1 Introduction

Minimal discs in hyperbolic space bounded by a quasicircle at infinity

Abstract.

We prove that the supremum of principal curvatures of a minimal embedded disc in hyperbolic three-space spanning a quasicircle in the boundary at infinity is estimated in a sublinear way by the norm of the quasicircle in the sense of universal Teichmüller space, if the quasicircle is sufficiently close to being the boundary of a totally geodesic plane. As a by-product we prove that there is a universal constant C independent of the genus such that if the Teichmüller distance between the ends of a quasi-Fuchsian manifold is at most C, then is almost-Fuchsian. The main ingredients of the proofs are estimates on the convex hull of a minimal surface and Schauder-type estimates to control principal curvatures.

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1. Introduction

Let be hyperbolic three-space and be its boundary at infinity. A surface in hyperbolic space is minimal if its principal curvatures at every point have opposite values. We will denote the principal curvatures by and , where is a nonnegative function on . It was proved by Anderson ([And83, Theorem 4.1]) that for every Jordan curve in there exists a minimal embedded disc whose boundary at infinity coincides with . It can be proved that if the supremum of the principal curvatures of is in , then is a quasicircle, namely is the image of a round circle under a quasiconformal map of the sphere at infinity.

However, uniqueness does not hold in general. Anderson proved the existence of a Jordan curve invariant under the action of a quasi-Fuchsian group spanning several distinct minimal embedded discs, see [And83, Theorem 5.3]. In this case, is a quasicircle and coincides with the limit set of . More recently in [HW13a] invariant curves spanning an arbitrarily large number of minimal discs were constructed. On the other hand, if the supremum of the principal curvatures of a minimal embedded disc satisfies then, by an application of the maximum principle, is the unique minimal disc asymptotic to the quasicircle .

The aim of this paper is to study the supremum of the principal curvatures of a minimal embedded disc, in relation with the norm of the quasicircle at infinity, in the sense of universal Teichmüller space. The relations we obtain are interesting for “small” quasicircles, that are close in universal Teichmüller space to a round circle. The main result of this paper is the following:

Theorem A.

There exist universal constants and such that every minimal embedded disc in with boundary at infinity a -quasicircle , with , has principal curvatures bounded by

Recall that the minimal disc with prescribed quasicircle at infinity is unique if . Hence we can draw the following consequence, by choosing :

Theorem B.

There exists a universal constant such that every -quasicircle with is the boundary at infinity of a unique minimal embedded disc.

Applications to quasi-Fuchsian manifolds

Theorem A has a direct application to quasi-Fuchsian manifolds. Recall that a quasi-Fuchsian manifold is isometric to the quotient of by a quasi-Fuchsian group , isomorphic to the fundamental group of a closed surface , whose limit set is a Jordan curve in . The topology of is . We denote by and the two connected components of . Then and inherit natural structures of Riemann surfaces on and therefore determine two points of , the Teichmüller space of . Let denote the Teichmüller distance on .

Corollary A.

There exist universal constants and such that, for every quasi-Fuchsian manifold with and every minimal surface in homotopic to , the supremum of the principal curvatures of satisfies:

Indeed, under the hypothesis of Corollary A, the Teichmüller map from one hyperbolic end of to the other is -quasiconformal for . Hence the lift to the universal cover of any closed minimal surface in is a minimal embedded disc with boundary at infinity a -quasicircle, namely the limit set of the corresponding quasi-Fuchsian group. Choosing , where is the constant of Theorem A, and choosing as in Theorem A (up to a factor which arises from the definition of Teichmüller distance), the statement of Corollary A follows.

We remark here that the constant of Corollary A is independent of the genus of .

A quasi-Fuchsian manifold contaning a closed minimal surface with principal curvatures in is called almost-Fuchsian, according to the definition given in [KS07]. The minimal surface in an almost-Fuchsian manifold is unique, by the above discussion, as first observed by Uhlenbeck ([Uhl83]). Hence, applying Theorem B to the case of quasi-Fuchsian manifolds, the following corollary is proved.

Corollary B.

If the Teichmüller distance between the conformal metrics at infinity of a quasi-Fuchsian manifold is smaller than a universal constant , then is almost-Fuchsian.

Indeed, it suffices as above to pick , which is again independent on the genus of . By Bers’ Simultaneous Uniformization Theorem, the Riemann surfaces determine the manifold . Hence the space of quasi-Fuchsian manifolds homeomorphic to , considered up to isometry isotopic to the identity, can be identified to . Under this identification, the subset of composed of Fuchsian manifolds, which we denote by , coincides with the diagonal in . Let us denote by the subset of composed of almost-Fuchsian manifolds. Corollary B can be restated in the following way:

Corollary C.

There exists a uniform neighborhood of the Fuchsian locus in such that .

We remark that Corollary A is a partial converse of results presented in [GHW10], giving a bound on the Teichmüller distance between the hyperbolic ends of an almost-Fuchsian manifold in terms of the maximum of the principal curvatures. Another invariant which has been studied in relation with the properties of minimal surfaces in hyperbolic space is the Hausdorff dimension of the limit set. Corollary A and Corollary B can be compared with the following theorem given in [San14]: for every and there exists a constant such that any stable minimal surface with injectivity radius bounded by in a quasi-Fuchsian manifold are in provided the Hausdorff dimension of the limit set of is at most . In particular, is almost Fuchsian if one chooses . Conversely, in [HW13b] the authors give an estimate of the Hausdorff dimension of the limit set in an almost-Fuchsian manifold in terms of the maximum of the principal curvatures of the (unique) minimal surface. The degeneration of almost-Fuchsian manifolds is also studied in [San13].

The main steps of the proof

The proof of Theorem A is composed of several steps.

By using the technique of “description from infinity” (see [Eps84] and [KS08]), we construct a foliation of by equidistant surfaces, such that all the leaves of the foliation have the same boundary at infinity, a quasicircle . By using a theorem proved in [ZT87] and [KS08, Appendix], which relates the curvatures of the leaves of the foliation with the Schwarzian derivative of the map which uniformizes the conformal structure of one component of , we obtain an explicit bound for the distance between two surfaces and of , where is concave and is convex, in terms of the Bers norm of . The distance goes to when approaches a circle in .

A fundamental property of a minimal surface with boundary at infinity a curve is that is contained in the convex hull of . The surfaces and of the previous step lie outside the convex hull of , on the two different sides. Hence every point of lies on a geodesic segment orthogonal to two planes and (tangent to and respectively) such that is contained in the region bounded by and . The length of such geodesic segment is bounded by the Bers norm of the quasicircle at infinity, in a way which does not depend on the chosen point .

The next step in the proof is then a Schauder-type estimate. Considering the function , defined on , which is the hyperbolic sine of the distance from the plane , it turns out that solves the equation

(\ref{lap 2u hyp})

where is the Laplace-Beltrami operator of . We then apply classical theory of linear PDEs, in particular Schauder estimates, to the equation in order to prove that

where and is expressed in normal coordinates centered at . Recall that is the Laplace-Beltrami operator, which depends on the surface . In order to have this kind of inequality, it is then necessary to control the coefficients of . This is obtained by a compactness argument for conformal harmonic mappings, adapted from [Cus09], recalling that minimal discs in are precisely the image of conformal harmonic mapping from the disc to . However, to ensure that compact sets in the conformal parametrization are comparable to compact sets in normal coordinates, we will first need to prove a uniform bound of the curvature. For this reason we will assume (as in the statement of Theorem A) that the minimal discs we consider have boundary at infinity a -quasicircle, with .

The final step is then an explicit estimate of the principal curvatures at , by observing that the shape operator can be expressed in terms of and the first and second derivatives of . The Schauder estimate above then gives a bound on the principal curvatures just in terms of the supremum of in a geodesic ball of fixed radius centered at . By using the first step, since is contained between and the nearby plane , we finally get an estimate of the principal curvatures of a minimal embedded disc only in terms of the Bers norm of the quasicircle at infinity.

All the previous estimates do not depend on the choice of . Hence the following theorem is actually proved.

Theorem C.

There exist constants and such that the principal curvatures of every minimal surface in with a -quasicircle, with , are bounded by:

(1)

where , is a quasiconformal map, conformal on , and denotes the Bers norm of .

Observe that the estimate holds in a neighborhood of the identity (which represents circles in ), in the sense of universal Teichmüller space. Theorem A is then a consequence of Theorem C, using the well-known fact that the Bers embedding is locally bi-Lipschitz.

Organization of the paper

The structure of the paper is as follows. In Section 2, we introduce the necessary notions on hyperbolic space and some properties of minimal surfaces and convex hulls. In Section 3 we introduce the theory of quasiconformal maps and universal Teichmüller space. In Section 4 we prove Theorem A. The Section is split in several subsections, containing the steps of the proof. In Section 5 we discuss how Theorem B, Corollary A, Corollary B and Corollary C follow from Theorem A.

Acknowledgements

I am very grateful to Jean-Marc Schlenker for his guidance and patience. Most of this work was done during my (very pleasent) stay at University of Luxembourg; I would like to thank the Institution for the hospitality. I am very thankful to my advisor Francesco Bonsante and to Zeno Huang for many interesting discussions and suggestions. I would like to thank an anonymous referee for many observations and advices which highly improved the presentation of the paper.

2. Minimal surfaces in hyperbolic space

We consider (3+1)-dimensional Minkowski space as endowed with the bilinear form

(2)

The hyperboloid model of hyperbolic 3-space is

The induced metric from gives a Riemannian metric of constant curvature -1. The group of orientation-preserving isometries of is , namely the group of linear isometries of which preserve orientation and do not switch the two connected components of the quadric . Geodesics in hyperbolic space are the intersection of with linear planes of (when nonempty); totally geodesic planes are the intersections with linear hyperplanes and are isometric copies of hyperbolic plane .

We denote by the metric on induced by the Riemannian metric. It is easy to show that

(3)

and other similar formulae which will be used in the paper.

Note that can also be regarded as the projective domain

Let us denote by the region

and we call de Sitter space the projectivization of ,

Totally geodesic planes in hyperbolic space, of the form , are parametrized by the dual points in .

In an affine chart for the projective model of , hyperbolic space is represented as the unit ball , using the affine coordinates . This is called the Klein model; although in this model the metric of is not conformal to the Euclidean metric of , the Klein model has the good property that geodesics are straight lines, and totally geodesic planes are intersections of the unit ball with planes of . It is well-known that has a natural boundary at infinity, , which is a 2-sphere and is endowed with a natural complex projective structure - and therefore also with a conformal structure.

Given an embedded surface in , we denote by its asymptotic boundary, namely, the intersection of the topological closure of with .

2.1. Minimal surfaces

This paper is mostly concerned with smoothly embedded surfaces in hyperbolic space. Let be a smooth embedding and let be a unit normal vector field to the embedded surface . We denote again by the Riemannian metric of , which is the restriction to the hyperboloid of the bilinear form (2) of ; and are the ambient connection and the Levi-Civita connection of the surface , respectively. The second fundamental form of is defined as

if and are vector fields extending and . The shape operator is the -tensor defined as . It satisfies the property

Definition 1.

An embedded surface in is minimal if .

The shape operator is symmetric with respect to the first fundamental form of the surface ; hence the condition of minimality amounts to the fact that the principal curvatures (namely, the eigenvalues of ) are opposite at every point.

An embedded disc in is said to be area minimizing if any compact subdisc is locally the smallest area surface among all surfaces with the same boundary. It is well-known that area minimizing surfaces are minimal. The problem of existence for minimal surfaces with prescribed curve at infinity was solved by Anderson; see [And83] for the original source and [Cos13] for a survey on this topic.

Theorem 2 ([And83]).

Given a simple closed curve in , there exists a complete area minimizing embedded disc with .

A key property used in this paper is that minimal surfaces with boundary at infinity a Jordan curve are contained in the convex hull of . Although this fact is known, we prove it here by applying maximum principle to a simple linear PDE describing minimal surfaces.

Definition 3.

Given a curve in , the convex hull of , which we denote by , is the intersection of half-spaces bounded by totally geodesic planes such that does not intersect , and the half-space is taken on the side of containing .

Hereafter denotes the Hessian of a smooth function on the surface , i.e. the (1,1) tensor

Sometimes the Hessian is also considered as a (2,0) tensor, which we denote (in the rare occurrences) with

Finally, denotes the Laplace-Beltrami operator of , which can be defined as

Observe that, with this definition, is a negative definite operator.

Proposition 4.

Given a minimal surface and a plane , let be the function . Here is considered as a signed distance from the plane . Let be the unit normal to , the shape operator, and the identity operatior. Then

(4)

as a consequence, satisfies

($\star$)
Proof.

Consider the hyperboloid model for . Let us assume is the plane dual to the point , meaning that . Then is the restriction to of the function defined on :

(5)

Let be the unit normal vector field to ; we compute by projecting the gradient of to the tangent plane to :

(6)
(7)

Now , where is the Levi-Civita connection of , namely the projection of the flat connection of , and so

Moreover, and thus

which proves (4). By taking the trace, (($\star$)) follows. ∎

Corollary 5.

Let be a minimal surface in , with a Jordan curve. Then is contained in the convex hull .

Proof.

If is a circle, then is a totally geodesic plane which coincides with the convex hull of . Hence we can suppose is not a circle. Consider a plane which does not intersect and the function defined as in Equation (5) in Proposition 4, with respect to . Suppose their mutual position is such that in the region of close to the boundary at infinity (i.e. in the complement of a large compact set). If there exists some point where , then at a minimum point , which gives a contradiction. The proof is analogous for a plane on the other side of , by switching the signs. Therefore every convex set containing contains also . ∎

3. Universal Teichmüller space

The aim of this section is to introduce the theory of quasiconformal mappings and universal Teichmüller space. We will give a brief account of the very rich and developed theory. Useful references are [Gar87, GL00, Ahl06, FM07] and the nice survey [Sug07].

3.1. Quasiconformal mappings and universal Teichmüller space

We recall the definition of quasiconformal map.

Definition 1.

Given a domain , an orientation-preserving homeomorphism

is quasiconformal if is absolutely continuous on lines and there exists a constant such that

Let us denote , which is called complex dilatation of . This is well-defined almost everywhere, hence it makes sense to take the norm. Thus a homeomorphism is quasiconformal if . Moreover, a quasiconformal map as in Definition 1 is called -quasiconformal, where

It turns out that the best such constant represents the maximal dilatation of , i.e. the supremum over all of the ratio between the major axis and the minor axis of the ellipse which is the image of a unit circle under the differential .

It is known that a -quasiconformal map is conformal, and that the composition of a -quasiconformal map and a -quasiconformal map is -quasiconformal. Hence composing with conformal maps does not change the maximal dilatation.

Actually, there is an explicit formula for the complex dilatation of the composition of two quasiconformal maps on :

(8)

Using Equation (8), one can see that and differ by post-composition with a conformal map if and only if almost everywhere. We now mention the classical and important result of existence of quasiconformal maps with given complex dilatation.

Measurable Riemann mapping Theorem. Given any measurable function on there exists a unique quasiconformal map such that , and almost everywhere in .

The uniqueness part of Measurable Riemann mapping Theorem means that every two solutions (which can be thought as maps on the Riemann sphere ) of the equation

differ by post-composition with a Möbius transformation of .

Given any fixed , -quasiconformal mappings have an important compactness property. See [Gar87] or [Leh87].

Theorem 2.

Let and be a sequence of -quasiconformal mappings such that, for three fixed points , the mutual spherical distances are bounded from below: there exists a constant such that

for every and for every choice of , . Then there exists a subsequence which converges uniformly to a -quasiconformal map .

3.2. Quasiconformal deformations of the disc

It turns out that every quasiconformal homeomorphisms of to itself extends to the boundary . Let us consider the space:

where if and only if . Universal Teichmüller space is then defined as

where is the subgroup of Möbius transformations of . Equivalently, is the space of quasiconformal homeomorphisms which fix , and up to the same relation .

Such quasiconformal homeomorphisms of the disc can be obtained in the following way. Given a domain , elements in the unit ball of the (complex-valued) Banach space are called Beltrami differentials on . Let us denote this unit ball by:

Given any in , let us define on by extending on so that

The quasiconformal map such that fixing , and , whose existence is provided by Measurable Riemann mapping Theorem, maps to itself by the uniqueness part. Therefore restricts to a quasiconformal homeomorphism of to itself.

The Teichmüller distance on is defined as

where the infimum is taken over all quasiconformal maps and . It can be shown that is a well-defined distance on Teichmüller space, and is a complete metric space.

3.3. Quasicircles and Bers embedding

We now want to discuss another interpretation of Teichmüller space, as the space of quasidiscs, and the relation with the Schwartzian derivative and the Bers embedding.

Definition 3.

A quasicircle is a simple closed curve in such that for a quasiconformal map . Analogously, a quasidisc is a domain in such that for a quasiconformal map .

Let us denote . We remark that in the definition of quasicircle, it would be equivalent to say that is the image of by a -quasiconformal map of (not necessarily conformal on ). However, the maximal dilatation might be different, with . Hence we consider the space of quasidiscs:

where the equivalence relation is if and only if . We will again consider the quotient of by Möbius transformation.

Given a Beltrami differential , one can construct a quasiconformal map on , by applying Measurable Riemann mapping Theorem to the Beltrami differential obtained by extending to on . The quasiconformal map obtained in this way (fixing the three points , and ) is denoted by . A well-known lemma (see [Gar87, §5.4, Lemma 3]) shows that, given two Beltrami differentials , if and only if . Using this fact it can be shown that is identified to , or equivalently to the subset of which fix , and .

We will say that a quasicircle is a -quasicircle if

It is easily seen that the condition that is a -quasicircle is equivalent to the fact that the element of the first model which corresponds to has Teichmüller distance from the identity .

By using the model of quasidiscs for Teichmüller space, we now introduce the Bers norm on . Recall that, given a holomorphic function with in , the Schwarzian derivative of is the holomorphic function

It can be easily checked that , hence the Schwarzian derivative can be defined also for meromorphic functions at simple poles. The Schwarzian derivative vanishes precisely on Möbius transformations.

Let us now consider the space of holomorphic quadratic differentials on . We will consider the following norm, for a holomorphic quadratic differential :

where is the Poincaré metric of constant curvature on . Observe that behaves like a function, in the sense that it is invariant by pre-composition with Möbius transformations of , which are isometries for the Poincaré metric.

We now define the Bers embedding of universal Teichmüller space. This is the map which associates to the Schwarzian derivative . Let us denote by the norm on holomorphic quadratic differentials on obtained from the norm on , by identifying with by an inversion in . Then

is an embedding of in the Banach space of bounded holomorphic quadratic differentials (i.e. for which ). Finally, the Bers norm of en element is

The fact that the Bers embedding is locally bi-Lipschitz will be used in the following. See for instance [FKM13, Theorem 4.3]. In the statement, we again implicitly identify the models of universal Teichmüller space by quasiconformal homeomorphisms of the disc (denoted by ) and by quasicircles (denoted by ).

Theorem 4.

Let . There exist constants and such that, for every in the ball of radius for the Teichmüller distance centered at the origin (i.e. ),

We conclude this preliminary part by mentioning a theorem by Nehari, see for instance [Leh87] or [FM07].

Nehari Theorem. The image of the Bers embedding is contained in the ball of radius in , and contains the ball of radius .

4. Minimal surfaces in

The goal of this section is to prove Theorem A. The proof is divided into several steps, whose general idea is the following:

  1. Given , if is small, then there is a foliation of a convex subset of by equidistant surfaces. All the surfaces of have asymptotic boundary the quasicircle . Hence the convex hull of is trapped between two parallel surfaces, whose distance is estimated in terms of .

  2. As a consequence of point (1), given a minimal surface in with , for every point there is a geodesic segment through of small length orthogonal at the endpoints to two planes , which do not intersect . Moreover is contained between and .

  3. Since is contained between two parallel planes close to , the principal curvatures of in a neighborhood of cannot be too large. In particular, we use Schauder theory to show that the principal curvatures of at a point are uniformly bounded in terms of the distance from of points in a neighborhood of .

  4. Finally, the distance from of points of in a neighborhood of is estimated in terms of the distance of points in from , hence is bounded in terms of the Bers norm .

It is important to remark that the estimates we give are uniform, in the sense that they do not depend on the point or on the surface , but just on the Bers norm of the quasicircle at infinity. The above heuristic arguments are formalized in the following subsections.

4.1. Description from infinity

The main result of this part is the following. See Figure 1.

Proposition 1.

Let . Given an embedded minimal disc in with boundary at infinity a quasicircle with , every point of lies on a geodesic segment of length at most orthogonal at the endpoints to two planes and , such that the convex hull is contained between and .

Figure 1. The statement of Proposition 1. The geodesic segment through has length , for , and this does not depend on .
Remark 2.

A consequence of Proposition 1 is that the Hausdorff distance between the two boundary components of is bounded by . Hence it would be natural to try to define in such a way a notion of thickness or width of the convex hull:

However, a bound on the Hausdorff distance is not sufficient for the purpose of this paper. It will become clear in the proof of Theorem C and Theorem A, and in particular for the application of Lemma 15, that the necessary property is the existence of two support planes which are both orthogonal to a geodesic segment of short length through any point .

We review here some important facts on the so-called description from infinity of surfaces in hyperbolic space. For details, see [Eps84] and [KS08]. Given an embedded surface in with bounded principal curvatures, let be its first fundamental form and the second fundamental form. Recall we defined its shape operator, for the oriented unit normal vector field (we fix the convention that points towards the direction in ), so that . Denote by the identity operator. Let be the -equidistant surface from (where the sign of agrees with the choice of unit normal vector field to ). For small , there is a map from to obtained following the geodesics orthogonal to at every point.

Lemma 3.

Given a smooth surface in , let be the surface at distance from , obtained by following the normal flow at time . Then the pull-back to of the induced metric on the surface is given by:

(9)

The second fundamental form and the shape operator of are given by

(10)
(11)
Proof.

In the hyperboloid model, let be the minimal embedding of the surface , with oriented unit normal . The geodesics orthogonal to at a point can be written as

Then we compute

The formula for the second fundamental form follows from the fact that . ∎

It follows that, if the principal curvatures of a minimal surface are and , then the principal curvatures of are

(12)

In particular, if , then is a non-singular metric for every . The surfaces foliate and they all have asymptotic boundary .

We now define the first, second and third fundamental form at infinity associated to . Recall the second and third fundamental form of are and .

(13)
(14)
(15)
(16)

We observe that the metric and the second fundamental form can be recovered as

(17)
(18)
(19)

The following relation can be proved by some easy computation:

Lemma 4 ([Ks08, Remark 5.4 and 5.5]).

The embedding data at infinity associated to an embedded surface in satisfy the equation

(20)

where is the curvature of . Moreover, satisfies the Codazzi equation with respect to :

(21)

A partial converse of this fact, which can be regarded as a fundamental theorem from infinity, is the following theorem. This follows again by the results in [KS08], although it is not stated in full generality here.

Theorem 5.

Given a Jordan curve , let be a pair of a metric in the conformal class of a connected component of and a self-adjoint -tensor, satisfiying the conditions (20) and (21) as in Lemma 4. Assume the eigenvalues of are positive at every point. Then there exists a foliation of by equidistant surfaces , for which the first fundamental form at infinity (with respect to ) is and the shape operator at infinity is .

We want to give a relation between the Bers norm of the quasicircle and the existence of a foliation of by equidistant surfaces with boundary , containing both convex and concave surfaces. We identify to by means of the stereographic projection, so that correponds to the lower hemisphere of the sphere at infinity. The following property will be used, see [ZT87] or [KS08, Appendix A].

Theorem 6.

Let be a Jordan curve. If is the complete hyperbolic metric in the conformal class of a connected component of , and is the traceless part of the second fundamental form at infinity , then is the real part of the Schwarzian derivative of the isometry , namely the map which uniformizes the conformal structure of :

(22)

We now derive, by straightforward computation, a useful relation.

Lemma 7.

Let be a quasicircle, for . If is the complete hyperbolic metric in the conformal class of a connected component of , and is the traceless part of the shape operator at infinity , then

(23)
Proof.

From Theorem 6, is the real part of the holomorphic quadratic differential . In complex conformal coordinates, we can assume that

and , so that

and finally

Therefore . Moreover, by definition of Bers embedding, , because is a holomorphic map from which maps to . Since

this concludes the proof. ∎

We are finally ready to prove Proposition 1.

Proof of Proposition 1.

Suppose again is a hyperbolic metric in the conformal class of . Since by Lemma 4, we can write , where