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ASYMPTOTIC PLATEAU PROBLEM

BARIS COSKUNUZER

ABSTRACT. This is a survey of old and recent results about the asymptoticPlateau problem. Our aim is to give a fairly complete pictureof the field, andpresent the current situation.

1. INTRODUCTION

The asymptotic Plateau problem in hyperbolic space basically asks the existenceof an area minimizing submanifoldΣ ⊂ H

n+1 asymptotic to given submanifoldΓ ⊂ Sn

∞(Hn+1). In this survey article, we will cover old and recent resultsonthe problem. Most of the time, we will give the essential ideas of the proofs. Ouraim is to give a nice expository introduction for the interested researchers, and topresent a picture of this growing field.

CONTENTS

1. Introduction 12. Preliminaries 23. Existence 33.1. Absolutely Area Minimizing Submanifolds 33.2. Fixed Topological Type 44. Boundary Regularity at Infinity 65. Number of Solutions 95.1. Uniqueness and Finiteness Results 95.2. Nonuniqueness Results 136. CMC Hypersurfaces 136.1. Existence 156.2. Boundary Regularity at Infinity 166.3. Number of Solutions 176.4. Foliations of Hyperbolic Space 187. Further Results 197.1. Properly Embeddedness 197.2. The Global Structure 217.3. Intersections 227.4. Renormalized Area 23References 24

The author is partially supported by EU-FP7 Grant IRG-226062, TUBITAK Grant 109T685 andTUBA-GEBIP Award.

1

2 BARIS COSKUNUZER

2. PRELIMINARIES

In this section, we will overview the basic results which we will use in the fol-lowing sections. First, we will give the definitions of area minimizing surfaces.First set of the definitions are about compact submanifolds.The second set of thedefinitions are their generalizations to the noncompact submanifolds.

Definition 2.1. (Compact Case) LetD be a compact disk in a manifoldX. Then,D is anarea minimizing diskin X if D has the smallest area among the disks inX with the same boundary. LetS be a compact submanifold with boundary in amanifoldX. Then,S is anabsolutely area minimizing submanifoldin X if S hasthe smallest volume among all submanifolds (no topologicalrestriction) with thesame boundary inX. The absolutely area minimizing surfaces and hypersurfacescan be defined likewise.

Definition 2.2. (Noncompact Case) Anarea minimizing plane(least area plane)is a complete plane in a manifoldX such that any compact subdisk in the plane isan area minimizing disk inX. Let∆ be a complete submanifold in a manifoldX.Then,∆ is anabsolutely area minimizing submanifoldin X if any compact part(codimension-0 submanifold with boundary) of the∆ is an absolutely area mini-mizing hypersurface inX. The absolutely area minimizing surfaces, hypersurfacesand hyperplanes can be defined likewise.

Definition 2.3. A minimal surface (submanifold or hypersurface)in a manifoldXis a surface (submanifold or hypersurface) whose mean curvature vanishes every-where.

Note that the mean curvature being0 is equivalent to be locally area minimizing[10]. Hence, all area minimizing surfaces and hypersurfaces are also minimal.

Definition 2.4. (Convex Hull) LetA be a subset ofSn∞(Hn+1). Then theconvex

hull of A, CH(A), is the smallest closed convex subset ofHn+1 which is asymp-

totic toA. Equivalently,CH(A) can be defined as the intersection of all supportingclosed half-spaces ofHn+1 [23].

Note that∂∞(CH(A)) = A for anyA ⊂ Sn∞(Hn+1) (Note that this is a special

property ofHn+1, see [32]). In general, we say a subsetΣ of X has the convexhull property if it is in the convex hull of its boundary inX, i.e.Σ ⊂ CH(∂Σ). Inspecial case, ifΣ is a complete and noncompact hypersurface inH

n+1, then we sayΣ has convex hull property if it is in the convex hull of its asymptotic boundary, i.e.Σ ⊂ CH(∂∞Σ). The minimal hypersurfaces inHn+1 have convex hull property.

Lemma 2.1. [5] LetΣ be a minimal submanifold inHn+1 with ∂∞Σ = Γ. ThenΣ ⊂ CH(Γ).

The idea is quite simple. LetΣ be a minimal submanifold inHn+1 with ∂∞Σ =Γ. Let K be a nonsupporting halfspace inHn+1, i.e. ∂∞K ∩ Γ = ∅. SinceKis a halfspace inHn+1, we can foliateK with geodesic planes whose asymptoticboundaries are in∂∞K. Then, by maximum principle [10], K ∩ Σ = ∅, andhenceΣ ⊂ CH(Γ). We should also note that instead of smooth submanifolds, if

ASYMPTOTIC PLATEAU PROBLEM 3

one deals with area minimizing rectifiable currents, or stationary varifolds, whichmight have some singularities, for this type of results, oneneedsstrong maximumprinciple results which applies to these settings due to Simon [47], Solomon-White[49], Ilmanen [33] and Wickramasekera [57].

Throughout the paper,Hn+1 will represent the hyperbolicn + 1-space.Hn+1

has a natural compactificationHn+1 = Hn+1 ∪ Sn

∞(Hn+1) whereSn∞(Hn+1) is

the sphere at infinity ofHn+1. If Σ is a subset ofHn+1, the asymptotic boundaryof Σ, say∂∞Σ, can be defined as∂∞Σ = Σ−Σ whereΣ is closure ofΣ in Hn+1

in the Euclidean metric. In the remaining of the paper, it is mostly a good idea toimagineHn+1 in the Poincare ball model.

3. EXISTENCE

There are basically2 types of existence results for the asymptotic Plateau prob-lem. The first type is the existence of absolutely area minimizing submanifolds inX for a given asymptotic boundary in∂∞X. In this type, there is no topologicalrestriction on the submanifolds. The other type is the fixed topological type. Thearea minimizing submanifold with the given asymptotic datashould also be in thegiven topological type.

3.1. Absolutely Area Minimizing Submanifolds.

By using geometric measure theory methods, Michael Anderson solved the as-ymptotic Plateau problem for absolutely area minimizing varieties for any dimen-sion and codimension in [5].

Theorem 3.1. [5] Let Γp → Sn∞(Hn+1) be an embedded closed submanifold in

the sphere at infinity ofHn+1. Then there exists a complete, absolutely area mini-mizing locally integral p+1-currentΣ in H

n+1 asymptotic toΓp at infinity.

Proof: (Sketch) LetΓp be an embedded closed submanifold inSn∞(Hn+1).

First, Anderson proves a monotonicity formula for stationary p+1-currents suchthat the ratio between the volume of a stationary p+1-current restricted to ar-ballin H

n+1 and the volume of p+1-dimensionalr-ball is nondecreasing inr ([5],Theorem 1). Then, he defines a sequence of closed submanifoldsΓp

t in Hn+1 such

thatΓpt ⊂ ∂Bt(0) andΓp

t → Γp.Let Σt be an area minimizing integralp-current with∂Σt = Γt [24]. Then

by using the monotonicity formula, he gives a lower bound forthe volume ofΣt

restricted tor-ball, i.e.cr < ||Σt|Br ||. Also, by using the area minimizing propertyof Σt, he easily gives an upper boundCr for the volume ofΣt restricted tor-ball.Then,cr < ||Σt|Br || < Cr. Hence, by using compactness theorem for integralcurrents (See [24], [43]), he gets a convergent subsequence forΣi for eachr-ball. Then, by using diagonal subsequence argument, he extracts a convergentsubsequenceΣij → Σ whereΣ is an area minimizing integral p+1-current with∂∞Σ = Γp.

4 BARIS COSKUNUZER

Remark3.1. This result is one of the most important results in the field. This semi-nal paper can be considered as the beginning of the study of the asymptotic Plateauproblem. Later, we will see various generalizations of thisresult to different set-tings. Note that the embeddedness assumption on the given asymptotic boundaryis very essential. In [36], Lang constructed immersed examples inSn

∞(Hn+1) withno solutions to asymptotic Plateau problem.

Remark3.2. (Interior Regularity) By interior regularity results of geometric mea-sure theory [24], [43], whenp = n − 1, the currents in theorem are smoothly em-bedded hypersurfaces except for a singular set of Hausdorffdimensionn − 7. Inparticular whenp = n− 1 < 6, Σ is a smoothly embedded hypersurface inH

n+1.In the higher codimension case (p < n− 1), the interior regularity results say thatthe absolutely area minimizing currents are smoothly embedded p+1-submanifoldsin H

n+1 except for a singular set of Hausdorff dimensionp− 1.

Later, again by using geometric measure theory methods, Lang and Bangertgeneralized this result to Gromov hyperbolic Hadamard manifolds with boundedgeometry, and some other special cases in [35], [36], and [8] (See also [26]).

Theorem 3.2. [35] Let X be a Gromov hyperbolic Hadamardn-manifold withbounded geometry. LetΓ be ap dimensional closed submanifold in∂∞X. Thenthere exists a complete, absolutely area minimizing locally integral p+1-currentΣin X asymptotic toΓp at infinity.

Note that the varieties constructed in theorems above are absolutely area mini-mizing, and has no topological restrictions on them. Another interesting case is thefixed topological type.

3.2. Fixed Topological Type.

In above result, Anderson got absolutely area minimizing varieties asymptotic togiven submanifold in the asymptotic sphere. As there is no topological restrictionon the objects, we have no idea about the topological properties of them.

Another interesting case in Plateau problem is the fixed topological type. Thequestion is to find the smallest area surface in the given topological type with thegiven boundary. Its generalization to the asymptotic Plateau problem is natural.

On the other hand, hyperbolic3-manifolds, and essential2-dimensional sub-manifolds in them are very active research area. By essential, we meanπ1-injectivesurfaces, and they are very important tools to understand the structure of the hy-perbolic manifold by using geometric topology tools. At this point, when we passto the universal cover of the hyperbolic manifold and essential surfaces in them,the asymptotic Plateau problem in disk type becomes an important technique forconstruction of area minimizing representative of these essential surfaces in3-manifolds.

In [6], Anderson focused on the asymptotic Plateau problem in disk type, andgave an existence result in dimension3.

Theorem 3.3. [6] LetΓ be a simple closed curve inS2∞(H3). Then, there exists a

complete, area minimizing planeΣ in H3 with ∂∞Σ = Γ.

ASYMPTOTIC PLATEAU PROBLEM 5

The proof is very similar to the proof of the previous theorem. The basic differ-ence is instead of using area minimizing surfacesΣt with ∂Σt = Γt, he used thearea minimizingembeddeddisksDt with ∂Dt = Γt. The existence of the diskscomes from the solution of Plateau problem in disk type. However, the essentialpoint is that the disks are embedded and they are given by [4] (later more generalresult given by [41]). Hence, by using similar ideas, Anderson extracted a limitDi → Σ whereΣ is an area minimizing plane inHn+1 with ∂∞Σ = Γ.

Remark3.3. Note that this result is for just dimension3, it is not known if its gener-alization to higher dimensions is true or not. It might be possible to construct areaminimizing hyperplanes inHn+1 for any dimension, by generalizing these ideasand White’s results in [55] to replace the sequence of disksDi in Anderson’sproof with compact area minimizing hyperplanes inH

n+1.

Also, in [6], Anderson constructed special Jordan curves inS2∞(H3) such that

the absolutely area minimizing surface given by Theorem 3.1cannot be a plane([6], Theorem 4.5). Indeed, he constructed examples with genusg > g0 for anygiven genusg0. He also used these surfaces for some nonuniqueness resultswhichwe mention later.

In the same context, de Oliveira and Soret showed the existence of a completestable minimal surface inH3 for any given topological type of a surface withboundary. Also, they studied the isotopy type of these surfaces in some specialcases. The main difference with Anderson’s existence result is that Anderson startswith the asymptotic boundary data, and gives an area minimizing hypersurfacewhere there is no control on the topological type, while de Oliveira and Soret startswith a surface with boundary and constructs a stable minimalembedded surface ofthis type whose asymptotic boundary is essentially determined by the surface.

Theorem 3.4. [45] LetM be a compact orientable surface with boundary. Thenint(M) can be minimally, completely, properly and stably embeddedin H

3. Fur-thermore, the embeddings extends smoothly to an embedding from M to H3, thecompactification ofH3.

On the other hand, Gabai gave another construction for Theorem 3.3. Indeed, heneeded this results for more general metrics, and he gave a topological constructionfor such area minimizing planes in more general settings.

Theorem 3.5. [25] LetX beH3 with a different Riemannian metric induced froma metric on a closed3-manifold. LetΓ be a simple closed curve inS2

∞(X). Then,there exists aD2-limit laminationσ whose leaves are area minimizing planes inXwith ∂∞σ = Γ.

Proof: (Sketch) LetX beM whereM is the universal cover of a hyperbolic3-manifold M with any Riemannian metric. In a similar fashion to the Ander-son’s proof, Gabai starts with a sequence of area minimizingdisksDi in X with∂Di = Γi → Γ. To get a limiting plane here, instead of using the compactnesstheorem of geometric measure theory, he extracts some kind of Gromov-Hausdorfflimit σ of the sequenceDi by using minimal surface tools and techniques of [31].

6 BARIS COSKUNUZER

In particular, the sequenceDi of embedded disks in a Riemannian manifoldXconvergesto the laminationσ if

i) For any convergent sequencexni in X with xni

∈ Dniwhereni is a strictly

increasing sequence,limxni∈ σ.

ii) For anyx ∈ σ, there exists a sequencexi with xi ∈ Di andlim xi = x suchthat there exist embeddingsfi : D2 → Di which converge in theC∞-topology toa smooth embeddingf : D2 → Lx, wherexi ∈ fi(Int(D

2)), andLx is the leaf ofσ throughx, andx ∈ f(Int(D2)).

We call such a laminationσ aD2-limit lamination. Here, thetopologicallimit σis essentially all the limit points of a very special subsequence. Then, since locallythese are limits of area minimizing disks, by using the techniques of [31] he showsfirst that the leaves of the laminationσ are minimal planes. Then by using topolog-ical arguments, Gabai proves that these planes are not only minimal, but also areaminimizing. Then, he shows that this lamination must stay ina neighborhood ofthe convex hull ofΓ, i.e. σ ⊂ NC(CH(Γ)) whereCH(Γ) is the convex hull ofΓandC is a constant independent ofΓ. Then, he shows that∂∞σ = Γ and finishesthe proof.

Remark3.4. Until this paper, all the existence results on this problem came outvia the techniques of geometric measure theory. The disadvantage of geometricmeasure theory is that it is very powerful with absolutely area minimizing subman-ifolds, but you have to work very hard to get results in fixed topological type case.On the other hand, Gabai’s techniques are very natural for the fixed topologicaltype case as you can control the limiting process and limiting object topologically.

Later, the author generalized Gabai’s results to the Gromovhyperbolic3-spaceswith cocompact metric.

Theorem 3.6. [12] LetX be a Gromov hyperbolic3-space with cocompact metric.LetΓ be a simple closed curve inS2

∞(X). Then, there exists aD2-limit laminationσ whose leaves are area minimizing planes inX with ∂∞σ = Γ.

4. BOUNDARY REGULARITY AT INFINITY

After the above existence theorems, the next natural question was the regularityof the hypersurfacesΣ obtained as a solution of the asymptotic Plateau problem.By the interior regularity theorems of geometric measure theory,Σ is real analytichypersurface ofHn+1 away from a singular subset of Hausdorff dimensionn− 7.The question is the behavior of the hypersurfaces near infinity, i.e. the boundaryregularity at infinity. In other words, ifΣ is an area minimizing hypersurface inH

n+1, then what can be said about the boundary regularity ofΣ in Hn+1?The first main result about this problem came from Hardt and Lin in [29]. By

using geometric measure theory methods, they showed that near infinity, Σ is asregular as the asymptotic boundary forC1,α asymptotic boundary data.

Theorem 4.1. [29] Let Γ be aC1,α codimension-1 submanifold ofSn∞(Hn+1)

where0 < α ≤ 1. If Σ is a complete, absolutely area minimizing locally integraln-current inHn+1 with∂∞Σ = Γ. Then, nearΓ,Σ∪Γ is union ofC1,α submanifolds

ASYMPTOTIC PLATEAU PROBLEM 7

with boundary in Euclidean metric onHn+1. These submanifolds have disjointanalytic interiors, and they meetSn

∞(Hn+1) orthogonally atΓ.

Also, if we take the upper half space model forHn+1, thenRn × 0 ∪ ∞

would represent the asymptotic sphere. Then, for a givenC1 hypersurfaceΓ inR

n×0, there isρΓ with (Σ∪Γ)∩y < ρΓ is a finite union ofC1 submanifoldswith boundary which can be viewed as a graph overΓ× [0, ρΓ).

This result is very interesting as an area minimizing hypersurface inHn+1 hasbetter regularity near asymptotic boundary than in the interior. In other words, ifΣ is an area minimizing hypersurface inHn+1 with ∂∞Σ = Γ as above,Σ mighthave a singular set of Hausdorff dimensionn − 7, but this set must stay in thebounded part ofΣ as(Σ∪Γ)∩y < ρΓ is a finite union ofC1 submanifolds withboundary. In order to get this result, Hardt and Lin first get an interior regularityresult ”near infinity” by showing thatΣ can be expressed as a union of graphs offinitely many analytic functions on vertical planes tangentto Γ. Then by usingthis interior regularity ”near infinity” result, and hyperbolic isometries, they provethe regularity at boundary. In particular, if there was a sequence of singular pointsescaping to infinity (or converging to a point in asymptotic boundary), by rescalingΣ with hyperbolic isometries, they get new area minimizing hypersurfaces, and theimages of the singular points in these new area minimizing hypersurfaces wouldcontradict the earlier interior regularity result.

Later, by studying the following quasilinear, non-uniformly elliptic equationwhose solutions are minimal hypersurfaces in hyperbolic space, Lin and Tonegawagot higher regularity near asymptotic boundary. In the upper half space modelof Hn+1, let Ω ⊂ R

n × 0 be a domain andf : Ω → R+ be a function.

Considergraph(f) = Σf which defines a hypersurface inHn+1. The volume ofΣKf = Σf ∩ K × R

+ whereK is a compact subset ofΩ can be described asfollows:

vol(ΣKf ) =

∫

K

f−n√

1 + |∇f |2dx

Then, the corresponding Euler-Lagrange equation of this variational integralwould give the following Dirichlet problem:

∇f −fi.fj

1+|df |2 fij +nf= 0 in Ω

f > 0 in Ωf = 0 in ∂Ω

where|df |2 =∑n

i=1 f2i . In [6], Anderson showed the existence and the uniqueness

of the solution to this Dirichlet problem provided that∂Ω has nonnegative meancurvature with respect to inward normal inRn × 0.

If one wants to focus on the boundary regularity of the solution of this Dirichletproblem, an equivalent local description of the problem canbe given by consider-ing graph(f) near a point of the asymptotic boundary as a graph over a verticalplane which is tangent to the asymptotic boundary at the given point. In otherwords, letΓ = ∂Ω be at leastC1. Let P be the vertical tangent plane toΓ at

8 BARIS COSKUNUZER

p. By using hyperbolic isometries, we can assumep = 0 in Rn × 0 andP is

the plane(x, 0, y) ∈ Hn+1 | (x, 0) ∈ Rn andy ≥ 0. Then after scaling with

hyperbolic isometries, we can formulate the problem as follows: Letu : D → R

whereD = (x, 0, y) ∈ P | |x| ≤ 1 and0 ≤ y ≤ 1

y(∇u−ui.uj

1+|du|2uij)− n.uy = 0 in D

u(x, 0, 0) = ϕ(x)

whereu(x, 0, 0) = ϕ(x) is given byΓ nearp. Hence the question becomes whetheru is as smooth asϕ.

Lin studied first this quasi-linear degenerate elliptic partial differential equationin [39] and got the following result.

Theorem 4.2. [39] Let Γ be aCk,α codimension-1 submanifold ofSn∞(Hn+1)

where1 ≤ k ≤ n− 1 and0 ≤ α ≤ 1 or k = n and0 ≤ α < 1. If Σ is a completearea minimizing hypersurface inHn+1 with ∂∞Σ = Γ. Then, nearΓ, Σ ∪ Γ isunion ofCk,α submanifolds with boundary in Euclidean metric onHn+1.

Later, Tonegawa completed Lin’s results for higher regularity case by studyingfurther the above PDE, and finished off the problem by giving the following veryinteresting parity in [53].

Theorem 4.3. [53] Let Γ be aCk,α codimension-1 submanifold ofSn∞(Hn+1)

andΣ be a complete area minimizing hypersurface inHn+1 with ∂∞Σ = Γ. Let

k ≥ n+ 1 and0 < α < 1. Then,1. If n is even, thenΣ ∪ Γ is aCk,α submanifold with boundary nearΓ.2. If n is odd, thenΣ ∪ Γ may not be aCn+1 submanifold with boundary near

Γ in general.

This is a very interesting result as it gives a very subtle relation between thedimension and the asymptotic regularity of area minimizinghypersurfaces. In par-ticular, forn odd, Tonegawa gives a necessary and sufficient condition that Γ hasto satisfy in the form of a PDE in order to recoverCk,α regularity. Hence, whennis odd, ifΓ does not satisfy this PDE,Σ ∪ Γ cannot be smoother thanCn+1 eventhoughΓ is very smooth. Note also that in [53], Tonegawa studied a more generalform of the PDE above and generalized these results to Constant Mean Curvature(CMC) hypersurfaces inHn+1 (See Section 6.2).

For the higher codimension case (k < n), by the interior regularity results ofgeometric measure theory, the absolutely area minimizing k-currents are smoothlyembeddedk-submanifolds inHn+1 except for a singular set of Hausdorff dimen-sion k − 2. For the boundary regularity at infinity in this case, Lin also showedtheexistenceof an area minimizingk-currentΣ in H

n+1 which is as regular as theboundary at infinity, whereΓ = ∂∞Σ is aC1,α smooth closedk − 1-submanifoldin Sn

∞(Hn+1).

Theorem 4.4. [40] LetΓ be aC1,α smooth closedk−1-submanifold inSn∞(Hn+1).

Then there exists a complete area minimizingk-current inHn+1 with ∂∞Σ = Γ

such that nearΓ, Σ ∪ Γ is aC1,α submanifold with boundary in Euclidean metriconHn+1.

ASYMPTOTIC PLATEAU PROBLEM 9

Note that unlike the codimension-1 case, this higher codimension case does notsayany area minimizingk-current with asymptotic smooth asymptotic boundaryis boundary regular at infinity. This result only says the existence of such an areaminimizing current for any given smooth asymptotic data.

5. NUMBER OF SOLUTIONS

There are basically3 types of results on the number of solutions to the asymp-totic Plateau problem. The first type is the uniqueness results which classifies theasymptotic data with the unique solution to the asymptotic Plateau problem. Thenext type is the generic uniqueness and generic finiteness results which came outrecently. The last type can be called as the nonuniqueness results which constructsthe asymptotic data with more than one solution to the problem.

5.1. Uniqueness and Finiteness Results.

Next to the existence theorems, Anderson gave very interesting uniqueness andnonuniqueness results on minimal surfaces inH

3 and area minimizing hypersur-faces inHn+1 in [5] and [6]. Before visiting nonuniqueness results, we will listthe uniqueness results about the the asymptotic Plateau problem.

First, in [5], Anderson showed that if the given asymptotic boundaryΓ0 whichis a hypersurface bounding a convex domain inSn

∞(Hn+1), then there exists aunique absolutely area minimizing hypersurfaceΣ0 in H

n+1.

Theorem 5.1. [5] LetΓ0 be a hypersurface bounding a convex domain inSn∞(Hn+1).

Then, there exists a unique absolutely area minimizing hypersurfaceΣ0 in Hn+1

with ∂∞Σ0 = Γ0.

Proof: (Sketch) LetΓ0 be codimension-1 submanifold bounding a convexdomain inSn

∞(Hn+1), andΣ0 be an area minimizing hypersurface inHn+1 with∂∞Σ0 = Γ0 (Existence ofΣ0 is guaranteed by Theorem 3.1). AsΓ0 bounds aconvex domain inSn

∞(Hn+1), we can find a continuous family of isometriesϕtof Hn+1 such thatϕt(Γ0) = Γt whereΓt foliates wholeSn

∞(Hn+1). Similarly,if ϕt(Σ0) = Σt, then∂∞Σt = Γt, and asΣt images of continuous family ofisometries, it foliates wholeHn+1.

Hence, if there are two minimal hypersurfacesM1,M2 with ∂∞Mi = Γ0, oneof them (sayM2) is not a leaf of the foliation, andM2 must intersect a leafΣt0 ofthe foliation tangentially and by lying in one side.This contradicts to the maximumprinciple for minimal hypersurfaces.

Later, by using similar ideas, Hardt and Lin generalized this result to the codimension-1 submanifolds bounding star shaped domains inSn

∞(Hn+1) in [29].

Theorem 5.2. [29] Let Γ0 be a hypersurface bounding a star shaped domain inSn∞(Hn+1). Then, there exists a unique absolutely area minimizing hypersurface

Σ0 in Hn+1 with ∂∞Σ0 = Γ0.

10 BARIS COSKUNUZER

While these are the only known results on the number of solutions of the asymp-totic Plateau problem for a long time, many generic uniqueness results have comeout recently in both general case and fixed topological type case.

For the general case, the author showed that the space of closed codimension-1submanifoldsΓ in Sn

∞(Hn+1) bounding a unique absolutely area minimizing hy-persurfaceΣ in H

n is dense in the space of all closed codimension-1 submanifoldsin Sn

∞(Hn+1) by using a simple topological argument.

Theorem 5.3. [18] Let B be the space of connected closed codimension-1 sub-manifolds ofSn

∞(Hn+1), and letB′ ⊂ B be the subspace containing the closedsubmanifolds ofSn

∞(Hn+1) bounding a unique absolutely area minimizing hyper-surface inHn. ThenB′ is dense inB.

Proof: (Sketch) For simplicity, we will focus on the area minimizing planesin H

3. The general case is similar. LetΓ0 be a simple closed curve inS2∞(H3).

First, by using Meeks-Yau exchange roundoff trick, the author establishes that ifΓ1 andΓ2 are two disjoint simple closed curves inS2

∞(H3), andΣ1 andΣ2 arearea minimizing planes inH3 with ∂∞Σi = Γi, thenΣ1 andΣ2 are disjoint, too.Then, by using this result, he shows that for any simle closedcurveΓ in S2

∞(H3)either there exists a unique area minimizing planeΣ in H

3 with ∂∞Σ = Γ, or thereexist twodisjoint area minimizing planesΣ+,Σ− in H

3 with ∂∞Σ± = Γ0.Then, take a small neighborhoodN(Γ0) ⊂ S2

∞(H3) which is an annulus, andfoliate N(Γ0) by simple closed curvesΓt wheret ∈ (−ǫ, ǫ), i.e. N(Γ0) ≃Γ × (−ǫ, ǫ). By the above fact, for anyΓt either there exists a unique area min-imizing planeΣt, or there are two area minimizing planesΣ±

t disjoint from eachother. As disjoint asymptotic boundary implies disjoint area minimizing planes,if t1 < t2, thenΣt1 is disjoint andbelowΣt2 in H

3. Consider this collectionof area minimizing planes. Note that for curvesΓt bounding more than one areaminimizing plane, we have a canonical regionNt in H

3 between the disjoint areaminimizing planesΣ±

t .Now, the idea is to consider thethicknessof the neighborhoodsNt assigned to

the asymptotic curvesΓt. Let st be the length of the segmentIt of β (a fixedfinite length transversal curve to the collection) betweenΣ+

t andΣ−t , which is the

width of Nt assigned toΓt. Then, the curvesΓt bounding more than one areaminimizing planes have positive width, and contributes to the total thickness ofthe collection, and the curves bounding unique area minimizing plane has0 widthand do not contribute to the total thickness. Since

∑t∈(−ǫ,ǫ) st < C, the total

thickness is finite. This implies for only countably manyt ∈ (−ǫ, ǫ), st > 0, i.e.Γt bounds more than one area minimizing plane. For the remaining uncountablymanyt ∈ (−ǫ, ǫ), st = 0, and there exists a unique area minimizing plane for thoset. This proves the space of Jordan curves of uniqueness is dense in the space ofJordan curves inS2

∞(H3). Then, the author shows that this space is not only dense,but also generic. Also, this technique is quite general, andit can be generalized tomany different settings [15], [19].

ASYMPTOTIC PLATEAU PROBLEM 11

H

S

N

Ns

t

P

ΓΓ

+

Σs

Σo

p

sΓ

o

P

−

−

8

p+

Σ+t

−

β

t

2

3

Is

I t

FIGURE 1. A finite segment of geodesicγ intersects the collec-tion of area minimizing planesΣt in H

n asymptotic toΓt inS2∞(H3).

On the other hand, there has been important progress on the number of solu-tions to the asymptotic Plateau problem in fixed topologicaltype case. Recently in[13], the author showed a generic finiteness result forC3 smooth Jordan curves inS2∞(H3) for area minimizing planes inH3 by using geometric analysis and global

analysis methods. Later in [14], he improved this result to a generic uniquenessresult.

Theorem 5.4. [14] Let A be the space ofC3 simple closed curves inS2∞(H3).

Then there exists an open dense subsetA′ ⊂ A such that for anyΓ ∈ A′, thereexists a unique area minimizing planeΣ with ∂∞Σ = Γ.

Proof: (Sketch) In [13], by generalizing Tomi and Tromba’s global analytictechniques in [52] to hyperbolic setting, and by using Li and Tam’s powerful results[37] and [38], the author showed that the boundary restriction mapπ from the spaceof minimal maps fromD2 to H

3 with C3 asymptotic data to the space of theC3

immersions ofS1 into S2∞(H3) is Fredholm of index0. Hence, the derivative ofπ

is isomorphism for the generic curves.Fix a generic curveΓ in S2

∞(H3). By using the inverse function theorem, thereis a neighborhoodUΣ of a area minimizing planeΣ in π−1(Γ), mapping home-omorphically into a neighborhoodVΓ of Γ. By taking a pathα in VΓ, and byconsidering the corresponding pathπ−1(α) in UΣ, one can get a continuous familyof minimal planes with disjoint asymptotic boundaries aroundΣ. Then, the authorshows that this continuous family of minimal planes is indeed a foliation by areaminimizing planes of a neighborhood ofΣ. This implies the uniqueness of the areaminimizing plane inH3 spanningΓ. Then the author proves that the same is true

12 BARIS COSKUNUZER

for any curve in a neighborhood of a generic curve, and gets anopen dense subsetof theC3 Jordan curves inS2

∞(H3) with the uniqueness result.

Recently, in [2], Alexakis and Mazzeo generalized this result to any surface ofgenusk by using different methods. In [13], the author works with the space ofparametrizations of minimal planes inH3, and hence, in order to get a genericfiniteness result, he needs to deal with different parametrizations of the same min-imal plane. In [2], Alexakis and Mazzeo showed that ifMk is the moduli space ofall complete minimal surfaces of genusk in H

3 with asymptotic boundary curveaC3,α simple closed curve inS2

∞(H3), andξ is the space ofC3,α simple closedcurve inS2

∞(H3), then the boundary restriction mapπk : Mk → ξ is Fredholmof index0 (see also Section 7.2). Moreover, they also showed thatπk is not onlyFredholm of index0, but also proper (Theorem 4.3 in [2]). Hence by Sard-Smaletheorem [48], this implies a generic finiteness result for minimal surfaces of genusk. In other words, for a genericC3,α simple closed curveΓ in S2

∞(H3), thereexist finitely many complete minimal surfacesΣ of genusk in H

3 with ∂∞Σ = Γ.Indeed, their result also applies to convex cocompact hyperbolic 3-manifolds, too.

Note that the above generic uniqueness result for area minimizing planes re-quires some smoothness condition on the curves. Later, the author improved hisresult by removing the smoothness condition. This time, theauthor uses topologi-cal methods instead of techniques of global analysis. The technique is essentiallysame with the area minimizing hypersurfaces case mentionedabove.

Theorem 5.5. [18] LetA be the space of simple closed curves inS2∞(H3) and let

A′ ⊂ A be the subspace containing the simple closed curves inS2∞(H3) bounding

a unique area minimizing plane inH3. Then,A′ is generic inA, i.e. A − A′ is aset of first category.

Remark5.1. Note that the same result is true for area minimizing surfaces inH3,

too [18]. By these results, the asymptotic Plateau problem generically has a uniquesolution in both area minimizing surfaces inH3 case and area minimizing planesin H

3 case. In higher dimensions, the closed codimension-1 submanifolds inSn∞(Hn+1) bounding a unique absolutely area minimizing hypersurfacein H

n+1

are only dense in the closed codimension-1 submanifolds inSn∞(Hn+1). However,

by using the similar ideas, by fixing the topological type of the closed codimension-1 submanifold inSn

∞(Hn+1), it might be possible to get some generic uniquenessresult, too.

Remark5.2. Notice that except the convex and star-shaped asymptotic boundarycases, all the uniqueness results on the asymptotic Plateauare about area mini-mizing surfaces or area minimizing planes. Unfortunately,the techniques used forthese results cannot be extended to the minimal surfaces or minimal planes cases.The main obstacle here is that while two area minimizing surfaces with disjointasymptotic boundaries must be disjoint, the same statementmay not be true forminimal surfaces. In any case, it would be interesting problem to study this casein order to understand whether the simple closed curves inS2

∞(H3) bounding aunique minimal surface (or plane) is dense in the space of simple closed curves

ASYMPTOTIC PLATEAU PROBLEM 13

in S2∞(H3) or not. The author believes that the similar statements are not true in

minimal surfaces (or planes) case.

5.2. Nonuniqueness Results.

Beside his existence results, Anderson also gave many different nonuniquenessresults for the asymptotic Plateau problem in the fixed topological type in [6].

Theorem 5.6. [6] There exists a simple closed curveΓ in S2∞(H3) such that there

are infinitely many complete minimal surfacesΣi in H3 with ∂∞Σi = Γ.

For the proof of this theorem, Anderson first constructs a simple closed curvesuch that the absolutely area minimizing surface given by his existence theorem isnot a plane (positive genus) (a similar construction can be found in [30]). Then, bymodifying this curve, he constructs a curveΓ with the same property such that it isalso a limit set for a quasi-Fuchsian groupΛ. Since the absolutely area minimizingsurfaceΣ is Λ invariant and has positive genus, this implies the area minimizingsurfaceΣ/Λ in the compact hyperbolic manifoldH3 is not π1-injective. Thisimplies that the absolutely area minimizing surfaceΣ in H

3 with ∂∞Σ = Γ musthave infinite genus. Then, by using this property, he shows that there exist infinitelymany completeminimalsurfaces asymptotic toΓ.

Note that this result shows nonuniqueness forminimal surfacesfor fixed topo-logical type. Later, the author show nonuniqueness for areaminimizing planes(surfaces) inH3 case. In particular, the author shows that there are simple closedcurves inS2

∞(H3) bounding more than one area minimizing plane (surface).

Theorem 5.7. [18] There exists a simple closed curveΓ in S2∞(H3) such that there

are more than one area minimizing plane (surface)Σi in H3 with ∂∞Σi = Γ.

Remark5.3. In the nonuniqueness results above, only Hass’ result givesan explicitexample of a simple closed curve inS2

∞(H3) bounding more than one minimalsurfaces inH3. All other results on nonuniqueness so far shows the existence ofsuch a curve, but it does not give one. So, it would be interesting to construct anexplicit simple closed curve inS2

∞(H3) bounding more than one area minimizingsurface (or plane).

Remark5.4. Although there are many examples of simple closed curves inS2∞(H3)

bounding more than one minimal surface or more than one area minimizing surface(or plane) inH3, there is no example in higher dimensions so far. It would be in-teresting to generalize the nonuniqueness results to higher dimensions by showingwhether there exists a closed codimension-1 submanifold inSn

∞(Hn+1) boundingmore than one absolutely area minimizing hypersurfaces inH

n+1.

6. CMC HYPERSURFACES

After many important results on minimal hypersurfaces in hyperbolic space,like existence, regularity, etc., the question of generalization of these results toconstant mean curvature (CMC) hypersurfaces was naturallyraised: For a given

14 BARIS COSKUNUZER

codimension-1 submanifoldΓ in Sn∞(Hn+1), does there exists a complete CMC

hypersurfaceΣ with specified mean curvatureH in Hn+1 and∂∞Σ = Γ?

For simplicity, from now on, we will call CMC hypersurfaces with mean curva-tureH asH-hypersurfaces.

Note that for this generalization of the asymptotic Plateauproblem, we needto assume that|H| < 1 (after fixing an orientation onHn+1). This is because itis impossible to have a completeH-hypersurfaceΣ in H

n+1 with |H| ≥ 1 and∂∞Σ = Γ as we can always find a horosphere (H = 1) in H

n+1 with tangentialintersection with such aΣ which contradicts to the maximum principle.

We should also note thatH-hypersurfaces inH3 with H = 1 andH > 1 are alsoan area of active research. A basic reference for CMC hypersurfaces in hyperbolicspace withH > 1 would be [34]. For the caseH = 1, we refer to Rosenberg’ssurvey [46], and Bryant’s seminal paper [9] where he showed that any minimalsurface inR3 is isometric to a CMC surface inH3 with H = 1.

We should point out that the generalization of area minimizing (or minimal)hypersurfaces to CMC hypersurfaces is quite natural. As we see the minimal hy-persurfaces (H = 0) as the critical points of the area functional, CMC hypersur-faces occurs as the critical points of some modification of the area functional witha volume constraint. In particular, letΣn be a compact hypersurface, boundinga domainΩn+1 in some ambient Riemannian manifold. LetA be the area ofΣ,andV be the volume ofΩ. Let’s varyΣ through a one parameter familyΣt, withcorresponding areaA(t) and volumeV (t). If f is the normal component of thevariation, andH is the mean curvature ofΣ, then we getA′(0) = −

∫Σ nHf , and

V ′(0) =∫Σ f wheren is the dimension ofΣ, andH is the mean curvature.

Now, letΣ be a hypersurface with boundaryΓ. We fix a hypersurfaceM with∂M = Γ, and defineV (t) to be the volume of the domain bounded byM andΣt. Now, we define a new functional as a combination ofA andV . Let IH(t) =A(t) + nHV (t). Note thatI0(t) = A(t). If Σ is a critical point of the functionalIH for any variationf , then this will imply Σ has constant mean curvatureH[28]. Note that critical point of the functionalIH is independent of the choice ofthe hypersurfaceM since if IH is the functional which is defined with a differenthypersurfaceM , thenIH − IH = C for some constantC. On the other hand,we will call Σ a minimizing CMC hypersurfaceif Σ is the absolute minimum ofthe functionalIH among hypersurfaces with the same boundary. From this pointof view, CMC hypersurfaces are natural generalization of minimal hypersurfacesand area minimizing hypersurfaces as the area functional isjust H = 0 case forthe functionalIH . This point of view is very useful and essential to generalize thegeometric measure theory methods developed for area minimizing case to CMCcase as in [53] and [3].

Now, we continue with the basic notions onH-hypersurfaces inHn+1. We fix acodimension-1 closed submanifoldΓ in Sn

∞(Hn+1). Γ separatesSn∞(Hn+1) into

two parts, sayΩ+ andΩ−. By using these domains, we will give orientation tohypersurfaces inHn+1 asymptotic toΓ. With this orientation, mean curvatureH

ASYMPTOTIC PLATEAU PROBLEM 15

is positive if the mean curvature vector points towards positive side of the hyper-surface, negative otherwise. The following fact is known asmaximum principle.

Lemma 6.1. [Maximum Principle] LetΣ1 andΣ2 be two hypersurfaces in a Rie-mannian manifold which intersect at a common point tangentially. If Σ2 lies inpositive side (mean curvature vector direction) ofΣ1 around the common point,thenH1 is less than or equal toH2 (H1 ≤ H2) whereHi is the mean curvature ofΣi at the common point. If they do not coincide in a neighborhoodof the commonpoint, thenH1 is strictly less thanH2 (H1 < H2).

The other important notion about CMC Hypersurfaces inHn+1 is the general-

ization of the convex hull property to this context. Now, letΓ be a codimension-1submanifold ofSn

∞(Hn+1) and orient all spheres accordingly. IfT is a roundn − 1-sphere inSn

∞(Hn+1), then there is a uniqueH-hypersurfacePH in Hn+1

asymptotic toT for −1 < H < 1 [44]. T separatesSn∞(Hn+1) into two parts

∆+ and∆−. Similarly, PH dividesHn+1 into two domainsD+H andD−

H with∂∞D±

H = ∆±. We will call these regions asH-shifted halfspaces. If the asymp-totic boundary of aH-shifted halfspace containsΓ, then we will call thisH-shiftedhalfspace assupportingH-shifted halfspace. i.e. if A ⊂ ∆+, thenD+

H is a sup-porting H-shifted halfspace. Then theH-shifted convex hullof Γ, CHH(Γ) isdefined as the intersection of all supporting closedH-shifted halfspaces ofHn+1.

Now, the generalization of convex hull property of minimal hypersurfaces inH

n+1 to H-hypersurfaces inHn+1 is as follows [15]. Similar versions of thisresult have been proved by Alencar-Rosenberg in [3], and by Tonegawa in [53].

Lemma 6.2. [53], [3], [12] LetΣ be aH-hypersurface inHn+1 where∂∞Σ = Γand |H| < 1. ThenΣ is in theH-shifted convex hull ofΓ, i.e.Σ ⊂ CHH(Γ).

6.1. Existence.

In the following decade after Anderson’s existence ([5], [6]) and Hardt-Lin’sregularity results ([29],[39]), there have been many important generalizations ofthese results to CMC hypersurfaces in hyperbolic space. In [53], Tonegawa gen-eralized Anderson’s techniques to this case, and proved existence for CMC hyper-surfaces by using geometric measure theory methods. In the same year, by usingsimilar techniques, Alencar and Rosenberg got a similar existence result in [3].

Theorem 6.3. [53], [3] LetΓ be a codimension-1 closed submanifold inSn∞(Hn+1),

and let |H| < 1. Then there exists a CMC hypersurfaceΣ with mean curvatureH in H

n+1 where∂∞Σ = Γ. Moreover, any such CMC hypersurface is smoothexcept a closed singularity set of dimension at mostn− 7.

We should also note that Nelli and Spruck showed existence ofa CMC hy-persurface asymptotic toC2,α codimension-1 submanifoldΓ which is the bound-ary of a mean convex domain inSn

∞(Hn+1) by using analytic techniques in [44].Later, Guan and Spruck generalized this result toC1,1 codimension-1 submani-folds bounding star shaped domains inSn

∞(Hn+1).

16 BARIS COSKUNUZER

Theorem 6.4. [27] Let Ω be a star shaped (mean convex in[44]) domain inSn∞(Hn+1) whereΓ = ∂Ω is C1,1 (C2,α in [44]) codimension-1 submanifold in

Sn∞(Hn+1). Then, for any0 < H < 1, there exists a complete smoothly embedded

CMC hypersurfaceΣ with mean curvatureH and∂∞Σ = Γ. Moreover,Σ can berepresented as a graph of a functionu ∈ C1,1(Ω) (u ∈ C2,α(Ω) in [44]).

Even though this second existence result is for fairly restricted class of asymp-totic boundary data (star shaped condition), the CMC hypersurfaces obtained aresmoothly embedded with no singularity in any dimension (unlike the first one), andthey can be represented as a graph likexn+1 = u for a functionu ∈ C1,1(Ω) in halfspace model forHn+1. We should also note that, in [1], Aiyama and Akutagawagave a completely different construction for CMC surfaces of disk type inH3 withasymptotic boundaryC1,α smooth simple closed curve inS2

∞(H3) by studying aDirichlet problem at infinity.

6.2. Boundary Regularity at Infinity.

Beside the existence results, in [53], Tonegawa studied the following quasi-linear degenerate elliptic PDE which is a more general form of the PDE in Section4 for H-hypersurfaces with|H| < 1, and got important regularity results for thesehypersurfaces near asymptotic boundary.

y(∇u−ui.uj

1+|du|2uij)− n(uy −H√

1 + |Du|2) = 0 in D

u(x, 0, 0) = ϕ(x)

For k ≤ n, Tonegawa generalized the Lin’s result for minimal hypersurfaces(H = 0) in [39].

Theorem 6.5. [53] LetΓ beCk,α codimension-1 submanifold inSn∞(Hn+1) where

1 ≤ k ≤ n − 1 and 0 ≤ α ≤ 1 or k = n and 0 ≤ α < 1. If Σ is a completeCMC hypersurface inHn+1 with ∂∞Σ = Γ, thenΣ ∪ Γ is a Ck,α submanifoldwith boundary inHn+1 nearΓ.

On the other hand, Tonegawa showed that for higher regularity case,H = 0case is fairly different form theH 6= 0 case. As we mentioned in Section 4, inH = 0 case, Tonegawa showed that whenn is even the higher regularity is alwaystrue, but whenn is odd, the higher regularity depends on the asymptotic boundaryΓ (Theorem 4.3). In theH 6= 0 case, Tonegawa got a very surprising result thatwhile the similar result is true forn = 2, it is not true forn = 4.

Theorem 6.6. [53] a. (n = 2 case)LetΓ be aCk,α smooth simple closed curve inS2∞(H3) with k ≥ n+ 1 = 3, 0 < α < 1. LetΣ be aH-hypersurface inH3 with

∂∞Σ = Γ. Then,Σ ∪ Γ is aCk,α submanifold with boundary nearΓ.

b. (n = 4 case)For n = 4, H 6= 0 and |H| < 1, there exists a smoothΓsuch thatΣ ∪ Γ is not aCn+1 = C5 submanifold with boundary whereΣ is aH-hypersurface with∂∞Σ = Γ.

ASYMPTOTIC PLATEAU PROBLEM 17

We should also note that by studying the PDE above, or by usingsome barrierarguments, it is not hard to show that the intersection angleθH between anH-hypersurface and the asymptotic boundarySn

∞(Hn+1) is arctan(√1−H2

H) [53]. In

other words, letΓ be a codimension-1 submanifold inSn∞(Hn+1), andΣ be aH-

hypersurface inHn+1 with ∂∞Σ = Γ. Then for anyp ∈ Γ, the angleθH betweenΣ ∪ Γ andSn

∞(Hn+1) atp would bearctan(√1−H2

H).

6.3. Number of Solutions.

By using analytic techniques, Nelli and Spruck generalizedAnderson’s unique-ness result for mean convex domains in area minimizing hypersurfaces case toCMC context in [44]. Then, Guan and Spruck extended Hardt and Lin’s unique-ness results for star-shaped domains in area minimizing hypersurfaces case to CMChypersurfaces in hyperbolic space in [27].

Theorem 6.7. [27] Let Ω be a star shaped (mean convex in[44]) domain inSn∞(Hn+1) whereΓ = ∂Ω is C1,1 (C2,α in [44]) codimension-1 submanifold in

Sn∞(Hn+1). Then, for any0 ≤ H < 1, there exists a unique complete CMC

hypersurfaceΣ with mean curvatureH and∂∞Σ = Γ.

On the other hand, the author got a generic uniqueness resultfor CMC hyper-surfaces by generalizing his methods in [18]. In particular, he defined the notionof minimizing CMC hypersurfacesas generalizations of area minimizing hypersur-faces. In other words, as minimal hypersurfaces are critical points of the area func-tional, and area minimizing hypersurfaces are not just critical but minimum pointsof the functional, the same generalization is defined for CMChypersurfaces in[15]. The CMC hypersurfaces are the hypersurfaces with constant mean curvatureand corresponds to critical points of the functionalIH(t) = A(t) + nHV (t), andminimizing CMC hypersurfacescorresponds to minimizers of the functionalIH .Note that the existence result Theorem 6.3 by Tonegawa and Alencar-Rosenbergindeed gives minimizing CMC hypersurfaces.

Theorem 6.8. [15] Let A be the space of codimension-1 closed submanifolds ofSn∞(Hn+1), and letA′ ⊂ A be the subspace containing the closed submanifolds

of Sn∞(Hn+1) bounding a unique minimizing CMC hypersurface with mean cur-

vatureH in Hn+1. ThenA′ is generic inA, i.e.A−A′ is a set of first category.

On the other hand, there is no result for nonuniqueness of CMChypersurfaces.In particular, there is no known example of a codimension-1 submanifoldΓ inSn∞(Hn+1) such thatΓ is the asymptotic boundary of more than one CMC hyper-

surface with mean curvatureH for any0 < H < 1. ForH = 0, Anderson [6],Hass [30], and the author [18] gave such examples. It might be possible to gener-alize these techniques to prove nonuniqueness in CMC case for anyH ∈ (−1, 1).

18 BARIS COSKUNUZER

6.4. Foliations of Hyperbolic Space.

While discussing the uniqueness of CMC hypersurfaces for a given asymp-totic data in asymptotic boundary, there is a related problem in the subject: Fora given codimension-1 closed submanifoldΓ in Sn

∞(Hn+1), does the family ofCMC hypersurfacesΣH with mean curvatureH foliatesHn+1 or not, where−1 < H < 1 and∂∞ΣH = Γ. This question is related with uniqueness questionas existence of such a foliation automatically implies the uniqueness of CMC hy-persurfaceΣH with mean curvatureH where∂∞ΣH = Γ by maximum principle.In the reverse direction, the author showed the following result.

Theorem 6.9. [20] LetΓ be aC2,α closed codimension-1 submanifold inSn∞(Hn+1).

Also assume that for anyH ∈ (−1, 1), there exists a unique CMC hypersurfaceΣH with ∂∞ΣH = Γ. Then, the collection of CMC hypersurfacesΣH withH ∈ (−1, 1) foliatesHn+1.

Proof: (Sketch) First, by using the boundary regularity results in[53] andsome cut-paste arguments similar to exchange roundoff trick, the author showsthat two different minimizingH-hypersurfaces with same asymptotic boundarymust be disjoint (See Figure 2). In particular, he proves that if Γ is aC2,α closedcodimension-1 submanifold inSn

∞(Hn+1), andΣH1and ΣH2

are minimizingCMC hypersurfaces inHn+1 with ∂∞ΣHi

= Γi and−1 < H1 < H2 < 1,thenΣH1

andΣH2are disjoint. Hence,ΣH for −1 < H < 1 is a disjoint family

of hypersurfaces inHn+1. Now, there are two points to check to show thatΣHfoliatesHn+1. First point is that there is no gap between the leaves ofΣH, andthe second point is thatΣH fills H

n+1.

Sn∞(Hn+1)

SR0

T2

T1

Q S1

S2

Ω1

Ω2

FIGURE 2. For0 < H1 < H2 < 1, S1 is aboveS2 near the boundaryof the ballBR0

(p) by [53].

ASYMPTOTIC PLATEAU PROBLEM 19

For the first point the idea is to use the assumption thatΓ bounds a uniqueH-hypersurface for anyH ∈ (−1, 1). If there was a gap between the familyΣHH∈(−1,H0] andΣHH∈(H0,1), then constructing a sequence of hypersurfacesSi such thatSi ⊂ ΣHi

whereHi ց H0 and∂Si → Γ. Then, by showing thatSi → Σ′

H0whereΣ′

H0is another minimizingH0-hypersurface with∂∞Σ′

H0= Γ,

he gets a contradiction asΓ must bound a uniqueH0-hypersurface inHn+1.For the second point, ifΣH family of hypersurfaces does not fillHn+1, then

by constructing a suitable horosphere in the unfilled region, and by using the max-imum principle, the author gets a contradiction.

Hence, by the uniqueness results in [27] and [44], for the star shaped asymptoticdata and mean convex asymptotic data, the above result givespositive answer forthe question. Note that in [22], Chopp and Velling studied this problem by usingcomputational methods, and had an interesting result that for many different typeof curves inS2

∞(H3), CMC surfaces gives a foliation ofH3.On the other hand, recently in [54], Wang showed that if a quasi-Fuchsian3-

manifoldM contains a minimal surface whose principle curvature is less than1,thanM admits a foliation by CMC surfaces by using volume preserving meancurvature flow. If we lift this foliation to the universal cover, we get a foliationof H3 by CMC surfaces with same asymptotic boundaryΓ whereΓ is a simpleclosed curve inS2

∞(H3) and it is the limit set of the quasi-Fuchsian3-manifoldM .However, the limit set of quasi-Fuchsian manifolds are far from being smooth, eventhey contain no rectifiable arcs ([7]). Existence of one smooth point in the limit setimplies the group being Fuchsian, which means the limit set is a round circle inS2∞(H3). Hence, in addition to smooth examples in [20], [54] gives completely

nonrectifiable simple closed curve examples where CMC hypersurfaces with thegiven asymptotic data foliate the hyperbolic space. Also in[54], Wang constructsa simple closed curveΓ in S2

∞(H3) (as limit set of a quasi-Fuchsian 3-manifold)which is similar to the one in [30], where there cannot be a foliation ofH

3 by CMCsurfaces with asymptotic boundaryΓ.

7. FURTHER RESULTS

Other than existence, regularity and number of solutions tothe asymptotic Plateauproblem, there have been other important features which arestudied.

7.1. Properly Embeddedness.

The properly embeddedness of the solution of the asymptoticPlateau problem isone of the interesting problems which is under investigation. Namely, the questionis whether a solution to the asymptotic Plateau problemΣ with ∂∞Σ = Γ whereΓ is a codimension-1 closed submanifold inSn

∞(Hn+1) is properly embedded, ornot? In other words, ifϕ : S → H

n+1 is an embedding withϕ(S) = Σ, then isϕ proper? i.e. whether the preimage of a compact subsetK of Hn+1, ϕ−1(K), iscompact inS for anyK.

20 BARIS COSKUNUZER

Anderson implicitly talks about this property in [5], and [6]. Gabai conjecturesthe existence of a properly embedded area minimizing plane in H

3 (and for anycocompact metric onH3) for any given simple closed curveΓ in S2

∞(H3). Later,Soma proved the existence of such an area minimizing plane inmore general sit-uation (Gromov hyperbolic spaces) in [50] and [51]. Later, the author gave analternative proof for Soma’s results in [16].

Theorem 7.1. [50], [51], [16] LetX be a Gromov hyperbolic3-space with cocom-pact metric, andS2

∞(H3) be the sphere at infinity ofX. LetΓ be a given simpleclosed curve inS2

∞(H3). Then, there exists a properly embedded area minimizingplaneΣ in X with ∂∞Σ = Γ.

In recent years, the properly embeddedness of the complete minimal surfaceshas also been in serious attack inR3 case. This is called as Calabi-Yau Conjec-ture for minimal surfaces, and has been shown by Colding and Minicozzi in [11].Later, the author showed an analogous result for hyperbolicspace. In particular, heshowed that for any area minimizing planeΣ in H

3 with asymptotic boundaryΓwhich is a simple closed curve with one smooth point, thenΣ is properly embeddedin H

3. The technique is very different from Colding and Minicozzi’s techniques.While Colding-Minicozzi relates intrinsic distances and extrinsic distances for em-bedded minimal surface inR3 by using very powerful analytical techniques, theauthor’s techniques are purely topological.

Theorem 7.2. [17] LetΣ be a complete embedded area minimizing plane inH3

with ∂∞Σ = Γ whereΓ is a simple closed curve inS2∞(H3) with at least one

smooth (C1) point. Then,Σ must be proper.

Proof: (Sketch) Assume thatΣ is a non-properly embedded area minimizingplane inH3 with ∂∞Σ = Γ whereΓ is a simple closed curve inS2

∞(H3) with atleast one smooth point. The author gets a contradiction by analyzing the disks inthe intersection ofΣ with the ballsBR(0) which exhaustH3. First, he shows thatfor sufficiently large genericR > 0, Σ ∩ BR(0) contains infinitely many disjointdisks. Then, he categorize these disks as separating and nonseparating dependingon their boundary in the annulusAR = CH(Γ) ∩ ∂BR(0) is essential or not.

Then, he establishes the Key Lemma which shows that the nonseparating disksin BR(0) must stay close to the boundary∂BR(0). In particular, he proves that ifDr is a nonseparating disk inBr(0)∩Σ, then there is a functionF which is a mono-tone increasing function withF (r) → ∞ asr → ∞, such thatd(0,Dr) > F (r)whered is the distance. He proves the Key Lemma by using a barrier argument(See Figure 3). In other words, by using the smooth point assumption, he provesthe existence of a least area annulusAr in H

3 with ∂∞Ar = Γ+r ∪ Γ−

r , whereΓ±r

are simple closed curves sufficiently close toΓ in opposite sides. Since they arearea minimizing, any nonseparating diskDr must stay in one side of the least areaannulusAr. As r → ∞ the distance from0 to Ar will give the desired function.Hence, this shows that nonseparating disks do not come closeto 0 point, and stayclose to the boundary∂Br(0).

ASYMPTOTIC PLATEAU PROBLEM 21

B (0)r

ΓCH( )Γ

Σ

B (0)r

Γ

A r

CH( )Γ

Σ

As

As

S (H )8

2 3

S (H )2 3

8

FIGURE 3. The least area annulusAs is used as barrier in theproof of Key Lemma.

Finally, the author proves the main result by using the Key Lemma as follows.A separating diskDR1

in Σ ∩ BR1(0) will be a subdisk in a nonseparating disk

ER2in Σ ∩ BR2

(0) whereR2 > R1. By choosingR2 appropriately and by usingthe fact that the separating diskDR1

is a subset of the diskER2, he shows that the

nonseparating diskER2comes very close to0 point, which is a contradiction.

We should add that it would be very interesting to generalizeColding-Minicozzi’sresult on Calabi-Yau conjecture for minimal surfaces to this context [11], i.e. re-lating the intrinsic distances with the extrinsic distancefor any embedded minimalsurface inH3.

Note that properly embeddedness of absolutely area minimizing hypersurfacesin H

n+1 is almost automatic. This is because a nonproperly embeddedarea min-imizing hypersurface inHn+1 would have an intersection of infinite volume witha sufficiently large compact ball inHn+1, which is impossible for absolutely areaminimizing hypersurfaces. Also, in a forthcoming paper [21], the author constructsexamples of non-properly embedded minimal planes inH

3.

7.2. The Global Structure.

On the other hand, the space of all solutions to the asymptotic Plateau prob-lem is another interesting problem, and its structure givespowerful global analysistools to get important results on the number of solutions to the asymptotic Plateauproblem . In particular, the author showed that the space of minimal planes inH

3 with asymptotic boundaryC3,α smooth simple closed curve is a manifold andits projection to the asymptotic boundary is a Fredholm map in [13]. By usingthese results, the author showed a generic uniqueness result (Theorem 5.4) forC3

smooth simple closed curves inS2∞(H3), [14].

22 BARIS COSKUNUZER

Very recently, by using different techniques, Alexakis andMazzeo generalizedauthor’s results to complete properly embedded minimal surfaces of any fixedgenus in convex cocompact hyperbolic3-manifolds (H3 is a special case).

Theorem 7.3. [2] LetX be a convex cocompact hyperbolic3-manifold, andMk(X)is the space of properly embedded minimal surfaces inX of genusk with asymp-totic boundaryC3,α simple closed curve in∂∞X. Let ξ be the space of allC3,α

curves in∂∞X. Then, bothMk(X) (M0(H3) case in[13]) and ξ are Banach

manifolds, and the projection mapπk : Mk(X) → ξ is a smooth proper Fredholmmap of index0.

Note that being Fredholm map of index0 is a very strong property, and it canbe considered as the map is locally one-to-one for generic points. Indeed, theyshowed thatπk is not only Fredholm of index0, but also proper. Hence, by usingthis result, they developed a powerfulZ-valued degree theory forπk as follows:

deg(πk) =∑

Σ∈π−1

k(Γ)

(−1)n(Σ)

whereΓ is a regular value ofπk andn(Σ) is the number of negative eigenvalues ofthe Jabobi operator−LΣ. By combining this degree theory with the techniques in[52] and [56], one can get very interesting results on complete minimal surfaces inH

3 (see Section 4 in [2]).

7.3. Intersections.

Another interesting property of the solutions to the asymptotic Plateau problemis that their intersections mostly controlled by their asymptotic boundary. In manycases, if the asymptotic boundaries are disjoint, then the solutions to the asymptoticPlateau problem are also disjoint.

Theorem 7.4. [18] LetΓ1 andΓ2 be two disjoint simple closed curves inS2∞(H3).

If Σ1 andΣ2 are area minimizing planes inH3 with ∂∞Σi = Γi, thenΣ1 andΣ2

are disjoint, too.

The idea of the proof for this case is quite simple. IfΣ1 ∩ Σ2 is not empty,then as asymptotic boundaries are disjoint, the intersection must contain a simpleclosed curveγ. Then,γ bounds a diskDi in Σi. By swaping the disks, we getarea minimizing planes with a folding curveγ. Hence, we can reduce the area bysmoothing out the curve and get a contradiction [42]. With slight modifications,this result can be generalized to absolutely area minimizing hypersurfaces.

Theorem 7.5. [18] LetΓ1 andΓ2 be two disjoint connected closed codimension-1submanifolds inSn

∞(Hn+1). If Σ1 andΣ2 are absolutely area minimizing hyper-surfaces inHn+1 with ∂∞Σi = Γi, thenΣ1 andΣ2 are disjoint, too.

To generalize the idea of previous theorem, first by using theregularity result ofHardt-Lin in [29] (Theorem 4.1), it can be showed thatΣ1 andΣ2 are separating.

ASYMPTOTIC PLATEAU PROBLEM 23

As asymptotic boundariesΓ1 andΓ2 are disjoint, the intersectionα stays in thecompact part, and asΣi separating,α separates a compact codimension-0 partSi

from Σi. Again by swaping these parts, one can get absolutely area minimizinghypersurfaces with codimension-1 singularity setα which contradicts to interiorregularity results of geometric measure theory.

On the other hand, these arguments cannot be applied to minimal submanifolds,or area minimizing submanifolds in a specified topological class. In the minimalsubmanifold case, the surgery argument completely fails asthere is no area factorto compare. The main problem with the fixed topological classcase is that theessential surgery argument in the proof is not working as after surgery one may notstay in the same topological class. In the absolutely area minimizing case there isno topological restriction.

There is a related conjecture which has important applications in3-manifoldtopology.

Disjoint Planes Conjecture:LetΓ1,Γ2 be simple closed curves inS2∞(X), where

X is a Gromov hyperbolic3-space with cocompact metric. IfΓ1 andΓ2 do notcross each other (i.e. They are the boundaries of disjoint open regions inS2

∞(X)),then any distinct area minimizing planesΣ1,Σ2 in X with asymptotic boundaryΓ1,Γ2 are disjoint.

Even though this conjecture is interesting in its own right,it has powerful topo-logical applications. The most important application is constructing the area min-imizing representative of an essential2-dimensional object in a3-manifold, likeincompressible surfaces, and genuine laminations. With this conjecture, if a2-dimensional embedded essential objectS in a Gromov hyperbolic manifoldMinduces aπ1-invariant family of circles∂∞S in S2

∞(M ), then by spanning the cir-cles with area minimizing planes, the conjecture would giveyou anπ1-invariantpairwise disjointfamily of area minimizing planes inM . Hence, by projectingdown the planes toM , it is possible to get theembeddedarea minimizing repre-sentative ofS in the3-manifoldM . Note that the author showed that the conjectureis generically true in [19].

7.4. Renormalized Area.

In [2], in addition to the study of the global structure of moduli spaces of com-plete minimal surfaces inH3 and aZ-valued degree theory on them (see Section7.2), Alexakis and Mazzeo defined a notion calledrenormalized areaA(Y ) forproperly embedded minimal surfacesY in H

3 (or more generally convex cocom-pact hyperbolic3-manifolds) where∂∞Y = Γ is aC3,α simple closed curve inS2∞(H3). They showed that if a minimal surface minimize renormalized area, it

must be an area minimizing surface.

Theorem 7.6. [2] LetΓ be aC3,α embedded curve inS2∞(H3). Suppose thatY1

andY2 are two properly embedded minimal surfaces inH3 with ∂∞Y1 = ∂∞Y2 =

24 BARIS COSKUNUZER

Γ. If Y1 is area minimizing inH3, thenA(Y1) ≤ A(Y2), and equality holds if andonly if Y2 is also an area minimizer.

Moreover, they also showed that the renormalized area functional A is con-nected with the Willmore functionalW, which is the total integral of the squareof the mean curvature, in the following way. The renormalized area functionalis defined for any convex cocompact hyperbolic3-manifold X. After modify-ing the metric onX in a suitable way such that it induces aZ2-invariant smoothmetric on the double ofX, say 2X, consider the double of any surfaceΣ inMk(X) (see Section 7.2), say2Σ, in 2X. Then, Alexakis and Mazzeo showedthatA(Σ) = −1

2W(2Σ) for anyΣ ∈ Mk(X).On the other hand, they also define an extended renormalized areaR which is

defined for all properly embedded surfacesY which intersectS2∞(H3) orthogo-

nally and∂∞Y = Γ is aC3,α simple closed curve inS2∞(H3). Then the extended

renormalized area behaves just like the area for these surfaces.

Theorem 7.7. [2] LetΓ be aC3,α closed curve inS2∞(H3). Then the infimum of

R(Y ) whereY ranges over the set of allC3,α surfaces with∂∞Y = Γ which inter-sectS2

∞(H3) orthogonally is attained only by absolutely area-minimizing surfaces.Also, ifY is a critical forR, thenY must be a minimal surface.

Notice that renormalized area behaves just like the area forthese infinite surfacesin many ways. Hence, many techniques from the compact area minimizing surfacescan be generalized to these surfaces with this new tool.

AcknowledgementsI would like to thank Urs Lang for very valuable remarks.

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DEPARTMENT OFMATHEMATICS, KOC UNIVERSITY, SARIYER, ISTANBUL 34450 TURKEY

E-mail address: bcoskunuzer@ku.edu.tr