Polyhedral surfaces in flat (2 + 1)-spacetimes and balanced cellulations on hyperbolic surfaces

1.1 Main statements

The Gauss map encodes an intricate information about how a surface is curved in space. For convex surfaces, the Gauss map can be defined without further regularity assumptions, but by dropping the smoothness, we accept that the Gauss map becomes not really a map, but a multivalued map. The first concern of the current article is a study of the Gauss map from convex polyhedral surfaces in flat ( 2 + 1 ) -spacetimes. In particular, we are interested how it translates their extrinsic geometry into the intrinsic geometry of the target, which in our setting is a hyperbolic surface. Let us first recall some details in the classical case of Euclidean 3-space R 3 .

Consider a (compact) convex polyhedron 𝑃 in R 3 . The Gauss map of the polyhedron is a multivalued map that takes a boundary point of 𝑃 and associates it with the set of outward unit normals to the support planes of 𝑃 at this point. Therefore, the Gauss map transforms the polyhedron 𝑃 to a convex geodesic cellulation of the round sphere S 2 ⊂ R 3 . This cellulation, denoted as 𝒢, is inherently dual to the face cellulation of the original polyhedron 𝑃. Here is how this duality is manifested.

• [nolistsep]

• The Gauss image of a face of 𝑃 is a vertex of 𝒢.

• For an edge 𝐸 of 𝑃, its Gauss image is an edge 𝑒 in 𝒢, and the length of 𝑒 in S 2 is equal to the exterior dihedral angle of 𝑃 at 𝐸.

• Finally, under the Gauss map, a vertex 𝑉 of the polyhedron 𝑃 becomes a convex spherical polygon. The area of this spherical polygon is precisely the singular curvature present at the vertex 𝑉.

We assign a positive weight, denoted as w e , to each edge 𝑒 of 𝒢, equal to the length of the respective edge in the original polyhedron 𝑃. Considering a vertex 𝑣 of 𝒢, we see that the weights w e of the incident edges satisfy the following balance condition:

Here U e is the unit tangent vector of S 2 to the edge 𝑒 at the vertex 𝑣. We will refer to this balanced convex cellulation ( G , w ) as “the Gauss image of 𝑃”, although this is a slight abuse of terminology. We will call such a geodesic convex cellulation over S 2 with positive weights satisfying the balance condition (1.1) a balanced cellulation over S 2 .

In the other direction, from a balanced cellulation ( G , w ) over S 2 , it is not hard to construct a convex polyhedron 𝑃 such that its Gauss image is ( G , w ) . Also, 𝑃 is unique up to Euclidean isometries; see e.g. [62, 63].

The induced intrinsic metric 𝑑 on the boundary of 𝑃 is a flat metric with conical singularities with positive curvature on the topological sphere S 2 . This metric can be also recovered from the Gauss image ( G , w ) of 𝑃. Indeed, the balance condition (1.1) means that we may associate to each vertex of ( G , w ) a flat polygon so that the edges have outward unit normals U e and lengths w e . Gluing these polygons along the combinatorics dual to 𝒢, we obtain a flat surface with conical singularities, which is isometric to 𝑑 by construction. We will say that the flat metric 𝑑 is dual to ( G , w ) ; see Figure 1.

Figure 1

The left-hand side of the picture is a part of a balanced cellulation, around a face 𝑓. The green arrows in this picture are unit tangent vectors to the edges of the cellulation. In the picture in the middle, Euclidean strips are glued to every edge appearing in the left-hand side picture, in a direction orthogonal to the direction of the edge, and of width the weight of the edge. By the balance condition, the empty parts between the strips may be filled by convex flat polygons (colored brown). Gluing those flat polygons by identifying two segments that bound the same strip produces a flat metric on the sphere (picture on the right-hand side). These faces are glued around a vertex, and the sum of the interior angles of the faces at this vertex is equal to the sum of the exterior angles of the spherical convex polygon 𝑓. By the Gauss–Bonnet Formula, this sum plus the area of 𝑓 is equal to 2 ⁢ π .

At first glance, it might appear that retrieving the flat metric from a balanced cellulation provides less comprehensive information compared to recovering the entire convex polyhedron. However, the renowned theorem of Alexandrov demonstrates that surprisingly it is sufficient to recover the metric alone.

From what was said above, Theorem 1.1 is equivalent to the following statement.

The first aim of the present article is to generalize the theorems above to higher genus. The generalization of the definition of balanced cellulation is immediate to any constant curvature surface. The dual metric of a balanced cellulation over a flat torus is a flat metric (without cone-singularities), and we will not consider this case.

We will say that a surface 𝑆 is of hyperbolic type if 𝑆 is a connected closed oriented surface of genus greater than 1, and that 𝑆 is a hyperbolic surface if it is of hyperbolic type and is endowed with a hyperbolic metric. For a hyperbolic surface, due to the Gauss–Bonnet Formula for hyperbolic convex polygons, the curvature at the cone points of the dual flat metric is negative. The first aim of the present article is to prove the following hyperbolic version of Theorem 1.2.

A simplest example of the dualization is given by the convex cellulation of the hyperbolic genus 2 surface obtained by identifying the side of a regular hyperbolic octagon. If weights 1 are provided on any of the edges, then in the tangent space at the unique vertex of the cellulation, we obtain a flat regular octagon. Identifying its sides, we get a flat metric with a unique cone singularity of curvature 4 ⁢ π .

There is also an extrinsic version of Theorem I, which is actually the main point of our focus. We want to embed the flat metric 𝑑 isometrically as a convex polyhedral surface 𝑃 in a natural 3-dimensional space 𝑀. To have abundance of polyhedral surfaces, 𝑀 should have constant curvature. As the metric is flat, 𝑀 should be flat, and as the singular curvature are negative, 𝑀 should be Lorentzian. The convexity allows to embed the universal covering of 𝑃 into Minkowski space R 2 , 1 so that the embedding is equivariant with respect to a holonomy map π 1 ⁢ S → Isom 0 ⁢ ( R 2 , 1 ) . As the Gauss image of 𝑃 must be isometric to ( S , h ) , the linear part 𝜌 of the holonomy is prescribed by ℎ. Here we are identifying the hyperbolic plane with its hyperboloid model in Minkowski space R 2 , 1 , and choosing a developing map, so the holonomy of ( S , h ) has values in the subgroup SO 0 ⁢ ( 2 , 1 ) of the group of linear isometries of Minkowski space.

We will consider spacetimes 𝑀 whose holonomy is obtained in the following way. For any element γ ∈ π 1 ⁢ S , we define τ ⁢ ( γ ) ∈ R 2 , 1 and set ρ τ ( γ ) x : = ρ ( γ ) x + τ ( γ ) for all x ∈ R 2 , 1 . The image ρ τ ⁢ ( π 1 ⁢ S ) is a subgroup of isometries of Minkowski space if and only if 𝜏 satisfies a cocycle condition. From a work of G. Mess [51, 4], it is known that there exists a maximal open set in Minkowski space on which ρ τ ⁢ ( π 1 ⁢ S ) acts freely and properly. This maximal open set has two convex connected components, one future-complete and one past-complete. The quotient of such a connected component gives an example of what is known as flat globally hyperbolic maximal Cauchy compact ( 2 + 1 ) -spacetime, which is a concept coming from General Relativity (in short, flat GHMC ( 2 + 1 ) spacetime – we will often skip the reference to the dimension in the text). Let us denote these quotients by Ω + ⁢ ( ρ , τ ) and Ω − ⁢ ( ρ , τ ) respectively.

For the details of this construction, we refer the reader to [51, 4, 13, 5] (where [13, 5] consider higher-dimensional generalizations).

The following theorem implies Theorem I after an application of the Gauss map.

By construction, ( 2 + 1 ) -spacetimes Ω ± ⁢ ( ρ , τ ) are marked in the sense that, for every Cauchy surface Σ ⊂ Ω ± ⁢ ( ρ , τ ) , there exists a natural homeomorphism S → Σ determined up to isotopy. The marked isometry between ( S , d ) and ∂ P in Theorem I ^′ means that this isotopy class contains an isometry (which is then unique).

Actually, Theorem I and Theorem I ^′ are equivalent, as from the balanced cellulation, it is easy to construct the spacetime and the polyhedral surface, in a way similar to the Euclidean case; see [35, 31].

The second aim of the present article is to prove the following Simultaneous Uniformization Theorem.

Theorem II ^′ means that, to a discrete and faithful representation ρ τ : π 1 ⁢ S → Isom 0 ⁢ ( R 2 , 1 ) , we attach a geometric object, a pair of convex polyhedral surfaces embedded into the respective spacetimes Ω ± ⁢ ( ρ , τ ) , and show that our representation plus the geometric object is completely determined by the intrinsic geometry of the surfaces. In Section 1.3, we review analogous results from other settings serving as a source of our motivation.

The cocycles 𝜏 introduced above have values in Minkowski space. But there is a natural identification Λ between Minkowski space and the Lie algebra of SO ⁢ ( 2 , 1 ) . If the Teichmüller space T ⁢ ( S ) of 𝑆 is seen as the space of the equivalence classes of discrete and faithful representations of π 1 ⁢ S into SO 0 ⁢ ( 2 , 1 ) , then applying the identification Λ, a cocycle is identified with a tangent vector to the Teichmüller space. In turn, Theorem I ^′ says that any flat metric 𝑑 over 𝑆 with negative singular curvatures defines a vector field X d over T ⁢ ( S ) , which will be shown to be C 1 . In the present article, we will prove the following theorem, which implies Theorem II ^′ . It is worth noting that if a convex polyhedral set 𝑃 in Minkowski space is ρ τ -invariant, then − P is ρ − τ -invariant, and of course, the induced intrinsic metrics over ∂ P and ∂ ( − P ) are related by an isometry.

Theorem II is proved by associating to the metric 𝑑 the total length function L d over the Teichmüller space. For any hyperbolic metric over 𝑆, it is equal to the sum of the weighted lengths of the edges of the corresponding balanced convex cellulation. It will turn out that L d is strictly convex and proper, and that its Weil–Petersson symplectic gradient is X d .

1.2 Sketch of the proof of Theorem I ^′ and related results

The original proof by A. Alexandrov of Theorem 1.1 introduced the now classical deformation method. Roughly speaking, the space of isometric classes of flat metrics with a given number 𝑛 of vertices and the space of isometric classes of convex polyhedra (in R 3 ) with 𝑛 vertices are endowed with suitable topologies that make them manifolds of the same finite dimension. A natural map ℐ from the latter to the former is given by considering the induced intrinsic metric on the boundary of a polyhedron. Theorem 1.1 says that ℐ is a bijection, and furthermore, Alexandrov proves that it is a homeomorphism [2]. A key point is to prove that ℐ is injective, which is a refinement of the famous Cauchy Theorem. The proofs of the latter results heavily use special topological properties of the sphere and cannot be directly generalized to surfaces of higher genus. Instead of directly proving that ℐ is injective, one may show that it is locally injective, by using a first-order version of the Cauchy Theorem and the Local Inverse Theorem. In the case of the sphere, the first-order version of the Cauchy Theorem can be formulated as follows: it is not possible to deform at first order a convex geodesic cellulation of S 2 without changing the face angles at first order [2, Theorem 10.3.2].

In the hyperbolic case, the analogue of the first-order Cauchy theorem is the following statement.

This theorem is proven in [42]. It seems to us, however, that the proof, unfortunately, contains a mistake, which, however, is fixable for convex cellulations. (I. Iskhakov in [42] claims to prove Theorem 1.3 also for non-convex cellulations, but in this case, we, unfortunately, do not see how to make his proof working.) Because of this and because for our purposes we need to slightly generalize this result, we give our exposition of the proof of Theorem 1.3 in Section 3.3 together with more details on the mistake in the original argument.

Now we describe more precisely our proof strategy for Theorem I ^′ . Fix a holonomy ρ : π 1 ⁢ S → SO 0 ⁢ ( 2 , 1 ) of a hyperbolic metric ℎ over 𝑆, and let V ⊂ S be a finite set of cardinality 𝑛. By P c ⁢ ( ρ , V ) , we denote the space of configurations of points in Ω + ⁢ ( ρ , τ ) for some 𝜏 that are in bijection with 𝑉 and are in strictly convex position. The convex polyhedra we are looking for are the convex hulls of such configurations. We will see in Lemma 2.11 and Lemma 3.8 that P c ⁢ ( ρ , V ) is a connected analytic manifold of dimension 6 ⁢ g − 6 + 3 ⁢ n , where 𝐠 is the genus of 𝑆. Heuristically, 3 ⁢ n corresponds to the position of the points, and 6 ⁢ g − 6 to the choice of the cocycle, since the space of equivalence classes of cocycles can be identified with the tangent space at ρ 0 of the Teichmüller space of 𝑆.

On the other hand, let M − ⁢ ( S , V ) be the space of flat metrics over 𝑆 with the set of cone points in bijection with 𝑉, and with negative singular curvatures, considered up to marked isometries fixing 𝑉 and isotopic to identity. It is well known that such an (isotopy class of) metric is determined by a point in T ⁢ ( S ) , the position of the elements of 𝑉 and their cone angles. In turn, M − ⁢ ( S , V ) is a connected analytic manifold of dimension 6 ⁢ g − 6 + 3 ⁢ n ; see Section 2.4 and Lemma 3.9.

We define the map I ρ from P c ⁢ ( ρ , V ) to M − ⁢ ( S , V ) that associates to any configuration of points in P c ⁢ ( ρ , V ) the induced intrinsic metric on the boundary of the convex hull of the marked points. It appears that it is a C 1 -map (Lemma 2.14), and Theorem 1.3 implies that d ⁢ I ρ is non-degenerate. By the Local Inverse Theorem, I ρ is a local diffeomorphism.

Section 3.4 establishes that I ρ is proper. The compactness result for the ambient spacetimes is inspired by the techniques of F. Bonsante in [13], and the compactness result for the position of vertices is based on a result of T. Barbot, F. Béguin and A. Zeghib that relates the systole of a Cauchy surface to its distance to the initial singularity [6].

With the help of few additional topological arguments, we obtain the following theorem (see Section 3.2).

There is the following cousin of Theorem I ^′ .

The spacetimes Ω + ⁢ ( ρ , 0 ) are often called Fuchsian. Theorem 1.5 was proved in [30] by the first author. Note that, in this reference, the fact that the map induced metric is C 1 is not established, but it is covered by Lemma 2.14 of the present article. Instead of Theorem 1.3 by Iskhakov, the proof of Theorem 1.5 is based on a first-order rigidity result proved in [61, Theorem 6.2]. Later, a variational proof of Theorem 1.5 was done in [20], using the Hilbert–Einstein functional. This variational approach was first used in the Euclidean case (see the introduction of [10]) and in many other situations. However, we do not know if the variational approach can be implemented in our general non-Fuchsian case. Note that [30] considers similar results for spherical and hyperbolic cone-metrics with negative singular curvatures, while the Fuchsian realizations of hyperbolic cone-metrics of positive singular curvatures were investigated in [59, 29] – which were thought as a polyhedral analogue of [47].

There is a smooth analogue of Theorem I ^′ where the flat metric is replaced by a Riemannian metric of negative curvature. The equivalent statement similar to Theorem I is about the existence of a Codazzi tensor on a hyperbolic surface, determined by the negatively curved metric on the same surface. It is proved in [67]; we refer to [64] for more details. An interesting point is that, even if the proof of [67] is by a deformation method, the solution is identified with a critical point of a functional.

In Theorem I, we start from a flat metric and a hyperbolic metric, and look for a suitable balanced cellulation. There is a related problem, when one starts from a topological cellulation with positive weights on the edges and, for any hyperbolic metric on the surface, finds a homotopic balanced cellulation, which can be considered as a discrete harmonic map. A proof is done for a different balance condition by minimizing an energy functional [23, 44]. The balance condition appearing there for a vertex 𝑣 is given by

where ℓ e is the length of 𝑒 for the hyperbolic metric, and it can be equal to 0. Note that the balance condition forces the geodesic realization to be convex: if it is not convex at a vertex, then all the U e will point to a same half-space, and as the weights are taken positive, the balance condition cannot be satisfied. Also some edges of the cellulations may degenerate to a point. When this does not happen, we call the weighted cellulation admissible. It is known that triangulations are admissible [23, 50]. It is possible to see that [44, Theorem 2.1] translates as a statement similar to our Theorem I ^′ , at least in the admissible case, but instead of the induced metrics, one prescribes combinatorics of the polyhedron and for each edge prescribes the value of the edge length divided by dihedral angle. See [37] for another discretization of the energy functional and [48] for related results.

We do not know if the result of Theorem I corresponds to some minimum of a functional. However, we will obtain as a byproduct of our proof of Theorem I ^′ some regularity properties that are needed in order to state Theorem II, as we will discuss in the next section.

Note that an alternative to prescribe a weighted cellulation is to prescribe a weighted multi-curve or, more generally, a measured lamination on a surface. To obtain a measured geodesic lamination from it once a hyperbolic metric is chosen on the surface is classical, and it has a clear 3 ⁢ d interpretation as it is the Gauss image of the initial singularity of a flat GHMC ( 2 + 1 ) -spacetime, which was first noted by Mess [51].

Theorem I ^′ can be considered as a part of a general paradigm that geometry of spacelike future-convex equivariant convex sets in R n , 1 might be seen as a Lorentzian analogue of theory of Euclidean convex bodies; see e.g. [35, 14]. See the next section for similar results to Theorem I ^′ in the hyperbolic or anti-de Sitter setting.

1.3 Sketch of the proof of Theorem II and related results

From the discussion above, one can associate to a balanced cellulation ( G , w ) over a hyperbolic surface ( S , h ) an element of T ρ ⁢ T ⁢ ( S ) , where 𝜌 is the class of the holonomy of ℎ. Indeed, one considers the cohomology class associated to the flat GHMC spacetime, into which the flat metric dual to ( G , w ) embeds. Theorem I ^′ says that then any flat metric 𝑑 over 𝑆, with negative singular curvatures, defines a tangent vector field X d over Teichmüller space. From the regularity of maps involved in the proof of Theorem I ^′ , it will follow that X d is C 1 (Lemma 4.2).

We will consider the total length function L d over T ⁢ ( S ) that associates to each hyperbolic metric ℎ the weighted sum of the lengths of the balanced convex cellulation defined by 𝑑 and ℎ. The first point is that the Weil–Petersson symplectic gradient of L d is − X d (Proposition 4.20). Note that this immediately implies a reciprocity formula as in [71, Theorem 2.11].

The main point will be Lemma 4.24, which says that the (Weil–Petersson) Hessian of L d is positive definite. The argument is inspired by [70]. In this article, M. Wolf computed the Hessian of the length of a closed geodesic, where the length is also considered as a function over Teichmüller space (that this function is convex was previously observed by S. Wolpert in [73]).

The proof of Theorem II is established by the analysis of the function L d 1 + L d 2 . This argument is inspired by a work of F. Bonahon [11], where an analogous result is shown with flat metrics replaced by measured laminations.

In the realm of flat GHMC ( 2 + 1 ) -spacetimes, this connection was further extended in [16], where it was proved that such spacetimes can be parameterized by filling pairs of measured laminations (and moreover, it was also extended to the anti-de Sitter case, i.e., the case of constant curvature −1). A measured lamination essentially serves as the Gauss image of the initial singularity of a spacetime. It can be seen as a degenerate analogue of Theorem I ^′ .

The energy of a closed geodesic was studied in [74] by S. Yamada. For a weighted graph filling the surface, the fact that the energy has a minimum was proved in [44] by T. Kajigaya and R. Tanaka. The results in [44] (especially [44, Theorem 2.3 and Theorem 2.5]), in the case we called admissible in the end of the preceding section, imply an analogue of Theorem II ^′ with weighted filling graphs instead of flat metrics, and the prescribed weights give the edges lengths of the polyhedron divided by the dihedral angles. What worked without restriction and what we are doing in the present article is to prescribe a flat metric instead of a weighted cellulation.

The smooth analogue of Theorem II ^′ was done in [64] by G. Smith (see [34] for a partial result). The proof uses a functional, which is heuristically the same as ours, in the sense that, when a respective surface is realized in a flat spacetime, the value of both functionals is the total mean curvature of the surface. In the case when 𝑑 is a hyperbolic metric, most of the relevant properties of a similar function L d were established in [15]. It is curious to notice that, despite the general outline of our proof is similar to the one in [64], the details of the arguments are quite different.

An interesting direction of further research would be to obtain a common generalization of the present article and [64], which would be about isometric embeddings of CAT(0)-metrics. The existence part of Theorem 1.5 in this setting was obtained in [32] by the first author and D. Slutskiy. The existence is obtained by approximation, and thus does not rely on uniqueness results. Uniqueness results for induced metrics on the boundary of an arbitrary convex set is a hard problem, even in Euclidean space; see discussion in [56].

As we already mentioned, Theorem II ^′ belongs to the stream of works describing a representation of a discrete group into a Lie group, endowed with some kind of a geometric object. In our very brief review, we mention only some results on representations of π 1 ⁢ S (for a longer albeit still non-exhausting survey, we refer to [33]). A model result is the famous Bers Simultaneous Uniformization Theorem [8] stating that, for any two Riemann surfaces, there exists an essentially unique Kleinian surface group producing these Riemann surfaces simultaneously as the quotients of its two domains of discontinuity in C ⁢ P 1 .

With a Kleinian surface group, one also associates its quotient of H 3 , a quasi-Fuchsian hyperbolic 3-manifold. W. Thurston conjectured that it is also uniquely determined by the bending lamination on the boundary of convex core. The realization part of this was established in [12] by F. Bonahon and J.-P. Otal, while the uniqueness part was just recently obtained in [25] by B. Dular and J.-M. Schlenker. In a quasi-Fuchsian hyperbolic 3-manifold, one can consider a totally convex subset, which is homeomorphic to S × [ − 1 , 1 ] . In the case of smooth strictly convex boundaries, such a set (together with the ambient manifold, and hence with the Kleinian group) is uniquely determined by either the first fundamental form of the boundary, or by the third fundamental form as shown by Schlenker in [60]. The respective realization result on the first fundamental form was shown earlier by F. Labourie in [46]. These results can be considered as generalizations of the classical Weyl Problem to the case of non-trivial topology (the Weyl Problem is a smooth counterpart to Theorem 1.1; see e.g. [40] for more information on it). The polyhedral counterparts to the results of Schlenker were recently established by the second author in [54, 55]. It is interesting that, in contrast to the flat “quasi-Fuchsian” case presented here, in the hyperbolic case, cone metrics may admit not so polyhedral, somewhat degenerate embeddings, which make the analysis more intricate. Particularly, a hyperbolic analogue of Theorem I ^′ is still open both in smooth and in polyhedral settings: the main progress is a related result on the realization of hyperbolic metrics without cone singularities, which corresponds to the famous Grafting Problem; see [58], and see [26] for the dual problem. We mention also that the recent article [21] mixes the results of Labourie–Schlenker with the result of Bers.

The last thing we want to remark is that a curious direction of current research deals with similar problems in the context of anti-de Sitter geometry, i.e., Lorentzian geometry of constant curvature −1. The interest to it is spanned by deep connections to Teichmüller theory; see e.g. [51, 18]. Plenty of mentioned problems still remain open in the anti-de Sitter context, though we refer to the work [66] of A. Tamburelli, settling the anti-de Sitter version of the mentioned result of Labourie. Also, very recently, Q. Chen and J.-M. Schlenker resolved in [22] an anti-de Sitter counterpart to Theorem I ^′ in the smooth setting. As for polyhedral surfaces in anti-de Sitter spacetimes, they exhibit degenerations similar to the hyperbolic case (contrary to the flat case), which is additionally aggravated by the lack of some necessary toolbox in the anti-de Sitter setting. Thereby, a thorough understanding of anti-de Sitter polyhedral surfaces is still the matter of further investigations.

Let us emphasize that, in our Minkowski case, convex hulls of finitely many points are polyhedral; see Definition 2.5 and Lemma 2.7. As mentioned above, this is not true in the hyperbolic or anti-de Sitter setting, mainly because the convex hull may meet the boundary of the convex core. In the hyperbolic case, however, if the dual induced metric is polyhedral, then the resulting convex hull of vertices is polyhedral in the sense of the present article [54]. A way to explain this is to consider our theorems from the present article not as results for induced metric on surfaces in Minkowski spacetimes, but results for dual induced metrics on surfaces in co-Minkowski quasi-Fuchsian manifolds, where a convex core enters the picture; see [7]. Such manifolds are modeled on half-pipe geometry, which was popularized by J. Danciger [24], and which is roughly the geometry of spacelike planes of Minkowski space. We, however, chose not to use this point of view in the present article.

2.1 Flat GHMC spacetimes

Recall the following definitions from the introduction.

We consider the coordinates ( x 0 , x 1 , x 2 ) in the Minkowski space R 2 , 1 such that, for two points x , y ∈ R 2 , 1 , their scalar product is expressed as ⟨ x , y ⟩ = − x 0 ⁢ y 0 + x 1 ⁢ y 1 + x 2 ⁢ y 2 .

Let 𝑆 be a surface of hyperbolic type. In all the article, we identify the hyperbolic plane H 2 with the future component of the two-sheeted hyperboloid in R 2 , 1 . For a hyperbolic metric ℎ over 𝑆, a developing map provides a homeomorphism from S ̃ onto H 2 ⊂ R 2 , 1 , and its holonomy gives a representation ρ : π 1 ⁢ S → SO 0 ⁢ ( 2 , 1 ) , where SO 0 ⁢ ( 2 , 1 ) is the connected component of identity in SO ⁢ ( 2 , 1 ) .

The group Isom 0 ⁢ ( R 2 , 1 ) is SO 0 ⁢ ( 2 , 1 ) ⋉ R 2 , 1 , and for any map τ : π 1 ⁢ S → R 2 , 1 satisfying the cocycle condition (twisted by 𝜌)

we obtain a representation ρ τ : π 1 ⁢ S → Isom 0 ⁢ ( R 2 , 1 ) .

A 𝜌-cocycle 𝜏 is a 𝜌-coboundary if there exists v ∈ R 2 , 1 such that τ ⁢ ( γ ) = ρ ⁢ ( γ ) ⁢ ( v ) − v . We then write this cocycle as τ v . If two 𝜌-cocycles τ , τ ′ differ by a 𝜌-coboundary τ v , then the actions of ρ τ and ρ τ ′ are conjugated by the translation by vector 𝑣; see e.g. [35]. We denote by Z 1 ⁢ ( ρ ) the space of 𝜌-cocycles and by B 1 ⁢ ( ρ ) ⊂ Z 1 ⁢ ( ρ ) the subspace of 𝜌-coboundaries. The quotient is denoted by H 1 ⁢ ( ρ ) . Abusing notation, we will denote by the same letter a cocycle 𝜏 and its equivalence class in H 1 ⁢ ( ρ ) , but the context should always clarify this ambiguity.

We denote by R ⁢ ( S ) the space of Fuchsian representations of π 1 ⁢ S into SO 0 ⁢ ( 2 , 1 ) , and we see its quotient by the conjugate action of SO 0 ⁢ ( 2 , 1 ) as the Teichmüller space T ⁢ ( S ) of 𝑆. Abusing again notation, we will denote by the same letter such a representation 𝜌 and its equivalence class in Teichmüller space.

The tangent space of R ⁢ ( S ) is standardly identified with the set of s ⁢ o ⁢ ( 2 , 1 ) -valued cocycles twisted by the adjoint action of 𝜌, and the tangent space of T ⁢ ( S ) at 𝜌 can be considered as the quotient by the respective space of coboundaries; see e.g. [38]. There is a natural identification between s ⁢ o ⁢ ( 2 , 1 ) and R 2 , 1 : for example, a spacelike vector 𝑣 is identified with an infinitesimal hyperbolic translation along the geodesic in the hyperbolic plane obtained by the intersection of the hyperboloid and the plane orthogonal to the vector, with the amount of displacement defined by the length of 𝑣. See e.g. [31] for an explicit description. This identification is SO 0 ⁢ ( 2 , 1 ) -equivariant, where SO 0 ⁢ ( 2 , 1 ) acts on s ⁢ o ⁢ ( 2 , 1 ) via the adjoint representation. Hence this identification induces an isomorphism between T ρ ⁢ T ⁢ ( S ) and H 1 ⁢ ( ρ ) .

For a 𝜌-cocycle 𝜏, there exist two disjoint open convex sets of R 2 , 1 , maximal for the inclusion, respectively future- and past-complete, on which ρ τ ⁢ ( π 1 ⁢ S ) acts freely and properly discontinuously [51, 4, 13, 5]. We denote them by Ω ̃ + ⁢ ( ρ , τ ) and Ω ̃ − ⁢ ( ρ , τ ) respectively. If τ v is a 𝜌-coboundary, then Ω ̃ ± ⁢ ( ρ , τ ) and Ω ̃ ± ⁢ ( ρ , τ + τ v ) differ by a translation.

We will denote by Ω ± ⁢ ( ρ , τ ) the quotients of Ω ̃ ± ⁢ ( ρ , τ ) by ρ τ ⁢ ( π 1 ⁢ S ) and will refer to them as to flat GHMC ( 2 + 1 ) -spacetimes. The construction provides a natural marking isomorphism between π 1 ⁢ Ω ± ⁢ ( ρ , τ ) and π 1 ⁢ S . By a marked isometry, we will mean an isometry respecting the marking isomorphism. When two representations are conjugated by an element of Isom 0 ⁢ ( R 2 , 1 ) , the resulting flat GHMC spacetimes differ by a marked isometry. We are interested in spacetimes up to marked isometry. Hence, sometimes, we define Ω ± ⁢ ( ρ , τ ) for ρ ∈ T ⁢ ( S ) and τ ∈ T ρ ⁢ T ⁢ ( S ) ≅ H 1 ⁢ ( ρ ) , meaning that we take representatives ρ ∈ T ⁢ ( S ) , τ ∈ Z 1 ⁢ ( ρ ) and the quotient of Ω ̃ ± ⁢ ( ρ , τ ) . The space T ⁢ T ⁢ ( S ) can be interpreted then as the space of flat GHMC spacetimes up to marked isometry. Equivalently, it can be considered as the space of flat GHMC Lorentzian structures on S × R up to isometry isotopic to the identity.

2.2 Convex polyhedral surfaces in flat spacetimes

We now study totally convex subsets in Ω + ⁢ ( ρ , τ ) . In this section, a spacetime is fixed, so we will mostly skip the mention of 𝜌 and 𝜏. By a totally convex subset of Ω + , we mean a subset that contains every geodesic segment between every two points. For K ⊂ Ω + , we denote by conv ⁡ ( K ) the convex hull of 𝐾, i.e., the inclusion-minimal totally convex subset containing 𝐾. In our setting, however, convexity is better described by looking inside a choice of Ω ̃ + in R 2 , 1 . A set 𝐶 is totally convex if and only if C ̃ is a convex subset of R 2 , 1 , so conv ⁡ ( K ) is the quotient of the convex hull of K ̃ .

For a convex set C ⊂ R 2 , 1 , we will say that a plane Π is weakly supporting to 𝐶 if 𝐶 belongs to a single closed halfspace bounded by Π. We will say that Π is supporting if additionally Π has a common point with the closure of 𝐶. We will say that a surface in R 2 , 1 is spacelike convex if it is the boundary of a convex set and has only spacelike supporting planes. A spacelike convex surface in Ω + is a surface such that its lift to Ω ̃ + is spacelike convex.

We will say that C ⊂ Ω + is future (complete) if I + ⁢ ( p ) ⊂ C for any point p ∈ C , where I + ⁢ ( p ) is the set of the endpoints of all the future-directed timelike curves from 𝑝. A Cauchy surface of Ω + is a surface (not necessarily smooth) such that every inextendible causal curve intersects it exactly once. Cauchy surfaces in Ω + are all homeomorphic (to 𝑆), hence compact; see e.g. [53]. (Note that, in [53], only timelike curves are used for the definition of Cauchy surfaces, which is weaker than our definition.) The embedding of the universal covering of a convex Cauchy surface to Minkowski space is the graph of a convex 1-Lipschitz function over the whole horizontal plane [13, Lemma 3.11], a fact that we will often use implicitly.

The cosmological time of p ∈ Ω + is the supremum of Lorentzian lengths of the past-directed causal curves emanating from 𝑝 in Ω + . For a lift p ̃ ∈ Ω ̃ + of 𝑝, this supremum is realized by a geodesic segment with an endpoint on the initial singularity of Ω ̃ + , which is basically the spacelike part of ∂ Ω ̃ + . This produces a C 1 -function on Ω + , the cosmological time ct : Ω + → R > 0 . For α > 0 , we will denote by L α the level set of the cosmological time for the value 𝛼. It is a C 1 spacelike convex Cauchy surface. We refer to [13] for details.

We will need the following basic results.

For the next result, we will make use of the spherical model of R 2 , 1 : a geodesic map sending it injectively to an open half-sphere of the standard sphere S 3 . By ∂ ∞ R 2 , 1 , we will denote its boundary in the spherical model, by ∂ ∞ 0 R 2 , 1 ⊂ ∂ ∞ R 2 , 1 , we will denote the subset of endpoints of the future-directed lightlike rays, and by R 2 , 1 ̄ , we will denote the compactification R 2 , 1 ∪ ∂ ∞ R 2 , 1 .

By Lemma 2.3, for a convex polyhedral subset 𝐶, its boundary ∂ C is a Cauchy surface, hence compact, and then it has a finite number of vertices.

We remark that the face decomposition of the boundary of a convex polyhedral subset 𝐶 is what we will call a cellulation. We would like to consider it pulled back to 𝑆. To this purpose, let Σ be a Cauchy surface in Ω + . We will need the following classical result.

This is particularly the case with the boundary of a convex polyhedral set 𝐶. Let 𝑉 be its set of vertices; by the homeomorphism above, abusing the notation, we identify it with a subset of 𝑆.

To see that the faces of 𝐶 are homeomorphic to the 2-disk, we note that the faces of C ̃ are compact (since ∂ C ̃ is spacelike convex) convex Euclidean polygons. We frequently do not distinguish between a cellulation and its equivalence class. After identifying ∂ C with 𝑆, we will refer to the corresponding cellulation as to the face cellulation of 𝐶.

2.3 Spacetimes with marked points

Let V ⊂ S be a finite set of cardinality 𝑛. The aim of the present section is to describe a topology on the following space.

By V ̃ , we denote the full preimage of 𝑉 in S ̃ equipped with the natural π 1 ⁢ S -action. By P ̃ ⁢ ( S , V ) , we denote the set of triples ( ρ , τ , f ̃ ) , where 𝜌 is a Fuchsian representation, τ ∈ Z 1 ⁢ ( ρ ) and f ̃ : V ̃ → Ω ̃ + ⁢ ( ρ , τ ) is an equivariant map. This is an open subset of the space T ⁢ R ⁢ ( S ) × ( R 2 , 1 ) V . The group Isom 0 ⁢ ( R 2 , 1 ) acts on T ⁢ R ⁢ ( S ) freely and properly by conjugation. By adding the usual action on ( R 2 , 1 ) V , we get an action on T ⁢ R ⁢ ( S ) × ( R 2 , 1 ) V , and P ̃ ⁢ ( S , V ) is invariant with respect to this action. Next, the group π 1 ⁢ S acts on T ⁢ R ⁢ ( S ) × ( R 2 , 1 ) V fiberwise, via ρ τ on the fiber { ρ } × { τ } × ( R 2 , 1 ) V . This action is free and properly discontinuous on the set { ρ } × { τ } × ( Ω ̃ + ⁢ ( ρ , τ ) ) V , the intersection of P ̃ ⁢ ( S , V ) with { ρ } × { τ } × ( R 2 , 1 ) V . These two actions commute and the quotient by them of P ̃ ⁢ ( S , V ) is exactly P ⁢ ( S , V ) , which is thus endowed with the structure of an analytic manifold of dimension 12 ⁢ g − 12 + 3 ⁢ n , where 𝐠 is the genus of 𝑆. We remark that we will need the analytic structure only in Section 4.1.

We remark that there is a natural projection map

forgetting the marked points.

2.4 The induced metric map

Let 𝑑 be a flat metric on 𝑆 with cone points. We denote by V ⁢ ( d ) the set of cone points of 𝑑. For a finite V ⊂ S , by M ̄ ⁢ ( S , V ) , denote the set of such metrics with V ⁢ ( d ) ⊂ V . The metrics are considered up to isometries fixing 𝑉 and isotopic to the identity. (Note that the isotopy does not have to fix 𝑉. In the next section, we will have to deal with isotopies fixing 𝑉; in this case, we will say isotopies relative to 𝑉.) We denote by M ⁢ ( S , V ) the subset of M ̄ ⁢ ( S , V ) such that V ⁢ ( d ) = V . Finally, by M ̄ − ⁢ ( S , V ) and M − ⁢ ( S , V ) , we denote the subsets of M ̄ ⁢ ( S , V ) , M ⁢ ( S , V ) respectively consisting of metrics with negative singular curvatures.

A metric 𝑑 of M ̄ ⁢ ( S , V ) can be triangulated so that the set of vertices of the triangulation is exactly 𝑉. Changing the lengths of the edges provides a chart of M ̄ ⁢ ( S , V ) valued in R E , where 𝐸 is the set of edges. If 𝐠 is the genus of 𝑆 and n = | V | , then as we are considering triangulations, | E | = 6 ⁢ g − 6 + 3 ⁢ n . Changes of triangulation provide analytic transition maps, endowing M ̄ ⁢ ( S , V ) with a structure of ( 6 ⁢ g − 6 + 3 ⁢ n ) -dimensional analytic manifold. By looking at affine maps between two triangles, it is easy to see that the topology induced by this analytic structure is the one of Lipschitz convergence of metric spaces. The sets M ⁢ ( S , V ) , M − ⁢ ( S , V ) are open subsets of M ̄ ⁢ ( S , V ) .

Take an element p ∈ P c ⁢ ( S , V ) . The marking allows us to associate its boundary with ( S , V ) up to a homeomorphism fixing 𝑉 and isotopic to identity (Claim 1; the isotopy does not have to fix 𝑉). By Lemma 2.7, the induced metric belongs to M − ⁢ ( S , V ) . This provides a map

called the induced metric map.

By P ̄ c ⁢ ( S , V ) ⊂ P ⁢ ( S , V ) , we denote the subset of injective configurations in convex position, i.e., when 𝑓 is injective and f ⁢ ( V ) ⊂ ∂ conv ⁡ ( f ⁢ ( V ) ) . Note that it differs from the closure of P c ⁢ ( S , V ) in P ⁢ ( S , V ) as the latter also include the configurations when some of the marked points collapsed. The map I ̄ : P ̄ c ⁢ ( S , V ) → M ̄ − ⁢ ( S , V ) is defined in an obvious manner, and its restriction to P c ⁢ ( S , V ) is ℐ. Pick p = ( ρ , τ , f ) ∈ P ̄ c ⁢ ( S , V ) ; denote by V ⁢ ( p ) ⊂ V its set of vertices, i.e., the inclusion-maximal set such that f ⁢ ( V ⁢ ( p ) ) is in a strictly convex position and conv ⁡ ( f ⁢ ( V ⁢ ( p ) ) ) = conv ⁡ ( f ⁢ ( V ) ) . It is well-defined, since it can be defined alternatively as the set of v ∈ V such that f ⁢ ( v ) ∉ conv ⁡ ( f ⁢ ( V \ v ) ) .

If the face cellulation of p ∈ P c ⁢ ( S , V ) is a triangulation, then clearly ℐ is analytic around 𝐩. Now we analyze the regularity of ℐ in a more general situation.

Since there are only finitely many cellulations subdividing 𝒞, from the definition of the topologies, it is evident that I ̄ is continuous. We go further and show the following lemma.

The proof is reminiscent to similar proofs in [54, 55].

3.1 A topological interlude

Our space P c ⁢ ( S , V ) is not simply connected. We are interested to give an interpretation to elements of its universal covering. To this purpose, we need a topological tool, Lemma 3.1, which we will prove in this section.

First, we describe the setting. Let V ⊂ S be a finite subset of size 𝑛 and let S ∗ V be the space of injective maps c : V ↪ S , i.e., the configuration space of 𝑛 points on 𝑆, where the points are marked by 𝑉. The inclusion map allows us to consider 𝑉 itself as a point of S ∗ V . Denote by H 0 ⁢ ( S ) the group of self-homeomorphisms S → S isotopic to the identity, denote by H 0 ⁢ ( S , V ) the subgroup of self-homeomorphism that fix 𝑉 and are isotopic to the identity relative to 𝑉, and we denote by C ⁢ ( S , V ) the space of left cosets H 0 ⁢ ( S ) / H 0 ⁢ ( S , V ) . We consider both homeomorphisms groups with the compact-open topology and the coset space with the induced topology. We have the evaluation map H 0 ⁢ ( S ) → S ∗ V given by the evaluation of ℎ at 𝑉, which is clearly continuous. It factors through H 0 ⁢ ( S , V ) as a map ev V : C ⁢ ( S , V ) → S ∗ V . This map is a local homeomorphism; see e.g. [9, Chapter 4.1] (note that, in [9, Chapter 4.1], the homeomorphisms are not required to be isotopic to the identity; however, the argument does not change). Hence C ⁢ ( S , V ) is a manifold of dimension 2 ⁢ n .

Every homeomorphism h : S → S determines a homeomorphism h V : S ∗ V → S ∗ V by c ↦ h ∘ c , and if ℎ is isotopic to the identity, then so is h V . Every self-homeomorphism of S ∗ V isotopic to the identity admits a well-defined lift to the universal cover via the homotopy-lifting property applied to an isotopy. So every such h V lifts to a homeomorphism h ̃ V : S ∗ V ̃ → S ∗ V ̃ . Fix a lift V ̂ of 𝑉 to the universal covering S ∗ V ̃ . Now we define a new evaluation map