1 Introduction

Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

Abstract.

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence in -dimensional Minkowski space, provided is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Ampère equation on the unit disc, with the boundary condition , for a smooth positive function and a bounded lower semicontinuous function.

We then prove that a domain of dependence contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function is in the Zygmund class. Moreover in this case the surface of constant curvature contained in has bounded principal curvatures, for every . In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of .

Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature , as varies in .

The authors were partially supported by FIRB 2010 project “Low dimensional geometry and topology” (RBFR10GHHH003). The first author was partially supported by PRIN 2012 project “Moduli strutture algebriche e loro applicazioni”. The authors are members of the national research group GNSAGA

1. Introduction

A classical theorem of Riemannian geometry states that if is an isometric immersion of the round sphere into Euclidean space, then it is the standard inclusion up to an isometry of . On the other hand, Hano and Nomizu ([19]) first proved that the analogous statement does not hold in Minkowski space . Namely there are isometric embeddings of the hyperbolic plane in which are not equivalent to the standard inclusion of the hyperboloid model of into .

A possible way to understand the rigidity in the Euclidean case makes use of the so-called support function: basically if is an isometric immersion, the image of must be a locally convex surface by the Gauss equation. In particular it turns out that the Gauss map is bijective, and the support function is defined as , . A simple computation shows that , where is the covariant Hessian on the sphere. By a wise use of the comparison principle it turns out that the difference of any two solutions must be the restriction on of a linear form on . This allows to conclude that every solution is of the form for some . Therefore the surface is the round sphere of radius centered at .

The support function can be analogously defined for a spacelike convex immersion . If is an isometric immersion, then satisfies the equation . However the maximum principle cannot be directly used in this context by the non-compactness of . This is a general indication that some boundary condition must be taken into account to determine the solution and the immersion .

The classical Minkowski problem in Euclidean space can be also formulated for Minkowski space. Given a smooth spacelike strictly convex surface in , the curvature function is defined as , , where is the Gauss map and is the scalar intrinsic curvature on . Minkowski problem consists in finding a convex surface in Minkowski space whose curvature function is a prescribed positive function . Using the support function technology, the problem turns out to be equivalent to solving the equation

(1)

Also in this case for the well-posedness of the problem some boundary conditions must be imposed.

Using the Klein model of , Equation (1) can be reduced to a standard Monge-Ampère equation over the unit disc . In particular solutions of (1) explicitly correspond to solutions of the equation

(2)

It should be remarked that the correspondence can degenerate in some sense. Indeed given a convex surface , the support function is defined only on a convex subset of . On the other hand, any convex function over corresponds to some convex surface in Minkowski space, but in general might contain lightlike rays. We say that a convex surface is a spacelike entire graph if , where is a function on the horizontal plane such that for all .

In [20], Li studied the Minkowski problem in Minkowski space in any dimension showing the existence and uniqueness of the solution of (2) imposing , for a given smooth . The result was improved in dimension by Guan, Jian and Schoen in [17], where the existence of the solution is proved assuming that the boundary data is only Lipschitz. The solutions obtained in both cases correspond to spacelike entire graphs.

A remarkable result in [20] is that under the assumption that the boundary data is smooth, the corresponding convex surface has principal curvatures bounded from below by a positive constant. As a partial converse statement, if has principal curvatures bounded from below by a positive constant, then the corresponding function extends to a continuous function of the boundary of .

In a different direction Barbot, Béguin and Zeghib ([2]) solved the Minkowski problem for surfaces invariant by an affine deformation of a cocompact Fuchsian group. Let be a cocompact Fuchsian group, and an affine deformation of ( is a group of affine transformations whose elements are obtained by adding a translation part to elements of ). If is a -invariant surface, its curvature function is -invariant. Barbot, Béguin and Zeghib proved that, given a positive -invariant function , there exists a unique solution of Minkowski problem which is -invariant.

If is the support function corresponding to some -invariant surface , combining the result by Li and the cocompactness of , it turns out that extends to the boundary of . It is not difficult to see that the extension on the boundary only depends on and in particular it is independent of the curvature function. However the result in [2] is not a consequence of the results in [20, 17], as it is not likely that is Lipschitz continuous. This gives an indication that in dimension results in [20, 17] are not sharp.

One of the goals of the paper is to determine the exact regularity class of the extension on of functions corresponding to surfaces with principal curvatures bounded from below.

Theorem 1.1.

Let be a continuous function. Then there exists a spacelike entire graph in whose principal curvatures are bounded from below by a positive constant and whose support function extends if and only if is in the Zygmund class.

Recall that a function is in the Zygmund class if there is a constant such that, for every ,

Functions in the Zygmund class are -Hölder for every , but in general they are not Lipschitz.

Theorem 1.1 implies that spacelike entire graphs of constant curvature and with a uniform bound on the principal curvatures correspond to functions whose extension to is Zygmund. We prove that also the converse holds. This gives a complete classification of such surfaces in terms of Zygmund functions.

Theorem 1.2.

Let be a function in the Zygmund class. For every there is a unique spacelike entire graph in of constant curvature and with bounded principal curvatures whose corresponding function extends .

The proof of Theorem 1.2 relies on a general statement we prove about solvability of Minkowski problem. We precisely prove the following Theorem.

Theorem 1.3.

Let be a lower semicontinuous and bounded function and for some . Then there exists a unique spacelike graph in whose support function extends and whose curvature function is .

In general we say that a convex function extends if for every . By convexity, if is continuous this condition is equivalent to requiring that is continuous up to the boundary and its boundary value coincides with .

Let us explain the geometric meaning of the boundary value of the support function of . As , regarded as the set of lightlike directions, parameterizes lightlike linear planes, the restriction of the support function on gives the height function of lightlike support planes of , where means that there is no lightlike support plane orthogonal to . It can be checked that, when is the graph of a convex function , the condition is also equivalent to requiring that

for every . The asymptotic condition is stated in the latter fashion for instance in [29] and [9], where the existence problem for constant mean curvature surfaces is treated.

In this paper we will consider future convex surfaces with bounded support function on . Geometrically this means that is contained in the future cone of some point .

It is also useful to consider convex objects more general than spacelike surfaces. A future-convex domain is defined as an open domain in which is the intersection of a family of future half-spaces with spacelike boundary planes. Given a future convex set, we consider a bigger domain obtained as the intersection of the future of the lightlike support planes of . From the Lorentzian point of view is the Cauchy development of the boundary of . A domain obtained in this way is called a domain of dependence. An immediate consequence of Theorem 1.3 is that for any domain of dependence and any there exists a unique convex surface in of constant curvature whose Cauchy development coincides with . More precisely we prove the following Theorem.

Theorem 1.4.

If is a domain of dependence contained in the future cone of a point, then is foliated by surfaces of constant curvature .

A problem that remains open is to characterize spacelike entire graphs with bounded curvature which are complete for the induced metric. If a surface has bounded principal curvatures, the Gauss map turns to be bi-Lipschitz, hence the surface is automatically complete. On the opposite side we construct an example of non complete entire graphs which constant curvature; in our example the boundary extension of the support function is bounded but not continuous. This result underlines a remarkable difference with respect to CMC surfaces. Indeed it was proved in [7, 29] that an entire CMC spacelike graph is automatically complete.

The surfaces we construct are invariant under a one-parameter parabolic group of isometries of fixing the origin, and are isometric to a half-plane in . This strategy goes back to Hano and Nomizu, who first exhibited non-standard immersions of the hyperbolic plane in as surfaces of revolutions, namely surfaces invariant under a one-parameter hyperbolic group fixing the origin.

Ingredients in the proofs

We will use a description of domains of dependence due to Mess. The key fact is that it is possible to associate to a domain of dependence a dual measured geodesic lamination of . It turns out that if is a domain of dependence, its support function is the convex envelope of its boundary value on . Heuristically, the graph of is a piece-wise linear convex pleated surface. The bending lines provide a geodesic lamination over , whereas a transverse measure encodes the amount of bending.

This correspondence will be crucial in the present work. Solving the Minkowski problem with a given boundary value is equivalent to finding a convex entire graph in whose curvature function is prescribed and whose Cauchy development is the domain of dependence determined by . If is the dual measured geodesic lamination of , we construct a sequence of measured geodesic laminations and Fuchsian groups such that is -invariant and converges to on compact subsets of in an appropriate sense. By a result of [2] for every it is possible to solve the Minkowski problem. The proof of Theorem 1.3 is obtained by taking -invariant curvature functions converging to the prescribed curvature function, and showing that solutions of Minkowski problem for the domain dual to with curvature function converge to a solution of the original Minkowski problem for .

The convergence of solutions is obtained first by showing the convergence of the support functions and then by proving that the surface dual to the limit support function is an entire graph.

For the first step a simple application of the maximum principle implies some a priori bounds of the support functions of in terms of the support function of . This allows to conclude that the support functions of converge to a convex function extending . Applying standard theory of Monge-Ampère equation we have that is a generalized solution of our problem. On the other hand, in dimension , Alexander-Heinz theorem implies that is strictly convex and by standard regularity theorem we conclude that is a classical solution.

The second step is more geometric. The key idea is to use - as barriers - the already mentioned constant curvature surfaces which are invariant under a -parameter parabolic group fixing a point . A similar approach (also in higher dimension) using surfaces invariant for a hyperbolic group is taken in [17]. Since the parabolic-invariant surfaces are entire graphs and their support function is constant on , they are very appropriate to show (by applying the comparison principe) that the boundary of the domain dual to a solution of (2) cannot contain lightlike rays. The argument works well under the hypothesis that the boundary value of the support function is bounded, leading to the proof of Theorem 1.3.

The proof of Theorem 1.1 is based on a relation we point out between convex geometry in Minkowski space and the theory of infinitesimal earthquakes introduced and studied in [16, 14, 24, 27]. In particular given we prove that the convex envelope of is explicitly related to the infinitesimal earthquake extending the field . From this correspondence we see that the dual lamination associated by Mess to the domain defined by is equal to the earthquake lamination. Using a result of Gardiner, Hu and Lakic ([14]) we deduce that is Zygmund if and only if the Thurston norm of the dual lamination is finite. Given a convex entire graph with principal curvatures bounded from below by a positive constant, we point out by a direct geometric construction in Minkowski space an explicit estimate on the Thurston norm of the dual lamination. By the above correspondence, this proves one direction of the statement of Theorem 1.1.

Conversely, we show that a spacelike entire graph of constant curvature with bounded dual lamination has bounded principal curvatures. This proves Theorem 1.2 and shows the other implication of Theorem 1.1. The proof is obtained by contradiction. The key fact is the following: if is a domain of dependence with bounded dual lamination and is any sequence of isometries of Minkowski space such that contains a fixed point with horizontal support plane at , then converges to a domain of dependence with bounded dual lamination. Now let be an entire graph of constant curvature whose domain of dependence has bounded dual lamination. If the principal curvatures of were not bounded, we could construct a sequence of isometries bringing back to a fixed point a sequence where the principal curvature are degenerating. The statement above implies that the domain of dependence of is converging to a domain with bounded dual lamination. Applying standard regularity theory of Monge-Ampère equations to the support function of one obtains an a priori bound on the second derivatives of the support function of , thus leading to a contradiction.

Discussions and possible developments

Recently several connections between Teichmüller theory and the geometry of spacelike surfaces in the Anti-de Sitter space have been exploited [21, 1, 5]. The key idea is that graphs of orientation-preserving homeomorphisms of are naturally curves in the asymptotic boundary of the Anti-de Sitter space spanned by spacelike surfaces. A natural construction allows to associate, to any convex surface spanning the graph of a homeomorphism , a homeomorphism extending . It turns out that is quasi-conformal if and only if the principal curvatures of are bounded, showing that in this case is quasi-symmetric.

Theorem 1.1 can be regarded as an infinitesimal version of this property. More generally we believe that the relation between the theory of convex surfaces in Minkowski space and the infinitesimal Teichmüller theory is an infinitesimal evidence of the above connection for Anti-de Sitter space. The correct geometric setting to understand this idea goes through the geometric transition introduced recently in [10, 12, 11]. In fact Danciger introduced the half-pipe geometry, which is a projective blow-up of a a spacelike geodesic plane in Anti-de Sitter space. This model turns out to be a natural parameter space for spacelike affine planes in Minkowski space. Regarding the Minkowski space as the blow-up of a point in Anti-de Sitter space, the correspondence between spacelike affine planes in Minkowski space and points in half-pipe geometry is the infinitesimal version of the projective duality between points and spacelike planes in the Anti-de Sitter space. Using this connection the graph of support functions are naturally convex surfaces in half-pipe geometry.

Let be a smooth family of convex surfaces, for , such that is a totally geodesic plane and the principal curvatures of are . Using the correspondence above one can prove that the boundary of is the graph of a differentiable family of quasi-symmetric homeomorphisms of such that .

The rescaled limit of in the half-pipe model is the graph of the support function of a convex surface in Minkowski space. It turns out that the principal curvatures of are the inverse of the derivatives of the principal curvatures of . In particular they are bounded from below by a positive constant is the principal curvatures of are . Moreover the support function at infinity of corresponds to the vector field on .

This gives an evidence of the fact that Theorem 1.1 is a rescaled version of the correspondence between quasi-symmetric homeomorphisms and convex surfaces in Anti de Sitter space with bounded principal curvatures.

We believe that even Theorem 1.2 is a rescaled version in this sense of the existence of a -surface (actually, a foliation by -surfaces, compare Theorem 1.4) with bounded principal curvature spanning a prescribed quasi-symmetric homeomorphism (for ). However we leave this problem for a further investigation.

Finally we mention some questions still open in the Minkowski case, which are left for future developments.

  • As already remarked, an interesting problem is to characterize complete spacelike entire graphs in Minkowski space in terms of the regularity of the boundary value of the support function. From Theorem 1.2 we know that the surface is complete if this regularity is Zygmund. On the other hand it is not difficult to construct an example of isometric embedding of in Minkowski space with unbounded principal curvatures, for which the regularity of the support function will necessarily be weaker than Zygmund.

  • Another interesting question is to solve Minkowski problem for domains of dependence which are not contained in the future of a point. This would include the case of domains of dependence whose support function is finite only on some subset of which is obtained as the convex hull of a subset of . It is simple to check that there is no solution for the constant curvature problem when contains , or points (the corresponding domains are the whole space, the future of a lightlike plane, or the future of a spacelike line). An existence result in case where is an interval is given in [17] with some assumption on the smoothness of the support function on . In our opinion the construction of the support function in this setting is not difficult to generalize, but the barriers we use to prove that the corresponding surfaces are entire graph seem to be ineffective.

Organization of the paper

In Section 2 we give a short review of the different techniques used in the paper. We first recall the main definitions of the theory of future convex sets in Minkowski space, introducing the support function and the generalized Gauss map. We relate those notions to Mess’ description of domains of dependence and dual laminations. Minkowski problem is then formulated in terms of Monge-Ampère equations and we collect the main result used in the paper. Finally we discuss the notion of infinitesimal earthquakes and how it is related to the infinitesimal theory of Teichmüller spaces.

In Section 3 we solve Minkowski problem for domains of dependence contained in the future cone of a point, proving Theorem 1.3. In the following section we prove the existence of the foliation by constant curvature surfaces of the same domains, as in Theorem 1.4. In Section 5 we finally prove Theorems 1.1 and 1.2. Those theorems are collected in a unique statement, see Theorem 5.1.

Finally in Appendix A we construct constant curvature surfaces invariant by a parabolic group. The discussion leads to consider two classes of surfaces: the first class (used in the proof of Theorem 1.3) includes surfaces which are entire graphs, whereas the second class gives surfaces which develop a lightlike ray. Although we do not need surfaces of the latter type in this paper, we include a short description for completeness.

Acknowledgements

The authors would like to thank Thierry Barbot, François Fillastre and Jean-Marc Schlenker for their interest and encouragement during several fruitful discussions.

2. Preliminaries and motivation

2.1. Convex surfaces in Minkowski space

We denote by the -dimensional Minkowski space, namely endowed with the bilinear quadratic form

We denote by the group of orientation preserving and time preserving isometries of . It is know that there is an isomorphism

where is the group of linear isometries of Minkowski product, is the connected component of the identity, and acts on itself by translations.

A vector is called timelike (resp. lightlike, spacelike) if (resp. , ). A plane is spacelike (resp. lightlike, timelike) if its orthogonal vectors are timelike (resp. lightlike, spacelike). An immersed differentiable surface in is spacelike if its tangent plane is spacelike for every point . In this case, the symmetric 2-tensor induced on by the Minkowski product is a Riemannian metric. For instance, the hyperboloid

is a spacelike embedded surface, and is indeed an isometric copy of hyperbolic space embedded in . An example of an embedded surface which is not spacelike - but that will still be important in the following - is the de Sitter space

It is easy to see that parametrizes oriented geodesics of . Indeed, for every point in , the orthogonal complement is a timelike plane in , which intersects in a complete geodesic, and coincides as a set with .

Given a point , denotes the future cone over , namely the set of points of which are connected to by a timelike differentiable path (namely, a path with timelike tangent vector at every point) along which the coordinate is increasing. It is easy to see that is a translate of

It is clear from the definition that parametrizes spacelike linear planes in . Hence for every spacelike surface , there is a well-defined Gauss map

which maps to the normal of at , i.e. the future unit timelike vector orthogonal to .

As we stated in the introduction, one of the aims of this paper is to classify isometric (or more generally homothetic) immersions of the hyperbolic plane into which are contained in the future cone over some point. Mess proved [21] that if the first fundamental form of a spacelike immersion is complete, then the image of the immersion is a spacelike entire graph. So we will deal with convex surfaces in which are of the form , where is a convex function satisfying the spacelike condition , where is the Euclidean gradient of . Notice however that if is a spacelike entire graph in general it might not be complete.

It is convenient to extend the theory to the case of convex entire graphs which are not smooth and possibly contain lightlike rays. Those correspond to convex functions such that almost everywhere. We will extend the notion of Gauss map to this more general class.

A future-convex domain in is a closed convex set which is obtained as the intersection of future half-spaces bounded by spacelike planes. If is a convex function satisfying the condition , then the epigraph of is a future-convex domain, and conversely the boundary of any future-convex domain is the graph of a convex function as above.

A support plane for a future-convex domain is a plane such that and every translate , for in the future of , intersects . A future-convex domain can admit spacelike and lightlike support planes. We define the spacelike boundary of as the subset

It can be easily seen that is a union of lightlike geodesic rays. So is a spacelike entire graph if and only if it does not contain lightlike rays.

We can now define the Gauss map for the spacelike boundary of a future-convex set. We allow the Gauss map to be set-valued, namely

By an abuse of notation, we will treat the Gauss map as a usual map with values in . The following proposition (see [22, 23]) has to be interpreted in this sense.

Proposition 2.1.

Given a future-convex domain in , the Gauss map of has image a convex subset of . If is a strictly convex embedded spacelike surface, then its Gauss map is a homeomorphism onto its image.

2.2. The support function

We now introduce another important tool for this work, which is the Lorentzian analogue of the support function of Euclidean convex bodies. Roughly speaking, the support function encodes the information about the support planes of a future-convex domain. This is essentially done by associating to a unit future timelike (or lightlike) vector the height of the support plane with normal unit vector .

Definition 2.2.

Given a future-convex domain in , the support function of is the function defined by

The following is an immediate property of support functions.

Lemma 2.3.

Let and be future-convex domains with support functions and . Then if and only if .

It is clear from the definition that is 1-homogeneous, namely for every . Moreover, is lower semicontinuous, since it is defined as the supremum of continuous functions. It is straightforward to check that, given an isometry in with linear part and translation part , the support function of is

(3)

We will mostly consider the restriction of to the Klein projective model of hyperbolic space, which is the disc

This restriction will be denoted by lower case letters, , and uniquely determines the 1-homogeneous extension . We will generally write instead of . Analogously, also the restriction of to the hyperboloid, denoted , can be uniquely extended to a 1-homogeneous function, and will be often used in the following.

Remark 2.4.

It is easy to relate the restrictions and of the support function to and respectively. Let us consider the radial projection defined by

Its inverse is given by

Since is 1-homogeneous, we obtain

A 1-homogeneous convex function is called sublinear.

Lemma 2.5 ([13, Lemma 2.21]).

Given a future-convex domain in , the support function is sublinear and lower semicontinuous. Conversely, given a sublinear function on (or equivalently every convex function on ), consider the lower semicontinuous extension , which is defined on as

Then is the support function of a future-convex domain, defined by

The support function of a future-convex domain is finite on the image of the Gauss map of , since for every point in there exists a support plane with normal vector . Observe that if . We will call support function at infinity the restriction of to . Given , if and only if there exists a lightlike support plane orthogonal to the lightlike vector . In this case is the intercept of on the -axis.

Example 2.6.

The support function of the hyperboloid is . Hence its restriction is constant, . The support function at infinity is finite, , by an easy computation. For some less elementary examples, see Remark 3.12 and Appendix A.

In this paper, we are mostly concerned with domains of dependence for which the support function at infinity is finite, and is actually bounded. Geometrically, this means that the domain is contained in the future cone over some point.

The following lemma will be useful to compute the value of support functions at infinity.

Lemma 2.7 ([26, Theorem 7.4,7.5]).

Let be a sublinear and lower semincontinuous function. Let be a spacelike line such that . Then .

Some of the geometric invariants of can be directly recovered from the support function. This is the content of next lemma. Recall that, given a embedded spacelike surface in , its shape operator is a -tensor which can be defined as

where is the Levi-Civita connection of the first fundamental form of , is the future unit normal vector field, and is any tangent vector in . The Gauss Theorem in this Lorentzian setting gives the following relation between the intrinsic curvature of the first fundamental form and its shape operator, which holds for every point :

Finally, we define the hyperbolic Hessian of a function as the -tensor

where is the Levi-Civita connection of . We denote by the Euclidean Hessian of a function defined on an open subset of . In the following, the identity operator is denoted by .

Lemma 2.8 ([13, §2.10, 2.13]).

Let be a future-convex domain in and let be its Gauss map.

  • If the support function is , then the intersection of with any spacelike support plane consists of exactly one point. The inverse of the Gauss map is well defined and is related to the support function of by the formula

    (4)

    where is the support function restricted to .

  • If the support function is and the operator is positive definite, then is a convex -surface. The inverse of its shape operator and its curvature are

    (5)
    (6)

    where is the point of obtained from by radial projection.

We will often abuse notation and write in place of for .

2.3. The boundary value of the support function of an entire graph

Let be the boundary of a future-convex domain in . Denote by the function defining as a graph and the support function. We want to show that

for every unitary vector . This clarifies that the asymptotic conditions defined for instance in [29, 9] coincide with those considered here and in [17, 20].

First consider the case where where and . In this case the support function is . A simple computation shows

Now consider the general case. Imposing that the point lies in the future of the support plane we get . So it is sufficient to prove that .

Fix and consider the lightlike plane . This plane must intersect the future of . Let be a point in this intersection. The cone is contained in the future of , hence , where is the graph function for as above.

In particular, using the computation above for ,

Imposing that lies on the plane ,

Therefore, for any ,

and this concludes our claim, since is arbitrary.

2.4. Cauchy surfaces and domains of dependence

Given a future-convex domain in , a Cauchy surface for is a spacelike embedded surface such that every differentiable inextensible causal path in (namely, such that its tangent vector is either timelike or lightlike at every point) intersects in exactly one point. Given an embedded surface in , the maximal future-convex domain such that is a Cauchy surface for is the domain of dependence of . It turns out that is obtained as intersection of future half-spaces bounded by lightlike planes which do not disconnect .

It is easy to prove the following lemma.

Lemma 2.9.

Let be the support function of a future-convex domain , with . Let be a convex embedded surface and let be the support function of . Then is a Cauchy surface for if and only if .

Domains of dependence can be characterized in terms of the support function, see [4, Proposition 2.21].

Lemma 2.10.

Let be a domain of dependence in , whose lightlike support planes are determined by the function . Then the support function of is the convex envelope , namely:

A useful example of support functions of Cauchy surfaces can be obtained by looking at the leaves of the cosmological time of the domain of dependence. Observe that a timelike distance can be defined for two points and in , by means of the definition

where the supremum is taken over all causal paths from to . This is not a distance though, because it satisfies a reverse triangle inequality; however, is achieved along the geodesic from to . Given an embedded spacelike surface , consider the equidistant surface

where of course . If the support function of restricted to is , then has support function (see for instance [13])

This can be applied also for , instead of an embedded surface. In this way, we obtain the level sets of the cosmological time, namely the function defined by

where the supremum is taken over all causal paths in with future endpoint . If is the support function of , the level sets of the cosmological time have support function on the disc . It can be easily seen that all leaves of the cosmological time of are Cauchy surfaces for (although only ). Indeed, the support functions can be computed:

Therefore they all agree with on .

2.5. Dual lamination

In this paper, we will adopt the following definition of measured geodesic lamination. The equivalence with the most common definition is discussed for instance in [24]. Let be the set of (unoriented) geodesics of . The space is identified to where the equivalence relation is defined by . Note that has the topology of an open Möbius strip. Given a subset , we denote by the set of geodesics of which intersect .

Definition 2.11.

A geodesic lamination on is a closed subset of such that its elements are pairwise disjoint geodesics of . A measured geodesic lamination is a locally finite Borel measure on such that its support is a geodesic lamination.

A measured geodesic lamination is called discrete if its support is a discrete set of geodesics. A measured geodesic lamination is bounded if

where the supremum is taken over all geodesic segments of lenght at most 1 transverse to the support of the lamination. The Thurston norm of a bounded measured geodesic lamination is

Elements of a geodesic lamination are called leaves. Strata of the geodesic lamination are either leaves or connected components of the complement of the geodesic lamination in .

In his groundbreaking work, Mess associated a domain of dependence to every measured geodesic lamination , in such a way that the support function of is linear on every stratum of . Although we do not enter into details here, the measure of determines the bending of . (Recall is the convex envelope of some lower semicontinuous function .) The domain is determined up to translation in , and is called dual lamination of .

Given and , we will denote by the domain of dependence having as dual laminations and as a support plane tangent to the boundary at .

We sketch here the explicit construction of . In the following, given the oriented geodesic interval in , is the function which assigns to a geodesic (intersecting ) the corresponding point in , namely, the spacelike unit vector in orthogonal to for the Minkowski product, pointing outward with respect to the direction from to . Then

(7)

is a point of the regular boundary of such that is a support plane for , for every such that the expression in Equation (7) is integrable. The image of the Gauss map of the regular boundary of is composed precisely of those which satisfy this integrability condition.

In the following proposition, we give an explicit expression for the support function of the domain of dependence we constructed. By an abuse of notation, given two points , we will denote by the geodesic interval of obtained by projecting to the hyperboloid the line segment from to .

Proposition 2.12.

Suppose is a domain of dependence in with dual lamination and such that the plane is a support plane for , for and . Then the support function of is:

(8)

Indeed, the expression in Equation (8) holds for by Equation (7). Since the expression is 1-homogeneous, it is clear that it holds for every . Using Lemma 2.7, the formula holds also for the lower semicontinuous extension to .

It is easily seen from Equation (8) that the support function (which is the restriction of to ) is affine on each stratum of . In [21, 3] it was proved that every domain of dependence can be obtained by the above construction. Hence a dual lamination is uniquely associated to every domain of dependence.

The work of Mess ([21]) mostly dealt with domains of dependence which are invariant under a discrete group of isometries , whose linear part is a cocompact Fuchsian group. We resume here some results.

Proposition 2.13.

Let be a domain of dependence in with dual lamination . The measured geodesic lamination is invariant under a cocompact Fuchsian group if and only if is invariant under a discrete group such that the projection of to is an isomorphism onto . In this case, assuming is a support plane for , for and , the translation part of an element is:

2.6. Monge-Ampère equations

Given a smooth strictly convex spacelike surface in , let be the support function of and let be its restriction to . Given a point , let . The curvature of is given by (see Lemma 2.8)

where is the Gauss map, which is a diffeomorphism. For -surfaces, namely surfaces with constant curvature equal to , the support function satisfies the Monge-Ampère equation

(9)

More generally, the Minkowski problem consists of finding a convex surface with prescribed curvature function on the image of the Gauss map. Given a smooth function , the support function of a surface with curvature solves the Monge-Ampère equation

(\ref{monge ampere constan curvature})

We review here some key facts of Monge-Ampère theory. Given a convex function for a convex domain in , we define the normal mapping of as the set-valued function whose value at a point is:

In general is a convex set; if is differentiable at , then . We define the Monge-Ampère measure on the collection of Borel subsets of :

where denote the Lebesgue measure on .

Lemma 2.14 ([30, Lemma 2.3]).

If is a function, then

In general, is the regular part of the Lebesgue decomposition of , where we set

Definition 2.15.

Given a nonnegative measure on , we say a convex function is a generalized solution to the Monge-Ampère equation

(10)

if for all Borel subsets . In particular, given an integrable function , is a generalized solution to the equation if and only if, for all ,

We collect here, without proofs, some facts which will be used in the following. Unless explicitly stated, the results hold in , although we are only interested in . Recall that, by Aleksandrov Theorem, a convex function on is twice-differentiable almost everywhere.

Lemma 2.16 ([30, Lemma 2.2]).

Given a sequence of convex functions which converges uniformly on compact sets to , the Monge-Ampère measure converges weakly to .

Theorem 2.17 (Comparison principle, [30, 18]).

Given a bounded convex domain and two convex functions defined on , if for every Borel subset , then

Corollary 2.18.

Given two generalized solutions to the Monge-Ampère equation on a bounded convex domain , if on , then on .

Theorem 2.19 ([8, Lemma 3], [25]).

Given a bounded convex domain , let be a solution to defined on which is constant on . There is an estimate on the second derivatives of at which depends only on

and on the distance of to .

The following property will be used repeatedly in the paper, and is a peculiar property of dimension