y)|e4u(y) dy.

Next, we want to work out the function h in the definition of normal metric. Let

γ(x) = w − β0 log |x|,

where β0 is a constant to be determined. It is easy to see that

(−∆)n2 γ = 0.

Integrating the follows equation over the set Br(0)−B1(0),

(−∆)n2w = 0

for r ≥ 1, and by divergence theorem, we obtain that

−∫∂Br(0)

(−∆)n2−1w(y) dσ = a+

b

rn−2. (3.65)

Hence, by direct calculation, we have

(−∆)n2−1γ = (−∆)

n2−1w − bnβ0r

2−n,

where bn 6= 0 is some constant depending only on n. Since w satisfies the estimate(3.64), we conclude that the constant term of the above equality (3.65) should vanish,i.e., a = 0. If we choose

β0 =b

bn,


we can see that

−∫∂Br(0)

(−∆)n2−1γ dσ = 0. (3.66)

In order to find the function h with properties in the definition of normal metric,we need to separate the cases according to whether n is odd or even.

We need to find the function h in the definition of the normal metric. Let βi, 1 ≤i ≤ p− 2 be the constants to be determined and consider the function γ1 defined by,

γ1(x) = γ(x)−p−2∑j=1

βj|x|2j

. (3.67)

It is easy to see that(−∆)

n2 γ1 = 0

in Rn\B1(0). By the choice of β0, we know that

−∫∂Br(0)

(−∆)n2−1γ1 dσ = 0. (3.68)

By integrating the equation (3.68) over the set Br(0)\B1(0), and using the diver-gence theorem to get,

−∫∂Br(0)

(−∆)n2−2γ1 dσ = a+

b

rn−2. (3.69)

By the estimate (3.64) of u, we conclude that a = 0 in the equation (3.69).

By the definition of the function γ1, we have

(−∆)n2−2γ1 = (−∆)

n2−2u− b0(n)β0r

4−n − b1(n)β1(n)β1r2−n, (3.70)

Hence, the equations (3.69) and (3.70) imply that the limit

limr→∞

rn−2−∫∂Br(0)

(−∆)n2−2u dσ

exists and is finite. Thus, we can choose β1 such that b1(n)β1 is equal to that limit.With such choice of β1, we obtain

−∫∂Br(0)

(−∆)n2−2γ1 dσ = 0. (3.71)

Repeat this procedure for other n2 − 3 times to determine all the coefficients for

2 ≤ i ≤ p− 2. With those choices of βi, we let h be the resulting function γ1.

Finally, h has the property we need for normal metric. By the definition of v andh, we can see that the metric g = e2ug0 is a normal metric in all dimension. Therefore,we finish the proof of Theorem 3.10. 2

3.4 Chern-Gauss-Bonnet formula for conformally flat manifolds 33

3.4 Chern-Gauss-Bonnet formula for conformally flat manifolds

In this section, we will work out the generalized Chern-Gauss-Bonnet integral for-mula for locally conformally flat n dimensional manifolds with n = 2m even. Recallfirst the definition of locally conformally flat manifolds with simple ends.

Definition 3.13 Suppose that (M, g) is a complete non-compact locally conformallyflat n dimensional manifold such that

M = N ∪ l⋃

k=1

Ek,

where (N, g) is a compact locally conformally flat Riemannian manifold with boundary

∂N =l⋃

k=1

Ek,

and each end Ek is simple in the sense that

(Ek, gk) = (Rn\B1(0), e2ukg0)

for some function uk. Such manifolds will said to be complete locally conformally flatn dimensional manifolds with simple ends.

As main result of this section, we should apply the local version result of last sectionto show the following theorem,

Theorem 3.14 Suppose that (M, g) is a complete, locally conformally flat n dimen-sional manifold with simple ends. Let the number of those ends be equal to l, and assumethe following conditions,

(a). the scalar curvature is non-negative at infinity on each end;

(b). the Qn curvature is absolutely integrable.

Then, we have the following formula,

χ(M)− 2

(n− 1)!ωn

∫MQn dvg =

l∑k=1

µk, (3.72)

where

µk = limr→∞

[∫∂Br(0) e

(n−1)uk dσ]nn−1

n(ωn)1

n−1∫Br(0)\B1(0) e

nuk dx. (3.73)

Remark. It follows from the definition of a locally conformally flat, complete man-ifold with simple ends that

M = N ∪ l⋃

k=1

Ek.

Since (M, g) is locally conformally flat, in locally coordinates, the Qn curvature canbe simply defined by


Qn = [(−∆)n2 u] · e−nu,

where g0 is flat metric on Rn.

Proof. First of all, the well-known Chern-Gauss-Bonnet formula (see [31]) for com-pact manifold N with boundary ∂N is given by,

χ(N) =n!

πmm!22m

∫NKn dvg +

1

πmm!22m

∫∂N

η∗Φ, (3.74)

where η is the inward normal rather than outward normal and Φ is n− 1 form on theunit tangent bundle of N such that

dΦ = n!Kndvg. (3.75)

By Thm 0.2 of T. Branson, P. Gilkey and J. Pohjanpelto [3], there exists a constantc 6= 0 and an n− 1 form Ψ such that,

Qndvg = cKndvg + dΨ. (3.76)

Notice that our Qn curvature do satisfy the condition of their theorem 0.2 in [3].Therefore, combine the equations (3.74) and (3.76) and Stokes theorem to obtain,

χ(N)

=n!

πmm!22m

∫NKn dvg +

1

πmm!22m

∫∂N

η∗Φ

=n!

πmm!22m

∫NQn dvg +

∫∂N

n!

πmm!22mη∗Φ− 1

πmm!22mΨ. (3.77)

Since this is true for all locally conformally flat manifolds, in particular, it holdstrue for Sn. Hence, we can conclude that the constant c satisfies the following equality,

n!

πmm!22m=

2

(n− 1)!ωn. (3.78)

Let us denote by Ω the n− 1 form

n!

πmm!22mη∗Φ− 1

πmm!22mΨ.

Notice that for our case, ∂N =⋃lk=1 ∂B1(0), and near the boundary ∂N of N , the

manifold looks like the product. Hence, it is natural to use cylindrical coordinates sothat Qne

nu can easily be expressed as the differential form as in above formula (3.74).To do so, let us denote the function u+ log |x| by v near the boundary of ∂N . It is nothard to see that we still have,

(−∆)n2 v = Qn(y)env. (3.79)

On each end, the standard metric can be written as

ds2 = e2t(dt2 + ds20)

in the cylindrical coordinates. The advantage to use the cylindrical coordinates is that,on one hand, the normal vector field along ∂B1(0) is nothing but ∂

∂t ; on the other hand,the Laplace operator has a natural decomposition in normal direction,

3.4 Chern-Gauss-Bonnet formula for conformally flat manifolds 35

(−∆) = −e−nt ∂∂t

[e(n−2)t ∂

∂t

]+ e−2t(−∆0), (3.80)

where (−∆0) is the Laplace operator with respect to the round metric on Sn = ∂B1(0).Therefore, we have

[(−∆)n2 v]entdt ∧ dθ1 ∧ · · · ∧ dθn−1 = de(n−2)t[(−∆)

n2−1v]dθ1 ∧ · · · ∧ dθn−1,(3.81)

where dθ1 ∧ · · · ∧ dθn−1 is the volume form on Sn = ∂B1(0).

Hence, in cylindrical coordinates, we have

Qndvg = de(n−2)t[(−∆)n2−1v]dθ1 ∧ · · · ∧ dθn−1. (3.82)

It follows from (3.82) and the definition of Ω that

dΩ =2

(n− 1)!ωnde(n−2)t[(−∆)

n2−1v]dθ1 ∧ · · · ∧ dθn−1 (3.83)

Therefore, by Thm 0.3 of T. Branson, P. Gilkey and J. Pohjanpelto [3], there existsan n− 2 form Λ such that

Ω =2

(n− 1)!ωne(n−2)t[(−∆)

n2−1v]dθ1 ∧ · · · ∧ dθn−1+ dΛ. (3.84)

Now on each end, ∂B1(0) is a closed manifold without boundary, by Stokes theorem,we obtain ∫

∂B1(0)Ω = [(−∆)

n2−1v]dθ1 ∧ · · · ∧ dθn−1, (3.85)

with t = 0.

Therefore, combining (3.81) and (3.85),

χ(N)

=2

(n− 1)!ωn

∫NQn dvg +

2

(n− 1)!ωn

∫N

∂

∂t[(−∆)

n2−1v]|t=0 dσ. (3.86)

The key point here is that for each end, we have∫∂B1(0)

∂

∂t[(−∆)

n2−1v]|t=0 dσ

=

∫∂B1(0)

∂[(−∆)n2−1u]

∂rdσ + b(n)(n− 2)ωn−1, (3.87)

where e−t ∂∂r and en−1t = rn−1.

Therefore, Theorem 3.10 implies that

limr→∞

∫∂Br(0)

∂[(−∆)n2−1uk]

∂r− b(n)(2− n)r1−ndσ = µk

(n− 1)!ωn2

. (3.88)

Then it follows from those observations that

χ(M) = χ(N) =2

(n− 1)!ωn

∫MQn dvg +

l∑k=1

µk.


2

Finally, we have generalized the Chern-Gauss-Bonnet formula to high even dimen-sional locally conformally flat Riemannian manifolds. Although we believe that Chern-Gauss-Bonnet formula holds true for a general open Riemannian, due to the complexityof the integrand of Chern-Gauss-Bonnet formula, it is hard to get those generalizations.

4

Controlling the number of ends by the mean curvature

The main result of this thesis is to find some conditions such that the manifoldM is embedded, and the key step is to find when there is only one end. Hence, in thischapter we will use the mean curvature to control the number of ends of manifolds withnon-negative scalar curvature. In the following three properties, we consider a completen dimensional Riemannian manifold. The following is a generalization from [21].

First we have the Sobolev inequality proved by Michael and Simon,

Lemma 4.1 Let Mn be a sub-manifold immersed in Rn+p. Then for any functionh ∈ C1

0 (M), we have

(

∫M|h|

nn−1dM)

n−1n ≤ C1(

∫M|∇h|dM + n

∫M|Hh|dM). (4.1)

Remark 4.2 By Castillon, etc, the constant C1 can be taken to

1/(nω1nn αn,p),

where the constant αn,p can be defined as follows:

Let E be an n-dimensional linear subspace of Rn+p. For any n-plane F ⊂ Rn+p, andthe orthogonal projection

q : F → E,

define a function KE(F ) = | det(q)|, where det(q) is taken in orthonormal basis of Fand E. Then the constant αn,p is defined by the following integral:

αn,p =1

vol(Gn,n+p)

∫Gn,n+p

KE(F )1ndE,

the integration is taken for the Haar measure of Gn,n+p. Since Gn,n+p is homogeniousand by the invariance of Haar measure, αn,p only depends on n, p.

Then we have a corollary of Lemma 4.1

Lemma 4.3 Let Mn be a sub-manifold immersed in Rn+p. Suppose that n‖H‖nC1 < 1where C1 is a constant in the above lemma. Then for any f ∈ C1

0 (M), we have

(

∫M|f |

2nn−2dM)

n−2n ≤ Cs

∫M|∇f |2dM, (4.2)

where Cs = ( C11−n‖H‖nC1

2(n−1)n−2 )2.

38 4 Controlling the number of ends by the mean curvature

Proof. For a function h as in the above lemma, by Holder inequality, one has∫M|hH|dM ≤ (

∫Mh

nn−1dM)

n−1n ‖H‖n. (4.3)

Hence, this inequality (4.3) and the above Lemma 4.1 imply:

(

∫Mh

nn−1dM)

n−1n ≤ C1

1− n‖H‖nC1

∫M|∇h|dM. (4.4)

Now for any ϕ ∈ C10 , we set h = ϕ

2(n−1)n−2 . Thus, by (4.4) we get:

(

∫Mϕ

2nn−2dM)

n−1n ≤ C1

1− n‖H‖nC1

2(n− 1)

n− 2

∫M|ϕ

nn−2∇ϕ|dM

≤ C1

1− n‖H‖nC1

2(n− 1)

n− 2(

∫M|ϕ|

2nn−2 )

12 (

∫M|∇ϕ|2dM)

12 .

Thus the lemma is proved. 2

The next lemma is to get the volume control from below which will be useful in thecourse of proof of our main results in this chapter.

Lemma 4.4 Suppose that Mn(n ≥ 3) is a complete noncompact immersed hyper-surface in Rn+1. Assume that nC1‖H‖n < 1, where C1 is again the constant givenin Lemma 4.1. Then

Vol (B(q, s)) ≥ sn(1−nC1‖H‖n), (4.5)

for any q ∈M , and all s ≥ 0.

Proof. Take an arbitrary point p ∈ M , without loss of generality, we may assumep = 0 ∈ Rn+1. In the following we let d(·, ·) be the distance function of Rn+1, and r(·, ·)the distance function of M with respect to the induced metric. We will write d(x), r(x)if the base point is 0. Obviously d ≤ r for any two points in M . Let γ be a minimalgeodesic from 0, then

∂d

∂r= lim

t→0

d(γ(s+ t))− d(γ(s))

t

≤ limt→0

d(γ(s+ t), γ(s))

t( by the triangle inequality)

≤ 1. (since d ≤ r). (4.6)

By a direct computation, one can show that

∆Md2(x) = 2n(1 +H〈η, x〉),

where η is out unit normal to the hyper-surface M and x is the position vector in Rn+1.In particular, |〈η, x〉| ≤ d(x) ≤ r(x).

Let B(s) be the geodesic ball of M , of radius s centered at 0. Integrating the aboveequation over B(s) and using (4.6) and Holder inequality, we obtain

2n vol(B(s)) ≤ 2s vol(∂B(s)) + 2n(

∫B(s)|H|ndµ)

1n s(vol(B(s)))

n−1n .

39

Since the Sobolev inequality (4.4) holds on M , we have the iso-perimetric inequality,namely

1− n‖H‖nC1

C1(vol(B(s)))

n−1n ≤ vol(∂B(s)). (4.7)

And also note that in any manifold,

vol(∂B(s)) =∂

∂rvol(B(r)) |r=s .

We thus obtain

∂

∂r|r=s ln(vol(B(r)))− n(1− nC1‖H‖n)

s(1− nC1(‖H‖n − (∫B(s) |H|ndµ)

1n ))

≥ 0.

Therefore, by integrating it over the interval (0, s) and taking the exponential to get

vol(B(s)) ≥ sn(1−nC1‖H‖n),

by observing that ‖H‖n − (∫B(s) |H|

ndµ)1n ≥ 0. 2

The following lemma is about the ends of a manifold and harmonic function on themanifold.

Lemma 4.5 Let M be a complete and noncompact n-dimensional immersed hyper-surfaces in Rn+1 satisfying n‖H‖nC1 < 1 where C1 is a constant given in the abovelemma. If M has at least two ends , then M admits a nonconstant bounded harmonicfunction with finite energy.

Proof. The proof is analogy to the proof of Lemma 2 in [4]. We will provide theargument here for completeness of this chapter. We first prove that for each compact setK ⊂M , every noncompact component F of M \K has infinite volume. Suppose Vol(F )were finite. By the fact that lim

s→∞sn(1−n‖H‖nC1) = ∞ , there would exist a sufficiently

large s0 such that

sn(1−n‖H‖nC1)0 > Vol(F ).

Choosing a point x0 ∈ F so that r(x0, ∂F ) > s0 would lead to

Vol(F ) ≥ Vol(Bx0(s0)) ≥ sn(1−nC1‖H‖n)0

which is a contradiction. Hence Vol(F ) =∞.Next let M be covered by an exhaustion Di, a collection of relatively compact

sub-manifolds with boundary, for example, take Di = B(0, i) ∩M where B(0, i) is the

ball in Rn+1 with radial i and center 0. Let M \Di = ∪sj=1F(i)j be the disjoint union of

connected components with s ≥ 2. Fix an i0 and let F(i0)1 and F

(i0)2 be any two ends,

then each has infinite volume. For each i ≥ i0, let ui : Di → R be the minimizer of

the energy functional1

2

∫Di

|dui|2dv among all functions u such that u|∂F

(i)1

= 1 and

u|∂F

(i)k

= 0 for each k ≥ 2.

Then by the maximum principle for harmonic functions, 0 ≤ ui ≤ 1. For anyj < i, we extend uj to uj : Di → R continuously such that uj = 1 or 0 on the


complement Di −Dj . Then uj has the same boundary condition as ui on ∂Di. Henceby the minimality of the energy E(ui) of ui over Di, one has the following monotonicity:∫

Di

|∇ui|2dv ≤∫Di

|∇uj |2dv =

∫Dj

|∇uj |2dv for i > j.

Thus there exists a constant c1 > 0 such that∫Di

|∇ui|2dv ≤ c1 for i > i0.

Therefore, we can find a harmonic function u on M such that

limi→∞

ui(x) = u(x), ∀x ∈M,

0 ≤ u ≤ 1 and

∫M|∇u|2dv ≤ c1.

Since n > 2, we substitute f = ui(1− ui), in the inequality (4.2)(∫Di

(ui(1− ui))2nn−2 dv

)n−2n

≤ Cs∫Di

2|∇ui|2(1− ui)2 + 2u2i |∇ui|2dv

≤ 4Cs

∫Di

|∇ui|2 ≤ 4C2s c1. (4.8)

Since Vol(Di) → ∞, by letting i → ∞, we find that if u is a constant, then u ≡ 0or u ≡ 1. If u ≡ 1, we choose φ = uiψ where

ψ =

1, on F

(i0)2 ,

0, on F(i0)k , k 6= 2,

|∇ψ| ≤ c2, 0 ≤ ψ ≤ 1, and |∇ψ| vanishes outside Di0 , then inequality (4.2) implies that(∫Di

(uiψ)2nn−2 dv

)n−2n

≤ Cs∫Di

2ψ2|∇ui|2 + 2u2i |∇ψ|2dv ≤ c3, (4.9)

where the constant c3 = 2C2s c1 + 2c2

sc22 ·Vol(Di0). It follows that(∫

F(i0)2 ∩Di

u2nn−2

i dv

)n−2n

≤ c3.

As i → ∞, we find that Vol(F(i0)2 )

n−2n =

(∫F

(i0)2

u2nn−2 dv

)n−2n

≤ c3, a contradiction.

Similarly, u ≡ 0 can not happen by replacing u and ui by 1−u and 1−ui, respectively insame argument. Consequently u is not a constant. This completes the proof of Lemma2. 2

With this Lemma 4.5, we have the following result,

41

Theorem 4.6 Let Mn(n ≥ 3) be a complete sub-manifold immersed in Rn+1 withnon-negative scalar curvature. If

(

∫M|H|ndv)

1n < C2, (4.10)

where

C2 =1

C1

( √n(n− 2)

2√

2(n− 1)2 + n√n(n− 2)

),

then M has only one end.

Proof. We argue by contradiction. By the construction of Lemma 4.5, we know thatif M is of more than one end, then there exists a nontrivial bounded harmonic functionu(x) on M which has finite total energy.

For such a harmonic function u, let f(x) = |∇u|. By Bochner’s formula, we obtain:

1

2∆f2 = |Hess u|2 +Ric(∇u,∇u). (4.11)

Next we prove the following inequality,

|Hess u|2 ≥ (1 +1

n− 1)|∇f |2. (4.12)

If |∇u| = 0, the above inequality of course holds. If |∇u| 6= 0, for any p ∈ M wechoose a normal coordinate around p such that ui(p) = 0 (i ≥ 2) and u1(p) = |∇u|(p).Since u is harmonic, we have

u11 = −(∑i 6=1

uii).

Then at p,

fj =uijui|∇u|

=u1ju1

u1= u1j .

Hence,|∇f |2 = u2

1j .

We can calculate that

|Hess u|2 − |∇f |2

= u2ij − u2

1j

≥∑i 6=1

u2i1 +

∑i 6=1

u2ii

≥∑i 6=1

u2i1 +

1

n− 1(∑i 6=1

uii)2

=∑i 6=1

u2i1 +

1

n− 1u2

11

≥ 1

n− 1

n∑i=1

u2i1

=1

n− 1|∇f |2,


since Hessian is symmetric. Therefore, we get (4.12)

|Hess u|2 ≥ (1 +1

n− 1)|∇f |2.

Apply F. Leung’s curvature estimate in [18] with k = 0 to get

Ricmin ≥1

n22(n− 1)n2H2 − n(n− 1)|B|2

−(n− 2)n|H|√

(n− 1)(n|B|2 − n2H2). (4.13)

Since R = n2H2− |A|2 is non-negative, we plug this into the above estimate (4.13),

Ricmin ≥1

n22(n− 1)n2H2 − n(n− 1)|B|2

−(n− 2)n|H|√

(n− 1)(n|B|2 − n2H2)

≥ 1

n22(n− 1)n2H2 − n(n− 1)(n2H2 −R)

−(n− 2)n|H|√

(n− 1)n[(n2H2 −R)− nH2]

≥ 1

n22(n− 1)n2H2 − n(n− 1)n2H2

−(n− 2)n|H|√

(n− 1)n(n2H2 − nH2)≥ −2(n− 1)(n− 2)H2.

i.e.Ricmin ≥ −2(n− 1)(n− 2)H2. (4.14)

With help of this estimate (4.14), Bochner’s formula takes the form,

f∆f + 2(n− 1)(n− 2)H2f2 ≥ 1

n− 1|∇f |2. (4.15)

Now let ϕ be a cut-off function such that

ϕ(x) =

1, if x ∈ Bp(r),0, if x ∈M \Bp(2r),

and

|∇ϕ| ≤ C

rwith C = 2.

Multiplying ϕ2 on both sides of the above inequality (4.15) and integrating by partswe can write it as

2(n− 1)(n− 2)

∫MH2f2ϕ2dσ − 2

∫M〈∇f,∇ϕ〉fϕdσ

≥ n

n− 1

∫M|∇f |2ϕ2dσ.

Using Schwartz inequality, for any positive number δ1 > 0, we have

2(n− 1)(n− 2)

∫MH2f2ϕ2dv +

1

δ1

∫Mf2|∇ϕ|2dv

≥(

n

n− 1− δ1

)∫M|∇f |2ϕ2dv. (4.16)

43

On the other hand, Sobolev inequality yields∫M|∇(fϕ)|2dv ≥ C−1

s

(∫M

(fϕ)2nn−2 dv

)n−2n

.

Simple calculation, together with Schwartz inequality, yields

(δ2 + 1)

∫M|∇f |2ϕ2dv (4.17)

≥ C−1s

(∫M

(fϕ)2nn−2 dv

)n−2n

− (1 +1

δ2)

∫Mf2|∇ϕ|2dv,

where δ2 is a positive real number which will be chosen later. Combining (4.16) and(4.17), we have

2(n− 1)(n− 2)

∫MH2f2ϕ2dv

≥( nn−1 − δ1)C−1

s

δ2 + 1

(∫M

(fϕ)2nn−2 dv

)n−2n

−(

1

δ1+

nn−1 − δ1

δ2

)∫f2|∇ϕ|2dv. (4.18)

Now applying Holder inequality to the left hand side of the above inequality we canhave

2(n− 1)(n− 2)

(∫M|H|ndv

) 2n(∫

M(fϕ)

2nn−2 dv

)n−2n

≥( nn−1 − δ1)C−1

s

δ2 + 1

(∫M

(fϕ)2nn−2 dv

)n−2n

−(

1

δ1+

nn−1 − δ1

δ2

)∫Mf2|∇ϕ|2dv.

Finally we have(1

δ1+

nn−1 − δ1

δ2

)∫Mf2|∇ϕ|2dv

≥

(( nn−1 − δ1)C−1

s

δ2 + 1− 2(n− 1)(n− 2)‖H‖2n

)(∫M

(fϕ)2nn−2 dv

)n−2n

.

Recall that Cs = ( C11−n‖H‖nC1

2(n−1)n−2 )2, thus if we choose

C2 =1

C1

( √n(n− 2)

2√

2(n− 1)2 + n√n(n− 2)

), (4.19)

then it is easy to see that if ‖H‖n < C2, then ‖H‖2nCs < n2(n−1)2(n−2)

. Thus we can

choose δ1 > 0 and δ2 > 0 small enough such that(( nn−1 − δ1)C−1

s

δ2 + 1− 2(n− 1)(n− 2)‖H‖2n

)≥ ε > 0.


Then we have(1

δ1+

nn−1 − δ1

δ2

)∫Mf2|∇ϕ|2dv ≥ ε

(∫M

(fϕ)2nn−2 dv

)n−2n

.

Letting r →∞ we will have ∫Mf

2nn−2 dv ≤ 0,

which implies that f ≡ 0 and therefore u is a constant function. The contradiction hereshows that M has at most one end.

5

Mean curvature and embedding

With the preparation of the previous chapters, in this chapter, we will study thegeometry and topology of locally conformally flat four dimensional manifolds from anextrinsic point of view. To be more precise, we will give the following theorem,

Theorem 5.1 Let M → R5 be a complete, simply connected, noncompact, locally con-formally flat hypersurface immersed into R5 with 16H2 − |A|2 to be non-negative and∆(16H2 − |A|2) ∈ H1(M). Assume that

(

∫M|H|4dv)

14 < C2, (5.1)

where

C2 =1

13C1.

Then M is embedded.

From this theorem, we can see that if the total mean curvature of the immersedlocally conformally flat four dimensional hyper-surface is bounded by a constant whichis independent of the hyper-surface and conditions on the second fundamental form,then the hyper-surface is actually an imbedded hyper-surface. This result show somerelations between the geometry and the topology of some kind of manifolds.

5.1 Decomposition of the conformal factor

In this section, we want to obtain the Chern-Gauss-Bonnet formula on locally con-formally flat open four dimensional manifolds related to the ends of the manifolds andfind a decomposition of the conformal factor u.

Let M be a complete, connected, non-compact, oriented, locally conformally flat fourdimensional manifold immersed in R5, i.e. M is a locally conformally flat hyper-surfaceof R5.

Choose a conformal parametrization

f : Ω∗ → Σ ⊂M,

whereΩ∗ = x ∈ R4, |x| > 1,

46 5 Mean curvature and embedding

and Σ is a neighborhood of an end of M . Under this conformal parametrization, wedenote the conformal metric by g = e2ug0, where g0 = |dx|2 is the standard flat metricon Euclidean space R4. First let us recall the definition of a conformal metric to be anormal metric.

The next definition of normal metric is the same as in [7].

Definition 5.2 The conformal metric e2ug0 on Ω∗ satisfying that the Q4 curvature isabsolutely integrable is said to be normal on Ω∗ if the following decomposition holds,

u(x) =1

8π2

∫Ω∗

log|y||x− y|

·Q4(y)e4u(y) dy + α log |x|+ h(x), (5.2)

where α is some constant and h( x|x|2 ) is some biharmonic function on the unit ball B.

With the above definition, for the conformal metric of a locally conformally flathyper-surface in R5 around each end, we can show that under some assumption onthe mean curvature and the second fundamental form, the conformal metric is normalaround each end. Actually, here we can show that u has a better decomposition.

Since 0 ≤ R = 16H2 − |A|2, we have 16H2 ≥ |A|2. Also,

16H2

= (λ1 + λ2 + λ3 + λ4)2

= λ21 + λ2

2 + λ23 + λ2

4 + 2(λ1λ2 + λ1λ3 + λ1λ4 + λ2λ3 + λ2λ4 + λ3λ4)

≤ 4(λ21 + λ2

2 + λ23 + λ2

4)

= 4|A|2.

Therefore,4H2 ≤ |A|2 ≤ 16H2.

Hence,|H| ∈ L4(M)⇐⇒ |A| ∈ L4(M).

Suppose Ω4 is the standard volume form of S4, then by Theorem 26.2.2 in [10],

K4 dσ = G∗(Ω4),

where dσ is the volume form of M and K4 is the Gaussian curvature of M . Hence,

K4e4u dx1 ∧ dx2 ∧ dx3 ∧ dx4 = G∗(Ω4).

Given the condition that |H| ∈ L4(M), we have |A| ∈ L4(M), so the Gauss mapG ∈W 1,4(M); then K ∈ H1(M).

If ∆R = ∆(16H2−|A|2) ∈ H1(M), then Q4 ∈ H1(M). Hence, we have the followingtheorem

Theorem 5.3 Suppose |H| ∈ L4(M), ∆(16H2 − |A|2) ∈ H1(M), and R = 16H2 −|A|2 ≥ 0. In local coordinates (Ω∗, e2ug0), the following decomposition holds,

u(x) = u0(x) + α log |x|+ h(x),

wherelimx→∞

u0(x) = 0,

and h is a biharmonic function on Ω∗ ∪ ∞.

5.1 Decomposition of the conformal factor 47

Proof. Consider the Q4 curvature equation

∆2u = Q4e4u. (5.3)

Similar to the argument of Theorem 3.2.1 in [25], equation (5.3) has a solution u0

such thatlimx→∞

u0(x) = 0, ∆u0 = o(r−2),

as r →∞.

If we set w = u−u0, then ∆2w = 0. Hence, ∆w is harmonic. By mean value property,∀x0 ∈ Ω∗, and ∀ r > 0, s.t. Br(x0) ⊂ Ω∗, i.e. r ≤ |x0| − 1, we have,

(∆w)(x0) = −∫Br(x0)

(∆w)(y)dy

=4

r−∫∂Br(x0)

∂w

∂r(y)dy

=4

r

∂

∂r−∫∂Br(x0)

w(y)dy (5.4)

Multiple (5.4) by r4 on both sides and integral on r to get,

r2

8(∆w)(x0) = −

∫∂Br(x0)

w(y)dy − limr→0−∫∂Br(x0)

w(y)dy

= −∫∂Br(x0)

w(y)dy − w(x0),

i.e,r2

8(∆w)(x0) + w(x0) = −

∫∂Br(x0)

w(y)dy. (5.5)

Multiple (5.5) by 4r3 on both sides and integral on r, we get

r6

12(∆w)(x0) + r4w(x0) =

1

ω4

∫Br(x0)

w(y)dy. (5.6)

Divide both sides of (5.6) by r4, we have

r2

12(∆w)(x0) + w(x0) = −

∫Br(x0)

w(y)dy.

Set

P (r) =r2

12(∆w)(x0) = −

∫Br(x0)

w(y)dy − w(x0). (5.7)

|( ddr−∫∂Br(x0)

u0(x)dσ)2 +−∫∂Br(x0)

∆u0(x)dσ|

≤ |( ddr−∫∂Br(x0)

u0(x)dσ)|2 + |−∫∂Br(x0)

∆u0(x)dσ|

≤ C11

r2, (5.8)


where C1 is a constant.

By the definition of P (r) and w,

−∫∂Br(x0)

u(x)dσ

= u(x0) +−∫∂Br(x0)

u0(x)dσ − u0(x0) + P (r).

Therefore,

[d

dr−∫∂Br(x0)

u(x)dσ]2 +−∫∂Br(x0)

∆u(x)dσ

= [d

dr−∫∂Br(x0)

u0(x)dσ]2 +−∫∂Br(x0)

∆u0(x)dσ

+∆P (r) + [d

drP (r)]2 + 2

d

drP (r) · [ d

dr−∫∂Br(x0)

u0(x)dσ]. (5.9)

Let us consider the scalar curvature equation under the conformal change g = e2ug0

on Ω∗,

∆u+ |∇u|2 = −R6e2u.

Since R = 16H2 − |A|2 ≥ 0, by Cauchy inequality, we have

[d

dr−∫∂Br(x0)

u(x)dσ]2 +−∫∂Br(x0)

∆u(x)dσ

= −∫∂Br(x0)

∆u(x)dσ + [−∫∂Br(x0)

d

dru(x)dσ]2

≤ −∫∂Br(x0)

∆u(x) + |∇u|2dσ

= −−∫∂Br(x0)

R

6e2udσ

≤ 0. (5.10)

Hence, by (5.8), (5.9) and (5.10), we get

r2(∆P (r) + [d

drP (r)]2) ≤ C2, (5.11)

where the constant C2 is independent of x0 and r.

By (5.7), we can calculate ∆P (r) = 23(∆w)(x0) and d

drP (r) = r6(∆w)(x0). If we

choose r = |x0|2 , then

|x0|2|(∆w)(x0)| ≤ C, (5.12)

where C is a constant independent of x0.

Let h(x) = w−α log |x|, it is easy to see that ∆2h(x) = 0. Take the spherical integralof the equation

∆2w = 0

for r ≥ 1,

5.1 Decomposition of the conformal factor 49

0 =

∫Br(0)−B1(0)

∆2w(y)dy

=

∫∂(Br(0)−B1(0))

d

dr(∆w(y))dσ

= ω3r3 d

dr−∫∂Br(0)

∆w(y)dσ −∫∂B1(0)

d

dr(∆w(y))dσ.

Henced

dr−∫∂Br(0)

∆w(y)dσ =1

ω3r3

∫∂B1(0)

d

dr(∆w(y))dσ. (5.13)

Integrate (5.13) on r on both sides to get,

−∫∂Br(0)

∆w(y)dσ −−∫∂B1(0)

∆w(y)dσ

= − 1

2ω3r2

∫∂B1(0)

d

dr(∆w(y))dσ +

1

2ω3

∫∂B1(0)

d

dr(∆w(y))dσ,

i.e.,

−∫∂Br(0)

∆w(y)dσ

= − 1

2ω3r2

∫∂B1(0)

d

dr(∆w(y))dσ

+[1

2ω3

∫∂B1(0)

d

dr(∆w(y))dσ −−

∫∂B1(0)

∆w(y) dσ]. (5.14)

By the estimate of ∆w, i.e. (5.12), we get the constant term of the above equality(5.14) should vanish. By direct calculation,

∆h = ∆w − 2α

r2.

Therefore, if we choose

α = − 1

4ω3

∫∂B1(0)

d

dr(∆w(y))dσ,

we obtain that

−∫∂Br(0)

∆hdσ = 0.

By the estimate of ∆w i.e. (5.12), we have

|x0|2|(∆h)(x0)| ≤ |x0|2|(∆h)(x0) +α

|x0|2|+ |α| ≤ C3.

Since |x0| ≥ 1,|(∆h)(x0)| ≤ C3,

where C3 is a constant.

Hence |∆h| is bounded on the neighborhood of ∞. Since ∆h is harmonic, by Thm4.10 in [1], ∆h is harmonic at infinity, so h is biharmonic on Ω∗ ∪∞. Therefore, finallywe get


u(x) = u0(x) + α log |x|+ h(x),

where limx→∞ u0(x) = 0, α is a constant and h is a biharmonic function on Ω∗∪∞. 2

By the decomposition of the conformal factor, we can show

Theorem 5.4 With the conformal parametrization

f : Ω∗ → R4.

we have the following limit,

limx→∞

|f(x)||x|m

=eλ

m,

whereλ = lim

x→∞h(x),m = α+ 1.

Proof. By the reflection with respect to the unit sphere, we map Ω∗ to the unitball B and take the the Taylor expansion of f = (f1, f2, f3, f4, f5) around 0,

f(x) = 〈fxi , x〉+O(|x|).

Since 〈fxi , fxj 〉 = 0, if i 6= j, and |fx1 | = |fx2 | = |fx3 | = |fx4 | = eu, we have〈f, fx1〉 = x1|fx1 |2 +O(|x|), and |f |2 = |x|2|fx1 |+O(|x|). Then by L’Hospital Law withrespect to x1 and by the decomposition of u, i.e. u(x) = u0(x) +α log |x|+ h(x), where

limx→∞

u0(x) = 0,

we can calculate that,

limx→0

|f ||x|α+1

= limx→0

(〈f, fx1〉|f |

)/[(α+ 1)|x|α x1

|x|]

= limx→0

(x1|fx1 |2

|x||fx1 |)/[(α+ 1)|x|α x1

|x|]

= limx→0

eu0(x)+h(x)|x|α

(α+ 1)|x|α

=eλ

α+ 1.

Hence, if we letλ = lim

x→∞h(x),m = α+ 1,

we have the formula

limx→∞

|f(x)||x|m

=eλ

m.

2

This in some sense indicate that f behaves like xm, if we consider R4 as the quater-nions H. Using this, we can apply the covering map

5.2 Immersion and Embedding 51

x→ xm

to make any end of the manifold M to be a simple end.

By Theorem 1.2 of [7], we have the Chern-Gauss-Bonnet formula on M ,

χ(M)− 1

8π2

∫MQ4 dvM =

k∑i=1

vi,

where

vi = limr→∞

[∫∂Br(0) e

3uidσ]43

4(2π2)13

∫Br(0)\B e

4uidx

is the isoperimetric ratio on each end.

By Theorem 5.4,

vi = limr→∞

[∫∂Br(0) e

3uidσ]43

4(2π2)13

∫Br(0)\B e

4uidx

= limr→∞

[∫∂Br(0) e

3(ui0+αi log |x|+hi(x))dσ]43

4(2π2)13

∫ r1

∫∂Bs(0) e

4(ui0+αi log |x|+hi(x))dxds

= limr→∞

[∫∂Br(0) |x|

3αidσ]43

4(2π2)13

∫ r1

∫∂Bs(0) |x|4αidxds

= limr→∞

r4(1+αi)(2π2)43

4(2π2)13

∫ r1

∫∂Bs(0) |x|4αidxds

= limr→∞

2π24(1 + αi)r3+4αi

4∫∂Br(0) |x|4αidx

= limr→∞

2π2(1 + αi)r3+4αi

2π2r3+4αi

= 1 + αi = mi.

Hence, the Chern-Gauss-Bonnet formula can be written as

χ(M)− 1

8π2

∫MQ4dvM =

k∑i=1

mi.

2

5.2 Immersion and Embedding

In [25], Muller and Sverak made some assumption on the normal of the secondfundamental form, and they proved the following results


Theorem 5.5 Let M → Rn be a complete, connected, non-compact surface immersedinto Rn. Assume that either ∫

M|A|2 dvM < 8π, n = 3,

or ∫M|A|2 dvM ≤ 4π, n ≥ 4.

Then M is embedded.

In this chapter, we will generalize this result to locally conformally flat four dimen-sional Riemannian manifolds, which immersed into R5. With the preparations of theprevious chapters, we can prove the main result of this thesis.

Proof of Theorem 5.1. Since the scalar curvature R = 16H2−|A|2 is non-negativeand

(

∫M|H|4dv)

14 < C2,

by Theorem 4.6, the manifold has only one simple end, i.e. the total number of endsm = 1. Then by the decomposition

u(x) = u0(x) + (m− 1) log |x|+ h(x),

wherelimx→∞

u0(x) = 0, h(x) biharmonic at infinity,

we obtain that eu = |fxi | is bounded on Ω∗, and f behaves like x at infinity. By Theorem1.6 and Theorem 3.5 in Chapter 6 in [29], there exists an injective developing map

Φ : M → S4,

since the scalar curvature R ≥ 0. Therefore, M is conformally equivalent to R4. Hence,we can extend the conformal parametrization f to the whole manifold, i.e.,

f : R4 →M.

eu = |fxi | is also bounded on the compact set R4 \ Ω∗. Therefore, for any two pointsx, y ∈ R4, we have

|f(x)− f(y)| ≈ C|x− y|.

Hence, there are no intersection points and for any open set V in M , we can find anopen set U in R5 such that V = M ∩ U . Therefore, M is embedded into R5. 2

6

Conclusions and further work

From the above, only the co-dimensional one case has been studied. Similar tothe surface case, we can also study higher co-dimensional case. For this higher co-dimensional case, we can follow the method in [25]. However, the difficulty is we do notknow which canonical form on G4,n can be pulled back to be Q4 dv, where Q4 is theQ4 curvature, and dv is the volume form of M . And another difficulty is that we donot know how to control the number of ends. If we can find such a form and a way todeal with the number of ends, we can do similar things as in [25].

54 6 Conclusions and further work

Appendices

55

57

In Euclidean space Rn, for a function f ∈ C2(Rn), and a ball Br(x0) ⊂ Rn whichcentered at the point x0 ∈ Rn with radius r, its boundary is a sphere ∂Br(x0). We havethe following differential formula:

(1).d

dr−∫∂Br(x0)

fdσ = −∫∂Br(x0)

df

drdσ,

(2). ∆(−∫∂Br(x0)

gdσ) = −∫∂Br(x0)

∆g(x)dσ.

Proof. (1). By changing variables, we get

−∫∂Br(x0)

df(y)

drdσy

=1

ωn−1rn−1

∫|ω|=1

d

drf(x0 + rω)rn−1dσω

=1

ωn−1

d

dr

∫|ω|=1

f(x0 + rω)dσω

=1

ωn−1

d

dr

∫∂Br(x0)

f(y)r1−ndσy

=d

dr−∫∂Br(x0)

fdσ,

i.e.,d

dr−∫∂Br(x0)

fdσ = −∫∂Br(x0)

df

drdσ.

(2). Let

g(r) = −∫∂Br(x0)

g(x)dσ,

then, by (1) and divergence theorem,

g′(r) =

d

dr−∫∂Br(x0)

g(x)dσ

= −∫∂Br(x0)

dg

drdσ

=1

ωn−1rn−1

∫Br(x0)

∆g(x)dx.

Hence,

g′′(r) =

d

dr[

1

ωn−1rn−1

∫Br(x0)

∆g(x)dx]

= − n− 1

ωn−1rn

∫∂Br(x0)

dg

drdσ +

1

ωn−1rn−1

∫∂Br(x0)

∆g(x)dσ

= −n− 1

r−∫∂Br(x0)

dg

drdσ +−

∫∂Br(x0)

∆g(x)dσ

= −n− 1

rg′(r) +−

∫∂Br(x0)

∆g(x)dσ,

58

i.e.,

−∫∂Br(x0)

∆g(x)dσ = g′′(r) +

n− 1

rg′(r).

In spherical coordinates of Rn, for a function h,

∆h =1

rn−1

d

dr

(rn−1dh

dr

)+

1

r2∆Sn−1h.

Therefore,

∆g(r) =1

rn−1

d

dr

(rn−1dg(r)

dr

)+

1

r2∆Sn−1 g(r)

=n− 1

r

dg(r)

dr+d2g(r)

dr2

= −∫∂Br(x0)

∆g(x)dσ.

Hence, we proved

∆(−∫∂Br(x0)

gdσ) = −∫∂Br(x0)

∆g(x)dσ.

2

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