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TRANSLATING TRIDENTS
XUAN HIEN NGUYEN
Abstract. In this paper, we describe the construction of new examples of
self-translating surfaces under the mean curvature flow. We find the new sur-faces by desingularizing the intersection of a grim reaper and a plane to obtain
approximate solutions, then we solve a perturbation problem to find the ex-
act solutions. Our work is inspired from Kapouleas’ construction of minimalsurfaces but differs from it by our more abstract and direct study of the linear
operator, via Fredholm operators.
Let X(·, t) : Mn → Rn+1 be a one-parameter family of immersions of n-dimensional smooth hypersurfaces into Rn+1. The family of surfacesMt = X(Mn, t)is a solution to the mean curvature flow if
∂
∂tX(p, t) = H(p, t), p ∈M, t > 0,
X(p, 0) = X0(p),(1)
where H(p, t) is the mean curvature vector of the surface Mt = X(M, t) at thepoint X(p, t) for some initial data given by the immersion X0.
The purpose of this paper is to describe the construction of new self-translatingsurfaces under the mean curvature flow. Self-translating surfaces do not changeshape and are translated by the mean curvature flow at constant velocity. Up torotation and scaling, they satisfy the equation
(2) H − e1 · ν = 0,
where H is the mean curvature, ν is a normal vector chosen so that H = Hν,and e1 is the unit vector 〈1, 0, 0〉 in cartesian coordinates in R3. The vector e1
characterizes the velocity of the translation. For our purpose, we will use a versionof the equation above rescaled by a factor 1/τ :
(3) H − τe1 · ν = 0,
where τ is a small constant. An example of self-translating curve is the grim reaperγ, given by the equation
γ = (x, z) ∈ R2 | τx = − log cos(τz).
A cylinder over the grim reaper γ×R is an example for n = 2. Altschuler and Wu [1]proved the existence of a paraboloid type self-translating surface in R3. Angenentand Velazquez [2] showed that self-translating surfaces can model the asymptoticbehavior at certain singularities of the mean curvature flow. In this paper, we
2000 Mathematics Subject Classification. Primary 53C44.Key words and phrases. mean curvature flow, self-translating, eternal solutions.This work is partially supported by the National Science Foundation, Grant No. DMS-0710701.
1
2 XUAN HIEN NGUYEN
provide new examples of self-translating surfaces, constructed by desingularizingthe intersection of a plane and a grim reaper cylinder (see Figure 1 ).
Theorem 1. There is a family of surfaces Mτ parametrized by τ ∈ (0, δτ ) withthe following properties:
(i) for τ ∈ (0, δτ ), Mτ is a complete embedded surface in R3 that is self-translating under mean curvature flow,
(ii) Two ends of Mτ decay exponentially to planes, and the two other endsdecay exponentially to a grim reaper cylinder,
(iii) as τ → 0, 1τMτ = ( 1
τ (x, y, z) | (x, y, z) ∈Mτ tends to the Scherk surfaceuniformly on compact subsets of R3.
Figure 1: Translating trident
Idea of the proof. The proof is inspired by the gluing techniques developed byKapouleas [8]. The strategy is similar to the one adopted by Kapouleas [9] andTraizet [14] to construct minimal surfaces. We will discuss how our approach differsfrom Kapouleas’ at the end of this section.
The first step in the proof is the explicit construction of an initial surface Σ[τ, b].The surface is found be replacing a neighborhood of the intersection of a grim reaperand a plane by a Scherk minimal surface Σ0 (see Section 2). We choose to work ona large scale using equation 3 so the Scherk surface is not rescaled, however, thegrim reaper cylinder is very large. In Section 3, we show that our initial surfaceΣ[τ, b] is a good approximate solution to (3) by estimating the quantity H − τe1 · νfor τ and b small.
To complete the construction, we find a function whose graph over Σ[τ, b] pro-vides an exact solution to (3). The problem involves solving a partial differentialfor graphs of functions over the whole initial surface Σ[τ, b]. The functions have todecay exponentially to preserve the behavior at the ends. The first step in Section4 is the study of the associated linear operator on pieces of the initial surface. Fromthis, we derive information on the linear operator on the whole surface. The hopeis to invert the linear operator on each piece separately, then glue the solutions to-gether to invert the linear operator globally. Unfortunately, some difficulties arisefrom the presence of small eigenvalues on the central piece (Scherk surface) andfrom the imposed exponential decay. Using the theory of Fredholm operators, we
TRANSLATING TRIDENTS 3
prove that the linear operator on the whole surface has a trivial kernel and a rangeof codimension 1. If we find a way to project any function to the range of thelinear operator, it will still be possible to invert it. For this purpose, we introducea function w that is shown not to be in the range; we can then project any functionto the range, by adding a well chosen multiple of w to it.
The artificial presence of this multiple of w has to be corrected within the con-struction itself. A dislocation or “unbalancing” of the Scherk surface can generate(or cancel) the function w. It means that the ends of the Scherk surface have to beslightly repositioned in our construction. In Section 5, we finish the proof by usinga fixed point theorem.
Our approach differs from Kapouleas’ construction for minimal surfaces in twopoints. First, Kapouleas’s minimal surfaces are asymptotic to catenoids, whichare not isometric to planes. He has to define a different metric for the behaviorat infinity and can not treat his initial surface as a simple perturbation of theoriginal Scherk surface Σ0 globally. In our case, the initial surfaces Σ[τ, b] arealmost isometric to the original Σ0. We can therefore treat the linear operatoron Σ[τ, b] as a small perturbation of a linear operator on the Scherk surface Σ0.Secondly, we use the theory of Fredholm operators to convert results for the linearoperator locally into information on the linear operator on the whole surface Σ[τ, b].It is a more abstract and more direct approach.
We believe the methods in this article can be adapted to desingularize the in-tersections of a configuration of grim reaper cylinders. Moreover, they can giveinsight into the construction of new self-similar surfaces under mean curvature flowby desingularizing the intersection of a cylinder and a plane. The study of thelinear operator on different pieces of an initial surface is done in [13] [12]; however,the global behavior of the linear operator is still not known. Self-similar surfacesare especially interesting because they model the behavior of the mean curvatureflow near its singularities if the blow up is fast [7].
The author would like to thank Sigurd Angenent for invaluable discussions aboutthis problem.
1. The Scherk surface and the Grim Reaper
1.1. Notations. Let us denote the half plane by H+ = (s, y) ∈ R2 | s ≥ 0.Let ei, i = 1, 2, 3 be the coordinate vectors in R3.We will use a smooth cut off function ψ[a, b], where ψ transitions from being 0 ata to being 1 at b.We work with weighted Holder spaces defined by the following norms:
Definition 2. The weighted Holder norms are defined by
‖φ : Ck,α(Ω, g, f)‖ := supx∈Ω
f−1(x)‖φ : Ck,α(Ω ∩B(x), g)‖,
where Ω is a domain, g is the metric with respect to which we take the Ck,α norm,f is the weight function and B(x) is the geodesic ball centered at x of radius 1.
1.2. Scherk surface. The Scherk minimal surface Σ0 is given by the equation
(4) Σ0 = (x, y, z) ∈ R3 | sin y = sinhx sinh z.
As x (z) goes to infinity, Σ0 tends exponentially to the xy-plane (yz-plane resp.).More precisely, we have the following lemma:
4 XUAN HIEN NGUYEN
Figure 2: Scherk surface
Lemma 3 (Proposition 2.4 [9]). For given ε ∈ (0, 10−3), there is a constant a =a(ε) > 0 and smooth functions σ : H+ → R and F : H+ → R3 with the followingproperties:
(i) F (s, y) = (σ(s, y), y, s+ a) ∈ Σ0,(ii) ‖σ : C5(H+, gH+ , e−s)‖ ≤ ε.
We call Σ0∩z ≥ a and Σ0∩z ≤ −a the top and bottom wings of the Scherksurface respectively.
The constant ε is chosen small enough, with implied adjustments if necessarythroughout the article. A small ε gives us the equivalence of all of the metrics g,g0 and gΣ considered on the wings of the initial surface (see Corollary 9). This inturn allows us to show that the different operators for the linear analysis in Section4 are just small perturbations of each others.
1.3. Grim Reaper. In the xz-plane, the classical grim reapers are given by
τx = − ln(cos(τz)), τ 6= 0.
for z 6= 0, and x = 0 if z = 0. When τ = 0, we just consider the line x = 0. Forfixed τ , the curve is smooth and a short computation shows it satisfies (3). Grimreapers are classical examples of self-translating curves under mean curvature flow.
The parametrization by arc length of the grim reapers in R3 is
~γτ (s) =(1τ
ln(
cosh(τs)), 0,
1τ
arctan(
sinh(τs)))
(5)
=: (γτ1(s), 0, γτ2(s)).
Let us define the grim reaper cylinders by
(6) Γτ = (γτ1(s), y, γτ2(s)), (s, y) ∈ R2.
1.4. Symmetries. All the surfaces considered throughout the article are invariantunder
- R1 = Rotation of 180 degrees about the x-axis,- R2 = Reflection with respect to the planes y = π
2 + kπ, k ∈ Z,- the action of the group G generated by P : y 7→ y + 2π.
TRANSLATING TRIDENTS 5
We will use the notation S for both the surface S and its quotient S/G.
2. Initial surface
The initial surface depends on the two parameters τ ∈ (−δτ , δτ ) and b ∈(−δb, δb), with δτ and δb to be determined.
The parameter τ keeps track of the scale at which we are working. At first glance,there are two possible approaches. One can solve the original equation (3) with agrim reaper Γ1 and a plane. This approach involves shrinking the Scherk surfaceby a factor τ before placing it at the intersection. We want the Scherk surface totake as little room as possible, so that the error generated for H − e1 · ν is small.As τ goes to 0, the curvature blows up, and the case τ = 0 is not well defined.It is therefore preferable to work on a larger scale. The second approach fixes theScherk surface and rescales the grim reaper by 1/τ (when τ → 0, the grim reaperapproaches a plane). We then construct solutions to equation (3), which could beshrunk to be solutions to (2). The number 1/τ is proportional to the number ofholes of the Scherk surface within a period 0 ≤ y ≤ 2π for the constructed solutionsof (2).
The role of the parameter b (and of the transformation Zb) is less evident. It isstrongly tied to the range of our linear operator. The linear operator we wish tosolve is not surjective, in fact, its range has codimension 1. To take care of this, weintroduce a function w that allows us to project any function down to the rangeof the linear operator. Of course, we have to be able to generate (or cancel) anymultiples of w. This is done in the construction through a change of the parameterb. The curious reader can read the discussion at the beginning of Section 4 for moredetails.
2.1. Definition of Σb and Zb. We first define a family of diffeomorphisms Zb :R3 → R3 parametrized by b ∈ (−2δb, 2δb), which rotate the top and bottom wingsof Σ0 by an angle b towards the positive x axis. The Zb’s satisfy the followingconditions
(i) Zb is the identity in the region (x, y, z) | |z| < 12 |x|,
(ii) in the region (x, y, z) | |z| > 2|x| and |z| > 1, Zb is a rotation by anangle b about the y-axis towards the positive x-axis,
(iii) Z0 is the identity.
2.2. Construction of the top wing. We describe the construction for the topwing; the bottom wing is obtained by using the symmetry with respect to R1. Thepicture below shows a cross section of the resulting surface.
The top wing of the surface Zb(Σ0) is asymptotic to a half plane given by thedirections (sin b)e1 + (cos b)e3 and e2, with boundary line Lb = (a sin b, y, a cos b).We attach a part of the grim reaper cylinder Γτ with the following steps:1a. If τ 6= 0, we find the line on Γτ along which the tangent plane to Γτ is parallelto the directions (sin b)e1 + (cos b)e3 and e2 ; then we move Γτ so that this line onΓτ matches Lb.1b. If τ = 0, we just take Γ0 to be the half plane asymptotic to Zb(Σ0) withboundary Lb.2. We now add the graph of the function σ from Lemma 3 to the grim reaper endΓτ , τ ∈ (−δτ , δτ ).3. We use a cut off function where 0 ≤ s ≤ 1 to render the final surface smooth. We
6 XUAN HIEN NGUYEN
Figure 3: Construction of the top wing
also cut off the graph of σ in the region 3δsτ−1 ≤ s ≤ 4δsτ−1, where the constantδs is chosen later.
To make the construction more precise, consider the following parametrization,which is just (6) under a well chosen translation:if τ 6= 0,
(7) r[τ, b](s, y) = (γτ1(s+ s0)− γτ1(s0) + a sin b, y, γτ2(s+ s0)− γτ2(s0) + a cos b),
where s0 = 1τ sinh−1(tan b). If τ = 0, r[0, b](s, y)(s sin b+ a sin b, y, s cos b+ a cos b).
A short computation using (5) shows that this value of s0 is exactly the one forwhich
∂r[τ, b]∂s
(0, y) = (sin b)e1 + (cos b)e3.
Let us denote the grim reaper end by Γ[τ, b] = r[τ, b](H+). Let δs be a smallconstant to be chosen later. We are now ready to define the map F [τ, b], whichdescribes the top end of our new surface:
Definition 4. Define F [τ, b] : H+ → R3 to be
F [τ, b](s, y) = ψ[1, 0](s)Zb F (s, y)
+ (1− ψ[1, 0](s))(r[τ, b](s, y) + ψs(s)σ(s, y)νΓ[τ,b](s, y)),
where F is as in Lemma 3, ψs(s) = ψ[4δsτ−1, 3δsτ−1](s) and νΓ[τ,b](s, y) is the unitnormal to Γ[τ, b] at the point r[τ, b](s, y) pointing towards e1.
The cut off function ψ[1, 0] (and therefore the region s ∈ [0, 1]) is used to attachthe end smoothly to our surface Zb(Σ0). For s ∈ [1, 3δsτ−1], the surface is thegraph of σ over the grim reaper cylinder Γ[τ, b]. The graph of σ is then cut off in
TRANSLATING TRIDENTS 7
the region s ∈ [3δsτ−1, 4δsτ−1], which is another transition region. For s > 4δsτ−1,we just have part of the grim reaper cylinder Γ[τ, b].
2.3. Definition of Z[τ, b] and Σ[τ, b].
Definition 5. Define Z[τ, b] : Σ0 → R3 in the following way(i) Z[τ, b] = Zb on Σ0 ∩ (x, y, z) | |z| ≤ a,
(ii) Z[τ, b] F = F [τ, b] on Σ0 ∩ z ≥ a,(iii) Z[τ, b] R1 = R1 Z[τ, b].
The new surface is denoted by Σ = Σ[τ, b] = Z[τ, b](Σ0). We will call the setΣ0 ∩(x, y, z) | |z| ≤ a, |x| ≤ a the core of Σ0; and its image under Z[τ, b] is calledthe core of Σ. For a surface S, where S can represent Σ0 or Σ, we will use thefollowing notations:
S≤0 := (core of S),
S≤s := S≤0 ∪ (set corresponding to s ∈ [0, s] on the four wings),
S≥s := S \ S<s.We denote by gΣ (g0) the metric on Σ = Σ[τ, b] (Σ0 resp.) induced by the Euclideanmetric in R3.
An inspection of the construction gives us the following lemma:
Lemma 6. There are constants δτ and δb such that for τ ∈ (−δτ , δτ ) and b ∈(−δb, δb), the map Z[τ, b] is a smooth immersion depending smoothly on its param-eters and the surface Σ[τ, b] is an embedded surface invariant under the symmetries1.4.
From now on, we assume that τ ∈ (−δτ , δτ ) and b ∈ (−δb, δb).
3. Estimation of H − τe1 · ν on the initial surface
In this section, we show that the initial surface obtained is a good approximatesolution. The presentation of this section is parallel to Section 4 in [9].
We find the exact solution to (3) among graphs of functions over Σ. Since thegraph of any function v also has to be invariant under the symmetries 1.4, v has tosatisfy
(8) v R1 = −v, v R2 = v, v P = v.
We define Holder spaces with symmetries
Definition 7.
Ck,α(Ω, g, f) = v ∈ Ck,α(Ω, g, f) | v satisfies the symmetries (8).
From now on, all the functions on Σ satisfy the symmetries (8).
3.1. Notations. We will use the same notation for functions, tensors and operatorson the asymptotic grim reaper, their pushforward by F [τ, b] r−1 to Σ≥1[τ, b],and their pushforward by F r−1 to Σ0, and vice versa. To avoid confusion,we use symbols without subscripts for the geometric quantities considered on theasymptotic grim reaper; we use symbols with subscripts Σ for their counterpartson Σ≥1, and we use symbols with subscripts 0 to describe quantities considered onthe original surface Σ0. For example, g denotes the metric on the asymptotic grimreaper (induced by its immersion in R3), it also denotes the pushforward of this
8 XUAN HIEN NGUYEN
metric to Σ≤1 or to Σ0; while gΣ denotes the metric on Σ≤1 induced by the metricin R3 or its pullback to the asymptotic grim reaper or to Σ0. Finally, g0 is themetric on Σ0 or its pushforward to Σ or to the asymptotic grim reaper.
3.2. Estimates.
Lemma 8. The following are valid on Σ≥1[τ, b]:(i) ‖A : Ck(g)‖ ≤ Cτ ,
(ii) ‖ν : Ck(g)‖ ≤ C,where A is the second fundamental form, ν is the unit normal vector, the orientationof which is determined by the orientation on Σ0, and all the constants C dependonly on k.
Proof. The estimates are clearly true for τ = 0. If τ 6= 0, the curvature at a points on the grim reaper is given by
|~γ′′(s+ s0)| = τ
cosh(τ(s+ s0)).
We therefore have the estimate (i). For the unit normal, we again have an explicitformula
ν(s, y) = ((cosh(τ(s+ s0)))−1, 0,− tanh(τ(s+ s0)))and the estimate follows easily.
Corollary 9. The following estimates are valid:
‖gΣ − g0 : C3(Σ0, g0, e−s))‖ ≤ Cε+ Cτ + Cb,
‖|AΣ|2 − |AΣ0 |2 : C3(Σ0, g0)‖ ≤ Cε+ Cτ + Cb.
In particular, g0 and gΣ are uniformly equivalent on Σ[τ, b] by assuming withoutloss of generality ε small enough.
Proof. We will first show that
(9) ‖gΣ − g : C3(Σ≥1[τ, b], g, e−s))‖ ≤ Cε.Indeed, the variation of a metric under a normal perturbation f = ψsσ is given by
gΣ,ij = gij − 2fAij + F ′ij ,
where the F ′’s are polynomials in fAij and fi with terms at least quadratic (seeAppendix A). Note that ∇f = σ∇ψs + ψs∇σ. Each term σ∇kψs is of the or-der of Cεe−s( τδs
)k since the derivative of ψs is supported in s ∈ (3δs/τ, 4δs/τ).Without loss of generality, we can choose τ so that τ/δs < 1 therefore ‖σ∇kψs :C5(Σ≥1[τ, b], g, e−s)‖ ≤ Cε and ‖f : C5(Σ≥1[τ, b], g, e−s)‖ ≤ Cε. The bound onA from Lemma 8 then gives us (9). The asymptotic grim reaper is isometric to aplane, so Lemma 3 implies that the metrics g and g0 are equivalent on Σ0,≥1, and
‖gΣ − g0 : C3(Σ0,≥1, g0, e−s)‖ ≤ Cε.
The discussion above, combined with the formula for the perturbation of |A|2 (seeAppendix A equation (21)) gives us
‖|AΣ|2 − |A|2 : C3(Σ≥1, g, e−s)‖ ≤ Cε.
We have ‖|AΣ|2 − |A0|2‖ ≤ ‖|AΣ|2 − |A|2‖ + ‖|A|2‖ + ‖|A0|2‖ therefore, by theequivalence of the metrics, Lemmas 3 and 8, we obtain
‖|AΣ|2 − |A|2 : C3(Σ0,≥1, g0)‖ ≤ Cτ + Cε.
TRANSLATING TRIDENTS 9
The estimates on Σ0,≤1 are valid thanks to the smooth dependence on the param-eters τ and b from Lemma 6.
From now on, we will assume that τ is small enough with respect to the constantγ ∈ (0, 1) from the Lemma below.
Lemma 10. For γ ∈ (0, 1), we have
‖HΣ − τe1 · νΣ : C2(Σ≥1[τ, b], g, e−γs)‖ ≤ Cτ,where HΣ is the mean curvature of Σ[τ, b].
Proof. First, note that the estimate is true for s ≥ 4δsτ . Let us now work in theregion s ∈ [3δs/τ, 4δs/τ ]. We have H − τe1 · ν ≡ 0 on the grim reaper cylinder,so by the variation formulas in Appendix A , HΣ − τe1 · νΣ has terms at leastlinear involving fAij , ∇f and ∇2
ijf (with f = ψsσ). We are on the support ofthe derivative of the cut-off function ψs, so these terms, their first and secondderivatives behave like ( τδs
)ke−s, 0 ≤ k ≤ 4. For s ≥ 3δs/τ , we can arrange( τδs
)ke−s ≤ e−s ≤ τe−γs to be true by taking τ small enough in terms of γ.In the region s ≤ 3δs/τ , we have ψs ≡ 1. The plane and the original Scherk
surface are minimal surfaces, so
0 = ∆gR2σ +Q,
where Q is the Qf defined by the equation (20) with f = σ, and every geometricquantity and tensor in Q is taken with respect to the flat metric on the asymptoticplane (to the original Scherk surface). Since H−τe1 ·ν vanishes on the grim reapercylinder,
HΣ − τe1 · νΣ = ∆gσ + |A|2σ +Q′ + τe1 · ∇σ + τe1 ·Qν ,
where Q′ is Qf defined by the equation (20) with f = σ and where every geometricquantity and tensor in Q′ is taken with respect to the metric on the asymptoticgrim reaper cylinder. The term Qν is an expression at least quadratic in ∇σ andσA. Since the asymptotic grim reaper with the metric g is isometric to a planewith flat metric,
HΣ − τe1 · νΣ = −Q+ |A|2σ +Q′ + τe1 · ∇σ + τe1 ·Qν ,
By Lemma 8, the fact that ‖σ : C5(Σ≥1, g, e−s)‖ ≤ ε and the equation (19) for the
perturbation of the unit normal in the appendix, we have
‖|A|2σ + τe1 · ∇σ + τe1 ·Qν : C2(Σ≥1, g, e−γs)‖ ≤ Cτ.
We are left with
Q′ −Q =G1√
1 +G1
− G′1√1 +G′1
+G2
1 +G1 +√
1 +G1
− G′21 +G′1 +
√1 +G′1
.
We can reduce the fractions to the same denominator and expand the numeratorsand the square roots in Taylor series. The expressions for the G′’s and G′’s compriseterms at least quadratic depending on ∇σ, ∇2σ, A, σA and σ∇2A. If a term inthe expansion of the numerators has either A, σA or σ∇2A, it can be bounded byCτ according to Lemma 8. We now claim that there is no term involving only ∇σand ∇2σ. The grim reaper is asymptotic to a flat plane, so G1 only differs from G′1by terms involving the second fundamental form A. Since the second fundamentalform vanishes on planes, setting A = 0 in the expression for G′1 gives us G1. The
10 XUAN HIEN NGUYEN
same property is true for the other G’s and G′’s. The G’s and corresponding G′’scontain exactly the same terms involving only ∇σ and ∇2σ. These terms cantherefore be paired and cancelled.
3.3. The definition of w. Let us fix the parameter τ ∈ (0, δτ ). Let Hb and νbbe the mean curvature and normal vector on the surface Zb(Σ0). The function wbelow expresses the change in the quantity H as we vary b.
Definition 11. Let w : Σ0 → R be defined by
(10) w :=d
db
∣∣∣∣b=0
Hb Zb.
We also denote by w the pushforward on Σ[τ, b] by Z[τ, b] of the function wdefined above.
Proposition 12. On Σ[τ, b], we have
‖HΣ − τe1 · νΣ − bw : C0,α(Σ, gΣ, e−γs)‖ ≤ C(τ + |b|2).
Proof. We first show the bound in the region Σ≤1. By the smooth dependence onthe parameters (τ, b) in Lemma 6, the metrics gΣ[τ,b] for τ ∈ (−δτ , δτ ), |b| ≤ δb areall equivalent in Σ≤1. We also have
‖HΣ[τ,b] −HΣ[0,b] : C0,α(Σ≤1[τ, b], gΣ[τ,b])‖ ≤ Cτ.The definition of w implies
‖HΣ[0,b] −HΣ[0,0] − bw : C0,α(Σ≤1[τ, b], gΣ[τ,b])‖ ≤ C|b|2.Of course, HΣ[0,0] = 0 and we finish the proof for Σ≤1[τ, b].
On Σ≥1[τ, b], the bound follows from Lemma 10 and the fact that w = 0 whens ≥ 1.
4. Linear operator
When a surface is perturbed in the normal direction by the graph of a smallfunction f , the linear change of the quantity H − τe1 · ν is
∆f + |A|2f + τe1 · ∇f.For the proof of this formula, see Appendix A. For each τ ∈ (−δτ , δτ ), let us definethe linear operator
Lτf = ∆gΣf + |AΣ|2f + τe1 · ∇f, f ∈ C2,α(Σ, gΣ, e−γs).
We will use Lτ to denote both the operator on Σ and the corresponding operatoron Σ0, where the Laplacian is taken with respect to gΣ pulled back to Σ0, andwhere we also pullback |AΣ| to Σ0.
This section is dedicated to the study of Lτ between Holder spaces of expo-nentially decaying functions. First note that, for τ small, the operator Lτ is aperturbation of the operator L0 = ∆gΣ + |AΣ|2. Moreover, the surface Σ is al-most isometric to the original Scherk surface Σ0, so we can treat L0 on Σ as aperturbation of L = ∆g0 + |AΣ0 |2 on Σ0.
We will show that the operator L : C2,α(Σ0, g0, e−γs) → C0,α(Σ0, g0, e
−γs) isa Fredholm operator of index −1. It is well known that L is the linear operatorassociated to small normal perturbations of the mean curvature H. Since the meancurvature is invariant under translations of the surface, the functions e1 · ν, e2 · ν
TRANSLATING TRIDENTS 11
and e3 · ν are all in the kernel of L. The functions e2 · ν and e3 · ν do not satisfyour imposed symmetries so they can not be considered. The function e1 · ν doessurvive the symmetries but is disqualified because it does not decay exponentiallyalong the top and bottom wings. The fact that e1 ·ν was “almost” an eigenfunctioncauses some problems with regards to the range: the range is the set of functionsh ∈ C0,α(Σ0, g0, e
−γs) for which Φ(h) =∫
Σ0h(e1 · ν) = 0, so it has codimension 1
The operator L : C2,α(Σ0, g0, e−γs)→ C0,α(Σ0, g0, e
−γs) is not surjective, there-fore not invertible. Our strategy now is to find a function w ∈ C0,α(Σ0, g0, e
−γs) forwhich Φ(w) 6= 0. This w will allow us to project any function E ∈ C0,α(Σ0, g0, e
−γs)onto the range of L. Indeed, E − Φ(E)
Φ(w)w is now in the range of L, so there is a vEfor which LvE = E−bw, with b = Φ(E)
Φ(w) . We have to be able to generate and cancelmultiples of w by the construction.
If S is a surface with boundary embedded in R3 and Y is a vector field thatperturbs S, the change in area of S is given by
δ(area(S)) =∫S
divY =∫S
div(Y ‖) + div(Y ⊥)
=∫∂S
N · Y −∫S
Hν · Y,
where Y ⊥ = Y · ν and Y ‖ = Y − Y ⊥ are the orthogonal and parallel componentsof Y , and N is the unit outward conormal. If Y = e1, the change in area for S is 0,hence
∫SH(e1 · ν) =
∫∂Sη · e1. This formula is called a balancing formula [10]. If
we define w to be the change of H as we vary b (see (10)), then w is not orthogonalto e1 · ν if the conormal vectors are bent towards e1 as b varies. It is exactly therole of the transformations Zb.
Once we fully understand the linear operator L, we invoke the Proposition 13below to prove results for L0 and Lτ .
4.1. Perturbation of operators. We use Ω to denote a smooth surface, whichcan represent Σ, a piece of Σ, or an infinite half cylinder. Denote by C2,α
0 (Ω, g, e−γs)the set of functions in C2,α(Ω, g, e−γs) that vanish on the boundary of Ω. Considerthe operators T and T : C2,α
0 (Ω, g, e−γs)→ C0,α(Ω, g, e−γs) defined by
T v = ∆gu+ du+ V · ∇u,
T v = ∆gu+ du+ V · ∇u,
where g and g are metrics, d and d are functions, and V and V are vector fields onΩ. Define
N = max‖g − g : C2(Ω, g, e−γs)‖, ‖d− d : C1(Ω, g)‖, ‖V − V : C1(Ω, g)‖.
Proposition 13. Assume that there is a function w ∈ C0,α(Ω, g, e−γs) with w notin the range of T . Assume also that for every E ∈ C0,α(Ω, g, e−γs), there exist aconstant bE and a function vE ∈ C2,α
0 (Ω, g, e−γs) such that T vE = E − bEw and
‖vE : C2,α(Ω, g, e−γs)‖ ≤ C‖E : C0,α(Ω, g, e−γs)‖,|bE | ≤ C‖E : C0,α(Ω, g, e−γs)‖.
12 XUAN HIEN NGUYEN
If N is small enough, then there exist also a constant bE and a function vE ∈C2,α(Ω, g, e−γs) such that T vE = E − bEw and
‖vE : C2,α(Ω, g, e−γs)‖ ≤ C‖E : C0,α(Ω, g, e−γs)‖,(11)
|bE | ≤ C‖E : C0,α(Ω, g, e−γs)‖.(12)
Proof. For small N , the metrics g and g are equivalent on Ω. Let us define themap R : C0,α(Ω, g, e−γs) → (R × C2,α
0 (Ω, g, e−γs)) by R(E) = (bE , vE). Considerthe iteration
(b0, v0) = R(E), (bn, vn) = R(T vn−1 − T vn−1).
We then have
‖v0 : C2,α(Ω, g, e−γs)‖ ≤ C‖E : C0,α(Ω, g, e−γs)‖,(13)
‖vn : C2,α(Ω, g, e−γs)‖ ≤ C‖T vn−1 − T vn−1 : C0,α(Ω, g, e−γs)‖,≤ C20 N‖vn−1 : C1,α(Ω, g, e−γs)‖.(14)
We prove the last inequality by recalling ∆gv − ∆gv =∑2i,k=1(Γkii − Γkii)∇kv,
where Γkij and Γkij are the Christoffel symbol associated to g and g respectively.An estimate analogous to (14) is true for bn. We now choose N small enough sothat C20N < 1 in the equation above. The series
∑∞i=0(bi, vi) converges to a limit,
denoted by (b∞, v∞), in R× C2,α0 (Ω, g, e−γs). We can take vE = v∞ and bE = b∞:
T v∞ =∞∑i=0
(T (vi) + (T − T )vi) = E − b0w +∞∑i=1
T vi + (T − T )vi−1
= E −∞∑i=0
biw = E − b∞w.
The estimate (11) follows from (13) and (14). The bound on bE is proved in asimilar way.
4.2. Fredholm property and index for L. We first prove that the operator L isa Fredholm operator on various pieces of the surface Σ0; then use a Mayer-Vietoristype exact sequence argument to show L is a Fredholm operator and to computeits index on the whole surface Σ0.
The main theorem of this section, Theorem 18, can also be proven using the re-sults of Lockhart and McOwen for elliptic operators on manifolds with cylindricalends [11]. They use weighted Sobolev spaces, which could be switched to weightedHolder spaces (with faster exponential decay) using elliptic regularity. Since The-orem 18 is a central argument to this article, we include a proof for completeness.
4.2.1. Linear operator on the wings. Let Wt = Σ0 ∩ z ≥ a be the top wing, andWb = Σ0 ∩ z ≤ −a the bottom wing. Denote by C2,α
0 (Wt, gΣ, e−γs) the functions
in C2,α(Wt, gΣ, e−γs) that vanish on the boundary of Wt.
Lemma 14. The linear operator L : C2,α0 (Wt, g0, e
−γs) → C0,α(Wt, g0, e−γs) is a
Fredholm operator of index −1.
Proof. The linear operator L on Wt is a perturbation of the Laplace operator onan infinite flat half cylinder. The property is true for the Laplace operator, and weuse Lemma 3 and Proposition 13 to prove it for L.
TRANSLATING TRIDENTS 13
What happens with the Laplace operator on an infinite half-cylinder is analogousto what happens with L on Σ0. The Laplacian on H+/G vanishes for h1 = Cand h2 = s. Only h2 takes the value 0 on the boundary, but h2 does not haveexponential decay, so it can not be in the kernel. Nevertheless, it plays a role forthe range. A formal computation using Fourier series shows that E is in the rangeof ∆ : C2,α
0 (Wt, g0, e−γs)→ C0,α(Wt, g0, e
−γs) if and only if
(15)∫ ∞
0
∫ 2π
0
sE(s, y)dyds = 0.
Corollary 15. For τ ∈ [0, δτ ), the linear operator L : C2,α0 (Wt ∪Wb, g0, e
−γs) →C0,α(Wt ∪Wb, g0, e
−γs) is a Fredholm operator of index −1.
Proof. Because of the symmetry with respect to R1, the behavior of any functionon the bottom wing Wb = Σ≥1 ∩ z ≤ 0 is dictated by its values on the topwing.
The right wing, Wr = Σ≥1∩x ≥ 0 contains part of the x-axis and the equation(15) is satisfied by any function in C0,α(Wr, gΣ, e
−γs) thanks again to the symme-tries. Therefore, on the right wing, the Fredholm operator L : C2,α
0 (Wr, gΣ, e−γs)→
C0,α(Wr, gΣ, e−γs) has index 0. The same applies to the left wing.
4.2.2. Linear operator on the core. Let us call the piece Σ0,≤2 the core. Denote byC2,α
0 (Σ0,≤2, gΣ) the set of functions in C2,α(Σ0,≤2, gΣ) that vanish on the boundary∂Σ0,≤2.
Lemma 16. The operator L = ∆g0 + |AΣ0 |2 : C2,α0 (Σ0,≤2, gΣ)→ C0,α(Σ0,≤2, gΣ) is
a Fredholm operator of index 0.
Proof. Σ0 is the original Scherk surface therefore, it is a minimal surface. Since Σ0
does not have any umbilics, the Gauss map ν : Σ0 → S2, p 7→ ν(p) is conformal [4].Moreover, we can compute the normal vector ν explicitly using equation (4) andfind
ν(x, y, z) =(
sinh zcosh z
,− − cos ycosh z coshx
,sinhxcoshx
), for (x, y, z) ∈ Σ0.
The images under the Gauss map of the curves z = ±(2 + a) ∩ ∂Σ0, x = ±(2 +a) ∩ ∂Σ0 are circles on S2 centered at (±1, 0, 0) and (0, 0,±1) respectively. Therotation R1 in Σ0 corresponds to the reflection across the yz-plane in S2 ⊂ R3, andthe reflections R2 corresponds to reflection across the xz-plane in S2. Therefore,the symmetries 1.4 allow us to define a bijective conformal continuous Gauss mapν from Σ0,≤2 to S2 minus four disks centered at (±1, 0, 0) and (0, 0,±1).
The pull-back ν∗gS2 of the metric on S2 is ν∗gS2 = 12 |AΣ0 |2g. A function on Σ0
is in the kernel of ∆g0 + |AΣ0 |2 if its pushforward by the Gauss map is in the kernelof ∆gS2 +2 on the sphere. If U is a subset of the sphere with smooth boundary, theLaplace operator from C2,α
0 (U)→ C0,α(U) is invertible; therefore ∆ + 2 is of index0, since the inclusion C2,α
0 (U)→ C0,α(U) is compact.
4.2.3. Linear operator on the whole surface Σ0. To show that L is a Fredholm oper-ator on the whole surface Σ0 we use the following lemma, which is a generalizationof the ”Snake Lemma” from [3]:
14 XUAN HIEN NGUYEN
Lemma 17. Suppose we have Banach spaces Xi, Yi and Zi, i = 0, 2, operatorsLB : B2 → B0, B = X,Y, Z, between these spaces, and maps αi, αi, βi and βi,i = 0, 2 such that the rows are exact in the two diagrams below.
0 −−−−→ X2α2−−−−→ Y2
α2−−−−→ Z2 −−−−→ 0
LX
y LY
y LZ
y0 −−−−→ X0
α0−−−−→ Y0α0−−−−→ Z0 −−−−→ 0
0 ←−−−− X2β2←−−−− Y2
β2←−−−− Z2 ←−−−− 0
LX
y LY
y LZ
y0 ←−−−− X0
β0←−−−− Y0β0←−−−− Z0 ←−−−− 0.
Suppose that β α = idX and α β = idZ .Suppose also that the operators LY α2 − α0LX , LZ α2 − α0LY , LXβ2 − β0LY andLY β2 − β0LZ are compact, i.e., the diagrams commute up to compact operators.If two of the operators LX , LY or LZ are Fredholm, then the third one is also aFredholm operator and
index (LX)− index (LY ) + index (LZ) = 0.
We will finish the study of the linear operator assuming Lemma 17, then give itsproof at the end of the section.
Theorem 18. The operator L : C2,α(Σ0, g0, e−γs) → C0,α(Σ0, g0, e
−γs) is a Fred-holm operator of index −1.
Proof. Let U and V be two subsets of Σ0, with smooth boundary and such thatthe distance between ∂U and ∂V is strictly positive. Lemma 17 allows us to findthe index of a Fredholm operator on the union U ∪ V if we know its behavior onU , V and on the intersection U ∩ V . For the Banach spaces, we take
X2 = C2,α0 (U ∪ V, g, e−γs) X0 = C0,α(U ∪ V, g, e−γs)
Y2 = C2,α0 (U, g, e−γs)× C2,α
0 (V, g, e−γs) Y0 = C0,α(U, g, e−γs)× C0,α(V, g, e−γs)
Z2 = C2,α0 (U ∩ V, g, e−γs) Z0 = C0,α(U ∩ V, g, e−γs).
The operators LX , LY and LZ are just the operator L between the different spaces.In particular, LY (u, v) = (Lu,Lv).
Let ϕU be a smooth function defined on U such that 0 ≤ ϕU ≤ 1, ϕU = 1 onU \V and ϕU = 0 on ∂U ∩V . The function ϕV is defined similarly with the addedrequirement ϕ2
U + ϕ2V = 1 on U ∪ V .
The maps αi and βi (i = 0, 2) are defined as follows
αi(f) = (ϕUf, ϕV f) αi(u, v) = ϕV u− ϕUvβi(u, v) = ϕUu+ ϕV v βi(f) = (ϕV f,−ϕUf).
It is easy to verify that these maps give us exact sequences, and that β α = idXand α β = idZ . We are left to check that the diagrams are commutative up tocompact operators. We have,
(LY α2 − α0 LX)f = (∆(ϕUf)− ϕU∆f,∆(ϕV f)− ϕV ∆f)
= (∇ϕU∇f + (∆ϕU )f,∇ϕU∇f + (∆ϕU )f).
TRANSLATING TRIDENTS 15
Since the support of ∇ϕU and ∇ϕV is compact in Σ0/G, the operator LY α2 −α0LX is compact. Similarly, one can prove that LZ α2 − α0LY , LXβ2 − β0LY andLY β2 − β0LZ are also compact.
We apply Lemma 17 a first time with U = Σ0,≤2 and V = Wt ∪Wb; then weattach the right and left wings by using Lemma 17 two more times. Note thatthe intersection of U ∩ V is a cylinder each time. The operator L on U ∩ V is aperturbation of the Laplace operator on a finite flat cylinder Ω. It is known that∆gR2 : C2,α
0 (Ω, g0) → C0,α(Ω, g0) is invertible. Therefore, L has Fredholm index0 on U ∩ V thanks to the estimates in Section 1.2. This remark, combined withCorollary 15 and Lemma 16, shows that L : C2,α(Σ0, g0, e
−γs)→ C2,α(Σ0, g0, e−γs)
is a Fredholm operator of index −1.
4.3. Kernel and range of the linear operator on Σ0. The functions in thekernel of L on the minimal surface Σ0 are in one to one correspondence with thefunctions in the kernel of ∆gS2 +2 on the sphere minus four points. These functionsare exactly the coordinate functions e1 ·ν, e2 ·ν and e3 ·ν. The imposed symmetrieson the surfaces (the graph on the surface Σ0 of any acceptable function still has to beinvariant underR1 andR2) eliminate the functions e2·ν and e3·ν. The function e1·νis not exponentially decaying along the top and bottom wings. Hence, the kernel istrivial. This means that the cokernel of L : C2,α(Σ0, g0, e
−γs) → C2,α(Σ0, g0, e−γs)
has dimension 1.We define a linear functional on C0,α(Σ0, g0, e
−γs) by
Φ(h) =∫
Σ0
h(e1 · ν)dµ, h ∈ C0,α(Σ0, g0, e−γs),
where dµ is the Hausdorff measure on Σ0.
Lemma 19. The map Φ vanishes on the range of L.
Proof. A function in the range can be written as Lf for some f ∈ C2,α(Σ0, g0, e−γs),
therefore∫Σ0
(∆g0f + |AΣ0 |2f)(e1 · ν)
=∫
Σ0
∇(∇fe1 · ν)−∇(f∇(e1 · ν)) + f∆g0(e1 · ν) + |AΣ0 |2fe1 · νdµ.
Since (∆g0 + |AΣ0 |2)(e1 · ν) = 0, we obtain∫Σ0
(∆g0f + |AΣ0 |2f)(e1 · ν) = limc→∞
∫∂Σ0,≤c
∇ηfe1 · ν − f∇ηe1 · νds = 0,
where η is the conormal on ∂Σ0,≤c pointing out of Σ0,≤c. The last equality is provedusing the fact that f and its derivatives have exponential decay.
A change of the parameter b generates the function w defined in (10). In Lemma20 below, we show that w does not belong to the range of L. Therefore, we canmodify any function E ∈ C0,α(Σ0, g0, e
−γs) by substracting a multiple of w so thatE − bEw is in the range of L.
Lemma 20. The function w defined above on Σ0 satisfies the following:(i) w has support in Σ0,≤0 and ‖w : C0,α(Σ0, g0, e
−γs)‖ ≤ C,
16 XUAN HIEN NGUYEN
(ii) ∫Σ0
w(e1 · ν0)dµg0 ≥1C> 0,
where dµg0 is taken with respect to the metric g0. The constant C isindependent of τ and b (it does depend on δτ and δb).
Proof. The property (1) follows from the construction and the smooth dependenceof Σ[τ, b] on its parameters.
We now define the vector field ξ(x) = ddb
∣∣b=0
Zb(x), x ∈ Σ0 to be the changein position induced by Zb.
The vector ξ is decomposed into its parallel component ξ‖ = ξ − (ξ · ν0)ν0 andnormal component ξ⊥ = (ξ ·ν0)ν0. The variation of the mean curvature Hb is givenby w = d
db
∣∣b=0
Hb = ξ‖(H) + (∆gΣ0+ |AΣ0 |2)(ξ · ν0). Since Σ0 is a minimal surface,
w = (∆gΣ0+ |AΣ0 |2)(ξ · ν0).
Using the divergence theorem and the fact that e1·ν is in the kernel of ∆gΣ0+|AΣ0 |2,
we obtain∫Σ0
w(e1 · ν0)dµg0 =∫
Σ0,≤c
w(e1 · ν0)dµg0
=∫
Σ0,≤c
((∆gΣ0+ |AΣ0 |2)(ξ · ν0))(e1 · ν0)dµg0
=∫∂Σ0,≤c
(e1 · ν0)∇N (ξ · ν0)− (ξ · ν0)∇N (e1 · ν0)dσ,
where c can be any positive number, N is the exterior conormal to ∂Σ0,≤c relativeto Σ0,≤c, and dσ is the measure on ∂Σ0,≤c. On the top and bottom wings, theScherk surface Σ0 can be written as the graph of σ over an asymptotic half plane(see 1.2). More precisely, at a point (σ(s, y), y, s+ a) on the top wing, we have
ν0(s, y) =1√
1 + σ2y + σ2
s
(1,−σy(s, y),−σs(s, y)),
ξ(s, y) = (s+ a, 0,−σ(s, y)).
We can estimate N by (0, 0, 1), with an error decaying exponentially as c goes toinfinity. A short computation gives∣∣∣∣∣(
∫∂Σ0,≤c
(e1 · ν0)∇N (ξ · ν0)− (ξ · ν0)∇N (e1 · ν0)dσ)− 4π
∣∣∣∣∣ ≤ Ce−c.Letting c tend to ∞, we obtain
∫Σ0w(e1 · ν0)dµg0 = 4π.
Theorem 18 and Lemma 20 therefore give us the following theorem.
Theorem 21. Given any function E ∈ C0,α(Σ0, g0, e−γs), there exist a constant
bE and a function vE ∈ C2,α(Σ0, g0, e−γs) such that
LvE = E − bEw.Moreover, bE = Φ(E)/Φ(w) and
‖vE : C2,α(Σ0, g0, e−γs)‖ ≤ C‖E : C0,α(Σ0, g0, e
−γs)‖,|bE | ≤ C‖E : C0,α(Σ0, g0, e
−γs)‖.
TRANSLATING TRIDENTS 17
Corollary 22. For δτ and δb small enough, the operator Lτ has Fredholm index−1. In addition, for any E ∈ C0,α(Σ, gΣ, e
−γs) there are a constant bE and a uniquefunction vE ∈ C2,α(Σ, gΣ, e
−γs) such that
Lτ vE = E − bEw,
‖vE : C2,α(Σ, gΣ, e−γs)‖ ≤ C‖E : C0,α(Σ, gΣ, e
−γs)‖,
|bE | ≤ C‖E : C0,α(Σ, gΣ, e−γs)‖.
Proof. The operator Lτ = ∆gΣ+|AΣ|2+τe1·∇ is a perturbation of L = ∆g0+|AΣ0 |2.The proof follows from the estimates in Corollary 9 and the Proposition 13.
4.4. Proof of lemma 17. First note that αβ is a projection, and βα = id−αβis its complementary projection. We can therefore split Yi (i = 0, 2) into
Yi = Ai ⊕Bi,
with Ai = range αi and Bi = range βi. Using this splitting, the operator LY canbe decomposed as
LY = αβLY αβ + βαLY αβ + αβLY βα+ βαLY βα,
where we omitted the indices for the maps α and β.
Claim 23. The operators βαLY αβ and αβLY βα are compact.
Proof of claim. We will give the proof for βαLY αβ. The operator βαLY αβ can betreated in a similar way.
βαLY αβ = (1− αβ)LY αβ = (1− αβ)(LY αβ − αβLY ) + (1− αβ)αβLY= (1− αβ)(LY αβ − αLXβ + αLXβ − αβLY )
= (1− αβ)(LY α− αLX)β + (1− αβ)α(LXβ − βLY ).
The operators LY α− αLX and LXβ − βLY are compact by hypothesis, hence theclaim.
From the claim, if two of the operators LY , αβLY αβ and βαLY βα are Fredholm,the third one is also Fredholm. Moreover, we have
(16) index LY = index αβLY αβ + index βαLY βα.
The exactness of the rows implies that Ai = range αi ∼= Xi and Bi = range βi ∼= Zi.The isomorphisms are given by the αi’s. We also know that βi is the inverse of αi.Consider the following commutative diagram,
0 −−−−→ X2α2−−−−→ A2 −−−−→ 0
β0LY α2
y yα0β0LY α2β2
0 −−−−→ X0α0−−−−→ A0 −−−−→ 0,
which shows that the operator α0β0LY α2β2 is Fredholm if and only if β0LY α2 isFredholm. We also have
β0LY α2 = β0(LY α2 − α0LX) + β0α0LX = β0(LY α2 − α0LX) + LX .
By hypothesis, LY α2−α0LX is compact, therefore α0β0LY α2β2 is Fredholm if andonly if LX is Fredholm and
(17) index α0β0LY α2β2 = index β0LY α2 = index LX .
18 XUAN HIEN NGUYEN
Similarly, we show that βαLY βα is a Fredholm operator if and only if LZ is Fred-holm, and
(18) index βαLY βα = index LZ .
Combining equations (16), (17) and (18), we complete the proof of Lemma 17.
5. Fixed point argument
For simplicity, we use the notations
‖v‖i = ‖v : Ci,α(Σ[τ, b], gΣ, e−γs)‖, i = 0, 2,
where the values of τ and b are implied by the domain of v. We will also need thefollowing preliminary estimate.
Proposition 24. If v ∈ C2,α(Σ[τ, b]) with ‖v‖2 smaller than a suitable constant,then the graph Σ[τ, b, v] of v over Σ[τ, b] is a smooth immersion. Moreover,
‖Hv − τe1 · νv − (H − τe1 · ν)− Lτv‖0 ≤ C‖v‖22,where H, Hv, ν and νv are the mean curvature and the unit normal of Σ[τ, b] andΣ[τ, b, v] pulled back to Σ[τ, b] respectively.
Proof. This follows from the estimates and calculations for the variations of H andν in (20) and (19).
We are now ready to prove the main theorem of this paper:
Theorem 1. There is a family of surfaces Mτ parametrized by τ ∈ (0, δτ ) withthe following properties:
(i) for τ ∈ (0, δτ ), Mτ is a complete embedded surface in R3 that is self-translating under mean curvature flow,
(ii) Two ends of Mτ decay exponentially to planes, and the two other endsdecay exponentially to grim reaper cylinders,
(iii) as τ → 0, 1τMτ = ( 1
τ (x, y, z) | (x, y, z) ∈Mτ tends to the Scherk surfaceuniformly on compact subsets of R3.
Proof. Take α′ ∈ (0, α) and define the Banach space
χ = f ∈ C2,α′(Σ[τ, 0]).Denote by Φb : Σ[τ, 0]→ Σ[τ, b] a family of smooth diffeomorphisms which dependsmoothly on b and satisfy the following conditions: for every f ∈ C2,α(Σ[τ, 0]) andf ′ ∈ C2,α(Σ[τ, b]) we have
‖f Φ−1b ‖2 ≤ C‖f‖2, ‖f ′ Φb‖2 ≤ C‖f ′‖2.
Let us fix τ and define
C = (b, u) ∈ R× χ : |b| ≤ ζτ, ‖u‖2 ≤ ζτ.Define v = u Φ−1
b . Let us denote by M = Σ[τ, b] and by Mv the graph of v overM . Define the function F ∈ C0,α(M) by
F(b, v) = Hv − τe1 · νv,where Hv and νv are the mean curvature and the unit normal of Mv respectivelypulled back to M . Proposition 24 asserts that
‖F(b, v)−F(b, 0)− LτM (v)‖0 ≤ Cζ2τ2.
TRANSLATING TRIDENTS 19
Applying Corollary 22 with E = F(b, v) − F(b, 0) − LτM (v), we find a functionvE ∈ C2,α(M, gΣ, e
−γs) and a constant bE such that LτMvE = E−bEw and ‖vE‖2 ≤Cζ2τ2, |bE | ≤ Cζ2τ2. Thus
F(b, v) = F(b, 0) + LτM (v) + LτMvE + bEw.
Again, Proposition 12 gives us
‖F(b, 0)− bw‖0 ≤ C(τ + |b|2) ≤ C(τ + ζ2τ2),
so there exist a function v0 and a constant b0 such that LτMv0 = F(b, 0)− bw− b0wwith ‖v0‖2 ≤ C(τ + ζ2τ2), |b0| ≤ C(τ + ζ2τ2). Hence,
F(b, v) = LτMv0 + LτMv + LτMvE + bEw + bw + b0w.
Define the map I : C → R× χ by I(b, u) = (−bE − b0, (−vE − v0) Φb).We now claim that I(C) ⊂ C. Indeed, we have
‖ − vE − v0‖2 ≤ C(τ + 2ζ2τ2)
| − bE − b0| ≤ C(τ + 2ζ2τ2).
Choosing ζ > 3C and τ < ζ−2, we get C(τ + 2ζ2τ2) < ζτ .The set C is clearly convex. It is a compact set of X by the choice of the
Holder exponent α′ < α and the imposed exponential decay of the functions f ∈ C.The map I is continuous by construction, therefore we can apply the SchauderFixed Point Theorem (p. 279 in [6]) to obtain a fixed point (bτ , uτ ) of I for everyτ ∈ (0, δτ ) with δτ small enough. The graph of uτ over the surface Σ[τ, bτ ] is thena self-translating surface. It is a smooth surface by the regularity theory for ellipticequations.
The properties (ii) and (iii) follow from the construction.
Appendix A. Change in metric, unit normal and mean curvatureunder a normal perturbation f
We quote some results from Appendix C in [8]. Suppose we have a C2 immersionof a surface X : M → R3. We write g,A,H and ν : M → R3 for the first andsecond fundamental forms, the mean curvature and the Gauss map respectively.Suppose f is a C2 real function on M . We write Xf : M → R3 for Xf ≡ X + fν.Then g, A, H and ν : M → R3 will denote the first and second fundamental forms,the mean curvature and the Gauss map respectively of Xf (if defined).
Let e1, e2, ν be a local orthonormal frame of R3 whose restriction to M has e1
and e2 tangent to M . Let e1 = ∇e1Xf , e2 = ∇e2Xf . We have
e1 = (1− fA11)e1 + (−fA12)e2 + f1ν,
e2 = (−fA12)e1 + (1− fA22)e2 + f2ν.
This implies that, if |fA| < 1, the Xf is an immersion. A calculation gives
ν =(−f1 + F1)e1 + (−f2 + F2)e2 + (1 + F3)ν√
1 + F4
= ν −∇f + F1e1 + F2e2 + F3ν +F5e1 + F6e2 + F7ν
1 + F4 +√
1 + F4
ν = ν −∇f + Qνf ,(19)
20 XUAN HIEN NGUYEN
where Qνf is defined by the equation above,
g11 = 1− 2fA11 + F ′11, g22 = −2fA12 + F ′12, g22 = 1− 2fA22 + F ′22,
g11 =1 + 2fA11 + F11
1 + F5, g22 =
2fA12 + F12
1 + F5, g22 =
1 + 2fA22 + F22
1 + F5,
where all the F ’s and F ′’s are polynomials in fAij and fi with terms at leastquadratic, and F3 has no terms in fi.
From now on, we use Φ to denote a term which can be either fA or∇f . We use an∗ to denote contraction with respect to g. Also, Gn stands for linear combinationswith universal coefficients of terms which are contractions with respect to g of atleast two Φ’s. Now Gn will be used to denote linear combinations (with universalcoefficients) of terms which are contractions of a number of - possibly none - Φ’swith one of the following:
(i) A ∗ Φ ∗ Φ,(ii) f∇A ∗ Φ,(iii) fA ∗ ∇2f ,(iv) ∇2f ∗ Φ ∗ Φ.
Notice that there are no quadratic terms of the form ∇f ∗∇2f , and that all termsare linear in the derivatives of at most second order of f . Denote by (·, ·) thestandard inner product in R3. Then
∇2eiej
Xf = (−fiAjk − fjAik − fAki,j)ek + (Aij + fij − fAikAjk)ν,
where we sum over repeated indices from 1 to 2. Therefore,
Aij = (∇2eiej
Xf , ν) =1√
1 + F4
(Aij + fij − fAikAjk + Gij),
where the Gij are similar to the G described above, but with a contraction missingso the last two indices are not contracted. This implies
H = gijAij = H + (∆f + |A|2f) +
(G1√
1 +G1
+G2
1 +G1 +√
1 +G1
)= H + (∆f + |A|2f) +Qf ,(20)
where we define Qf = G1√1+G1
+ G21+G1+
√1+G1
. We also compute
|A|2 =1
(1 +G2)(|A|2 + 2fjkAjk + 2fAijAjkAki
+ fjkfjk + G3 ∗A+ G4 ∗ ∇2f + G5 ∗ G6),
(21) |A|2 = |A|2 + 2fjkAjk + 2fAijAjkAki+
+G2(−|A|2 − 2fAijAjkAki) + fjkfjk + G7 ∗A+ G8 ∗ ∇2f + G9 ∗ G10
(1 +G2).
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TRANSLATING TRIDENTS 21
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Department of Mathematical Sciences, University of Cincinnati, OH, 45221
Current address: Department of Mathematical Sciences, University of Cincinnati, OH, 45221
E-mail address: [email protected]
