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

François Fillastre and Roman Prosanov

###### Abstract.

We first prove that given a hyperbolic metric $h$ on a closed surface $S$, any
flat metric on $S$ with negative singular curvatures isometrically embeds as a
convex polyhedral Cauchy surface in a unique future-complete flat globally
hyperbolic maximal (2+1)-spacetime whose linear part of the holonomy is given
by $h$. The Gauss map allows to translate this statement to a purely
2-dimensional problem of finding a balanced geodesic cellulation on the
hyperbolic surface, from which the flat metric can be easily recovered.

We show next that given two such flat metrics on the surface, there exists a
unique pair of future- and past-complete flat globally hyperbolic maximal
(2+1)-spacetimes with the same holonomy, in which the flat metrics embed
respectively as convex polyhedral Cauchy surfaces. The proof follows from
convexity properties of the total length of the associated balanced geodesic
cellulations over Teichmüller space.

###### Key words and phrases:

(2+1)-spacetimes; balanced cellulation; Teichmüller space; Gauss map;
Polyhederal surfaces; flat metrics; Codazzi tensors; length functional

###### Contents

  1. 1 Introduction
    1. 1.1 Main statements
    2. 1.2 Sketch of the proof of Theorem I’ and related results
    3. 1.3 Sketch of the proof of Theorem II’ and related results
  2. 2 Background
    1. 2.1 Flat quasi-Fuchsian spacetimes
    2. 2.2 Convex polyhedral surfaces in flat spacetimes
    3. 2.3 Spacetimes with marked points
    4. 2.4 The induced metric map
  3. 3 Proof of Theorem I’
    1. 3.1 A topological interlude
    2. 3.2 The induced metric map on fibers
    3. 3.3 Infinitesimal rigidity
    4. 3.4 Compactness
      1. 3.4.1 Convergence of ambient spacetimes
      2. 3.4.2 Convergence of marked points
  4. 4 Proof of Theorem II’
    1. 4.1 The Teichmüller tangent vector field associated to a flat metric
    2. 4.2 The total length functional
      1. 4.2.1 Definition
      2. 4.2.2 Smoothness
      3. 4.2.3 Properness
    3. 4.3 Pairing of Codazzi tensors and balanced convex cellulations
      1. 4.3.1 Background on Codazzi tensors
      2. 4.3.2 Traceless Codazzi tensor associated to a balanced convex cellulation
      3. 4.3.3 Weil–Petersson metric
    4. 4.4 First derivatives
    5. 4.5 Second derivative along Weil–Petersson geodesic
    6. 4.6 Proof of Theorem II’

##  1\. Introduction

###  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 $\mathbb{R}^{3}$.

Consider a (compact) convex polyhedron $P$ in $\mathbb{R}^{3}$. 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 that point. Therefore, the Gauss map transforms the polyhedron $P$ to a
convex geodesic cellulation of the round sphere
$\mathbb{S}^{2}\subset\mathbb{R}^{3}$. This cellulation, denoted as
$\mathscr{G}$, is inherently dual to the face cellulation of the original
polyhedron $P$. Here is how this duality is manifested:

  * •

The Gauss image of a face of $P$ is a vertex of $\mathscr{G}$.

  * •

For an edge $E$ of $P$ its Gauss image is an edge $e$ in $\mathscr{G}$, and
the length of $e$ in $\mathbb{S}^{2}$ 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
$\mathscr{G}$, equal to the length of the respective edge in the original
polyhedron $P$. Considering a vertex $v$ of $\mathscr{G}$ we see that the
weights $w_{e}$ of the incident edges satisfy the following balance condition:

$\sum_{e\ni v}w_{e}U_{e}=0~{}.$ (1)

Here $U_{e}$ is the unit tangent vector of $\mathbb{S}^{2}$ to the edge $e$ at
the vertex $v$. We will refer to this balanced convex cellulation
$(\mathscr{G},w)$, as “the Gauss image of $P$”, although this is a slight
abuse of terminology. We will call such a geodesic convex cellulation over
$\mathbb{S}^{2}$ with positive weights satisfying the balance condition (1) a
_balanced convex cellulation_ over $\mathbb{S}^{2}$.

In the other direction, from a balanced convex cellulation $(\mathscr{G},w)$
over $\mathbb{S}^{2}$, it is not hard to construct a convex polyhedron $P$
such that its Gauss image is $(\mathscr{G},w)$. Also $P$ is unique up to
Euclidean isometries, see e.g. [Sch77, Sch93].

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
$(\mathscr{G},w)$ of $P$. Indeed, the balance condition (1) means that we may
associate to each vertex of $(\mathscr{G},w)$ 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 $\mathscr{G}$, 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 $(\mathscr{G},w)$, see Figure 1.

Figure 1.  The left hand side of the picture is a part of a balanced convex
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 bounds 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 $2\pi$.

At first glance, it might appear that retrieving the flat metric from a convex
geodesic 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.

######  Theorem 1.1 (Alexandrov Theorem, extrinsic version).

Let $d$ be a flat metric with positive singular curvatures on the sphere
$S^{2}$. Then there exists a unique (up to isometries) convex polyhedron $P$
in Euclidean space such that the induced intrinsic distance on $\partial P$ is
isometric to $d$.

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

######  Theorem 1.2 (Alexandrov Theorem, intrinsic version).

Let $d$ be a flat metric with positive singular curvatures on the sphere
$S^{2}$. Then there exists a unique (up to isometries) balanced geodesic
cellulation $(\mathscr{G},w)$ on $\mathbb{S}^{2}$ whose dual metric is
isometric to $d$.

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

We will say that a surface $S$ is of _hyperbolic type_ if $S$ is a connected
closed oriented surface of genus $>1$, 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.

######  Theorem I (Hyperbolic Alexandrov Theorem, intrinsic version).

Let $d$ be a flat metric with negative singular curvatures on a surface $S$ of
hyperbolic type, and let $h$ be a hyperbolic metric on $S$. Then there exists
a unique balanced geodesic cellulation $(\mathscr{G},w)$ on $(S,h)$ whose dual
metric is isotopic to $d$.

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 gives a flat metric with a unique cone
singularity of curvature $4\pi$.

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 $\mathbb{R}^{2,1}$ so that the
embedding is equivariant with respect to a holonomy map
$\pi_{1}S\rightarrow{\rm Isom}_{0}(\mathbb{R}^{2,1})$. As the Gauss image of
$P$ must be isometric to $(S,h)$, the linear part $\rho$ of the holonomy is
prescribed by $h$. Here we are identifying the hyperbolic plane with its
hyperboloid model in Minkowski space $\mathbb{R}^{2,1}$, 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 $\gamma\in\pi_{1}S$ we define
$\tau(\gamma)\in\mathbb{R}^{2,1}$ and set
$\rho_{\tau}(\gamma):=\rho(\gamma)+\tau(\gamma)$. The image
$\rho_{\tau}(\pi_{1}S)$ is a subgroup of isometries of Minkowski space if and
only if $\tau$ satisfies a cocycle condition. From a work of G. Mess [Mes07,
ABB+07] it is known that there exists a maximal open set in Minkowski space on
which $\rho_{\tau}(\pi_{1}S)$ acts freely and properly. This maximal open
subset 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. Let us denote (the marked
isometry classes of) these quotients by $\Omega^{+}(\rho,\tau)$ and
$\Omega^{-}(\rho,\tau)$ respectively. For the details of this construction we
refer the reader to [Mes07, ABB+07, Bon05b, Bar05] (the last two consider
higher-dimensional generalizations).

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

######  Theorem I’ (Hyperbolic Alexandrov Theorem, extrinsic version).

Let $d$ be a flat metric with negative singular curvatures on a surface $S$ of
hyperbolic type, and let $\rho$ be the holonomy of a hyperbolic metric on $S$.
Then there exists a unique convex Cauchy polyhedron $P$ in a unique (up to
marked isometry) flat maximal (2+1)-spacetime $\Omega^{+}(\rho,\tau)$ such
that the induced intrinsic metric on $\partial P$ is marked isometric to $d$.

By construction (2+1)-spacetimes $\Omega^{\pm}(\rho,\tau)$ are marked in the
sense that for every Cauchy surface $\Sigma\subset\Omega^{\pm}(\rho,\tau)$
there exists a natural homeomorphism $S\rightarrow\Sigma$ determined up to
isotopy. The marked isometry between $(S,d)$ and $\partial P$ 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 convex
cellulation it is easy to construct the spacetime and the polyhedral surface,
in a way similar to the Euclidean case, see [FV16, FS20a].

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

######  Theorem II’.

Let $d_{+}$ and $d_{-}$ be flat metrics with negative singular curvatures on a
surface $S$ of hyperbolic type. Then there exists a unique (up to marked
isometries) pair of flat maximal (2+1)-spacetimes $\Omega^{+}(\rho,\tau)$ and
$\Omega^{-}(\rho,\tau)$ containing unique convex Cauchy polyhedra with induced
intrinsic metrics on the boundary marked isometric to $d_{+}$ and $d_{-}$
respectively.

The meaning of Theorem II’ is that to a discrete and faithful representation
$\rho_{\tau}:\pi_{1}S\rightarrow{\rm Isom}_{0}(\mathbb{R}^{2,1})$ we attach a
geometric object, a pair of convex polyhedral surfaces embedded into the
respective spacetimes $\Omega^{\pm}(\rho,\tau)$, 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 $\tau$ introduced above have values in Minkowski space. But there
is a natural identification $\Lambda$ between Minkowski space and the Lie
algebra of $SO(2,1)$. If the Teichmüller space $\mathcal{T}(S)$ of $S$ is seen
as the space of the equivalence classes of discrete and faithful
representations of $\pi_{1}S$ into $SO_{0}(2,1)$, then, applying the
identification $\Lambda$, 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
$\mathcal{T}(S)$, 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
$\rho_{\tau}$-invariant then $-P$ is $\rho_{-\tau}$-invariant, and of course
the induced intrinsic metrics over $\partial P$ and $\partial(-P)$ are related
by an isometry.

######  Theorem II.

Let $d_{1}$ and $d_{2}$ be two flat metrics over $S$ with negative singular
curvatures. Then $X_{d_{1}}$ and $-X_{d_{2}}$ intersect in the tangent bundle
of Teichmüller space at a unique point, and the intersection is transverse.

Theorem II’ is proved by associating to the metric $d$ the _total length
function_ $\operatorname{L}_{d}$ 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 $\operatorname{L}_{d}$ 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.D. 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 $\mathbb{R}^{3}$) with $n$ vertices are
endowed with suitable topologies that make them manifolds of the same finite
dimension. A natural map $\mathcal{I}$ from the former to the latter is given
by considering the induced intrinsic metric on the boundary of a polyhedron.
Theorem 1.1 says that $\mathcal{I}$ is a bijection, and furthermore,
Alexandrov proves that it is a homeomorphism [Ale05]. A key point is to prove
that $\mathcal{I}$ 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 $\mathcal{I}$ 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
$\mathbb{S}^{2}$ without changing the face angles at first order [Ale05,
Theorem 10.3.2].

In the hyperbolic case the analog of the first order Cauchy theorem is the
following statement.

######  Theorem 1.3 (Iskhakov, [Isk00]).

It is not possible to deform at first order a convex geodesic cellulation of a
hyperbolic surface without changing the face angles at first order.

This theorem is proven in [Isk00]. It seems to us, however, that the proof,
unfortunately, contains a mistake, which, however, is fixable for convex
cellulations. (I. Iskhakov in [Isk00] 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 $\rho:\pi_{1}S\to SO_{0}(2,1)$ of a hyperbolic metric $h$ over $S$,
and let $V\subset S$ be a finite set of cardinality $n$. By
$\mathcal{P}_{c}(\rho,V)$ we denote the space of configurations of points in
$\Omega^{+}(\rho,\tau)$ for some $\tau$ 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 2.10 and Lemma 3.8
that $\mathcal{P}_{c}(\rho,V)$ is a connected analytic manifold of dimension
$6\mathbf{g}-6+3n$, where $\mathbf{g}$ is the genus of $S$. Heuristically,
$3n$ corresponds to the position of the points, and $6\mathbf{g}-6$ to the
choice of the cocycle, since the space of equivalence classes of cocycles can
be identified with the tangent space at $\rho_{0}$ of the Teichmüller space of
$S$.

On the other hand, let $\mathcal{M}_{-}(S,V)$ 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 a (isotopy class of) metric
is determined by a point in $\mathcal{T}(S)$, the position of the elements of
$V$ and their cone angles. In turn, $\mathcal{M}_{-}(S,V)$ is a connected
analytic manifold of dimension $6\mathbf{g}-6+3n$, see Section 2.4 and Lemma
3.9.

We define the map $\mathcal{I}_{\rho}$ from $\mathcal{P}_{c}(\rho,V)$ to
$\mathcal{M}_{-}(S,V)$ that associates to any configuration of points in
$\mathcal{P}_{c}(\rho,V)$ 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
2.13), and Theorem 1.3 implies that $d\mathcal{I}_{\rho}$ is non-degenerate.
By the Local Inverse Theorem, $\mathcal{I}_{\rho}$ is a local diffeomorphism.

Section 3.4 establishes that $\mathcal{I}_{\rho}$ is proper. The compactness
result for the ambient spacetimes is inspired by the techniques of F. Bonsante
in [Bon05b], 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 [BBZ11].

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

######  Theorem 1.4.

$\mathcal{I}_{\rho}:\mathcal{P}_{c}(\rho,V)\to\mathcal{\mathcal{M}}_{-}(S,V)$
is a $C^{1}$-diffeomorphism.

There is the following cousin of Theorem I’. We call $\Omega^{+}(\rho,\tau)$
_Fuchsian_ is $\tau$ is trivial — by contrast, we will say _quasi-Fuchsian_ in
the general case, see also [BF20] for the choice of this terminology.

######  Theorem 1.5.

Let $d$ be a flat metric with negative singular curvatures on a surface $S$ of
hyperbolic type. Then there exists a unique convex Cauchy polyhedron $P$ in a
unique (up to marked isometries) future-complete Fuchsian flat spacetime such
that the induced intrinsic metric on $\partial P$ is marked isometric to $d$.

Theorem 1.5 was proved in [Fil11] 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 2.13 of the present article. Instead of Iskhakov
Theorem 1.3 the proof of Theorem 1.5 is based on a first-order rigidity result
proved in [Sch07, Theorem 6.2]. Later, a variational proof of Theorem 1.5 was
done in [Bru20], using the _Hilbert–Einstein functional_. This variational
approach was first used in the Euclidean case (see the introduction of
[BI08]), and in many other situations. However, we do not know if the
variational approach can be implemented in our quasi-Fuchsian case. Note that
[Fil11] 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
[Sch02, Fil07] —which were thought as a polyhedral analogue of [LS00].

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 [TV95], we refer to [Smi22] for more details. An interesting point
is that, even if the proof of [TV95] 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 convex 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 geodesic
cellulation, which can be considered as a discrete harmonic map. A proof is
done for a different balance condition by minimizing an energy functional
[CdV91, KT21]. The balance condition appearing there for a vertex $v$ is given
by

$\sum_{e\ni v}w_{e}U_{e}\ell_{e}=0,$

where $\ell_{e}$ 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. It is known that this cannot happen in the case of a
triangulation [CdV91, LWZ21]. See [GLM20] for another discretization of the
energy functional.

We don’t 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 quasi-Fuchsian
interpretation as it is the Gauss image of the initial singularity of a flat
GHMC (2+1)-spacetime, which was first noted by Mess [Mes07].

Theorem I’ can be considered as a part of a general paradigm that geometry of
spacelike future-convex equivariant convex sets in $\mathbb{R}^{n,1}$ might be
seen as a Lorentizan analogue of theory of Euclidean convex bodies, see e.g.
[FV16, BF17]. 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 convex cellulation
$(\mathscr{G},w)$ over a hyperbolic surface $(S,h)$ an element of
$T_{\rho}\mathcal{T}(S)$, where $\rho$ is the class of the holonomy of $h$.
Indeed, one considers the cohomology class associated to the flat quasi-
Fuchsian spacetime into which the flat metric dual to $(\mathscr{G},w)$
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_ $\operatorname{L}_{d}$ over
$\mathcal{T}(S)$ 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
$\operatorname{L}_{d}$ is $-X_{d}$ (Proposition 4.21). Note that this
immediately implies a _reciprocity formula_ as in [Wol82, Theorem 2.11].

The main point will be Lemma 4.24, which says that the (Weil–Petersson)
Hessian of $\operatorname{L}_{d}$ is positive definite. The argument is
inspired by [Wol12]. 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 [Wol87]).

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
[Bon05a], where an analogous result is shown with flat metrics replaced by
measured laminations.

In the realm of flat quasi-Fuchsian spacetimes this connection was further
extended in [BS12], where it was proved that flat quasi-Fuchsian 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 [Yam99] by S. Yamada. For a
weighted graph filling the surface the fact that the energy has a minimum was
proved in [KT21] by T. Kajigaya and R. Tanaka. It seems plausible that the
results in [KT21] might imply an analogue of Theorem II’ with weighted filling
graphs instead of flat metrics. However, as we explained in the end of the
preceding section, it is not clear what could be an analogue of Theorem II in
that case, because the resulting weights satisfying the balance condition at
each hyperbolic metric do not encode the dihedral angles of the resulting
polyhedral surface. We do not know if it is possible to have a Simultaneous
Uniformization Theorem prescribing a topological cellulation and dihedral
angles.

The smooth analogue of Theorem II was done in [Smi22] by G. Smith (see [FS20b]
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 $\operatorname{L}_{d}$ were established in
[BMS15]. It is curious to notice that despite the general outline of our proof
is similar to the one in [Smi22], 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 [Smi22], which would be about
isometric embeddings of CAT(0)-metrics. The existence part of Theorem 1.5 in
this setting was obtained in [FS19] by the first author and D. Slutskiy. The
existence is obtained by approximation, and thus do 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 [Pro23].

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 $\pi_{1}S$ (for a longer albeit still non-exhausting
survey we refer to [FS18]). A model result is the famous Bers Simultaneous
Uniformization Theorem [Ber60] stating that for any two Riemann surfaces there
exists a essentially unique Kleinian surface group producing these Riemann
surfaces simultaneously as the quotients of its two domains of discontinuity
in $\mathbb{CP}^{1}$.

With a Kleinian surface group one also associates its quotient of
$\mathbb{H}^{3}$, a quasi-Fuchsian hyperbolic 3-manifold. In the latter one
can consider a totally convex subset, which is homeomorphic to
$S\times[-1,1]$. 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 J.-M. Schlenker in [Sch06]. The
respective realization result on the first fundamental form was shown earlier
by F. Labourie in [Lab92]. 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. [HH06] for more
information on the Weyl Problem). The polyhedral counterparts to the results
of Schlenker were recently established by the second author in [Pro22a,
Pro22b]. 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 [SW02], and see [DW08]
for the dual problem. We mention also that the recent article [CS22] 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. [Mes07,
BS20]. Plenty of mentioned problems still remain open in the anti-de Sitter
context, though we refer to the work [Tam18] of A. Tamburelli, settling the
anti-de Sitter version of the mentioned result of Labourie. As for polyhedral
surfaces in anti-de Sitter spacetimes, they exhibit similar degenerations as
in 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 2.6. 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
[Pro22a]. 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 [BF20]. Such
manifolds are modeled on _half-pipe geometry_ , which was popularized by J.
Danciger [Dan13], and which is roughly the geometry of space-like planes of
Minkowski space. We, however, chose not to use this point of view in the
present article.

### Acknowledgments

The first author was supported by the ANR Géométrie et Analyse dans le cadre
Pseudo-Riemannien – GAPR. The second author was funded by the Austrian Science
Fund (FWF) ESPRIT grant ESP-12-N. This article was initiated during a visit of
the second author to the University of Montpellier. It was finished during
authors’ stay at the Erwin Schrödinger International Institute for Mathematics
and Physics, Vienna, at the thematic program Geometry beyond Riemann:
Curvature and Rigidity. They thank the institutions for their hospitality.

The first author wishes to express his gratitude to Andrea Seppi for fruitful
discussions related to the content of the present article.

##  2\. Background

###  2.1. Flat quasi-Fuchsian spacetimes

Recall the following definitions from the introduction.

######  Definition 2.1.

A surface $S$ is of _hyperbolic type_ if $S$ is a connected closed oriented
surface of genus $\mathbf{g}>1$, and $S$ is a _hyperbolic surface_ if it is of
hyperbolic type and endowed with a hyperbolic metric $h$.

We consider the coordinates $(x_{0},x_{1},x_{2})$ in the Minkowski space
$\mathbb{R}^{2,1}$ such that for two points $x,y\in\mathbb{R}^{2,1}$ their
scalar product is expressed as

$\langle x,y\rangle=-x_{0}y_{0}+x_{1}y_{1}+x_{2}y_{2}~{}.$

Let $S$ be a surface of hyperbolic type. In all the article we identify the
hyperbolic plane $\mathbb{H}^{2}$ with the future component of the two-sheeted
hyperboloid in $\mathbb{R}^{2,1}$. For a hyperbolic metric $h$ over $S$ a
developing map provides a homeomorphism from $\tilde{S}$ onto
$\mathbb{H}^{2}\subset\mathbb{R}^{2,1}$, and its holonomy gives a
representation $\rho:\pi_{1}S\to SO_{0}(2,1)$, where $SO_{0}(2,1)$ is the
connected component of identity in $SO(2,1)$.

The group $\mathrm{Isom}_{0}(\mathbb{R}^{2,1})$ is
$SO_{0}(2,1)\ltimes\mathbb{R}^{2,1}$, and for any map

$\tau:\pi_{1}S\to\mathbb{R}^{2,1}$

satisfying the _cocycle_ condition (twisted by $\rho$):

$\tau(\gamma\mu)=\rho(\gamma)(\tau(\mu))+\tau(\gamma)~{},$ (2)

we obtain a representation

$\rho_{\tau}:\pi_{1}S\to\mathrm{Isom}_{0}(\mathbb{R}^{2,1})~{}.$

A $\rho$-cocycle $\tau$ is a _$\rho$ -coboundary_ if there exists
$v\in\mathbb{R}^{2,1}$ such that

$\tau(\gamma)=\rho(\gamma)(v)-v~{}.$

We then write this cocycle as $\tau_{v}$. If two $\rho$-cocycles
$\tau,\tau^{\prime}$ differ by a $\rho$-coboundary $\tau_{v}$, then the
actions of $\rho_{\tau}$ and $\rho_{\tau^{\prime}}$ are conjugated by the
translation by vector $v$, see e.g. [FV16]. By $\mathcal{Z}^{1}(\rho)$ we
denote the space of $\rho$-cocycles and by
$\mathcal{B}^{1}(\rho)\subset\mathcal{Z}^{1}(\rho)$ the subspace of
$\rho$-coboundaries. The quotient is denoted by $\mathcal{H}^{1}(\rho)$.
Abusing notation, we will denote by the same letter a cocycle $\tau$ and its
equivalence class in $\mathcal{H}^{1}(\rho)$, but the context should always
clarify this ambiguity.

By $\mathcal{R}(S)$ we denote the space of Fuchsian representations of
$\pi_{1}S$ into $SO_{0}(2,1)$, and we see its quotient by the conjugate action
of $SO_{0}(2,1)$ as the Teichmüller space $\mathcal{T}(S)$ of $S$. Abusing
again notation, we will denote by the same letter such a representation $\rho$
and its equivalence class in Teichmüller space.

The tangent space of $\mathcal{R}(S)$ is standardly identified with the set of
$\mathfrak{so}(2,1)$-valued cocycles twisted by the adjoint action of $\rho$,
and the tangent space of $\mathcal{T}(S)$ at $\rho$ can be considered as the
quotient by the respective space of coboundaries, see e.g. [Gol84]. There is a
natural identification between $\mathfrak{so}(2,1)$ and $\mathbb{R}^{2,1}$:
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. [FS20a]
for an explicit description. This identification induces an isomorphism
between $T_{\rho}\mathcal{T}(S)$ and $\mathcal{H}^{1}(\rho)$.

For a $\rho$-cocycle $\tau$ there exist two disjoint open convex sets of
$\mathbb{R}^{2,1}$, maximal for the inclusion, respectively future- and past-
complete, on which $\rho_{\tau}(\pi_{1}S)$ acts freely and properly
discontinuously [Mes07, ABB+07, Bon05b, BF20]. We denote them
$\tilde{\Omega}^{+}(\rho,\tau)$ and $\tilde{\Omega}^{-}(\rho,\tau)$
respectively. If $\tau_{v}$ is a $\rho$-coboundary, then
$\tilde{\Omega}^{\pm}(\rho,\tau)$ and
$\tilde{\Omega}^{\pm}(\rho,\tau+\tau_{v})$ differ by a translation.

We will denote by $\Omega^{\pm}(\rho,\tau)$ the quotients of
$\tilde{\Omega}^{\pm}(\rho,\tau)$ by $\rho_{\tau}(\pi_{1}S)$ and will refer to
them as to _flat quasi-Fuchsian (2+1)-spacetimes_. The construction provides a
natural marking isomorphism between $\pi_{1}\Omega^{\pm}(\rho,\tau)$ and
$\pi_{1}S$. By a _marked isometry_ we will mean an isometry respecting this
isomorphism. E.g., if another representative of $\rho$ is chosen in the
quotient of $\mathcal{R}(S)$, then the resulting flat quasi-Fuchsian
spacetimes will differ by a marked isometry. We are interested in spacetimes
up to marked isometry. For our usual confusion of classes and elements, in the
notation $\Omega^{\pm}(\rho,\tau)$, $\rho$ and $\tau$ are meant as equivalence
classes, while they are meant as elements of those classes for
$\tilde{\Omega}^{\pm}(\rho,\tau)$.

###  2.2. Convex polyhedral surfaces in flat spacetimes

We now study (totally) convex subsets in $\Omega^{+}(\rho,\tau)$. In this
section a spacetime is fixed, so we will mostly skip the mention of $\rho$ and
$\tau$. By a _totally convex subset_ of $\Omega^{+}$ we mean a subset that
contains every geodesic segment between every two points. For
$K\subset\Omega^{+}$ by ${\rm conv}(K)$ we denote 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
$\tilde{\Omega}^{+}$ in $\mathbb{R}^{2,1}$. A set $C$ is totally convex if and
only if $\tilde{C}$ is a convex subset of $\mathbb{R}^{2,1}$, so ${\rm
conv}(K)$ is the quotient of the convex hull of $\tilde{K}$.

For a convex set $C\subset\mathbb{R}^{2,1}$ we will say that a plane $\Pi$ is
_weakly supporting_ to $C$ if $C$ belongs to a single closed halfspace bounded
by $\Pi$. We will say that $\Pi$ is _supporting_ if additionally $\Pi$ has a
common point with the closure of $C$. We will say that a surface in
$\mathbb{R}^{2,1}$ is _spacelike convex_ if it is the boundary of a convex set
and has only spacelike supporting planes. A spacelike convex surface in
$\Omega^{+}$ is a surface such that its lift to $\tilde{\Omega}^{+}$ is
spacelike convex.

We will say that $C\subset\Omega^{+}$ is _future (complete)_ if for any point
$p\in C$ we have $I^{+}(p)\subset C$, where $I^{+}(p)$ is the set of the
endpoints of all the future-directed timelike curves from $p$. A _Cauchy
surface_ of $\Omega^{+}$ is a surface (not necessarily smooth) such that every
inextendible causal curve intersects it exactly once. Cauchy surfaces in
$\Omega^{+}$ are all homeomorphic (to $S$), hence compact, see e.g. [O’N83].
(Note that in [O’N83] 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 $1$-Lipchitz function over the whole horizontal plane [Bon05b,
Lemma 3.11], a fact that we will often use implicitly.

The _cosmological time_ of $p\in\Omega^{+}$ is the supremum of Lorentzian
lengths of the past-directed causal curves emanating from $p$ in $\Omega^{+}$.
For a lift $\tilde{p}\in\tilde{\Omega}^{+}$ of $p$ this supremum is realized
by a geodesic segment with an endpoint on the initial singularity of
$\tilde{\Omega}^{+}$, which is basically the spacelike part of
$\partial\tilde{\Omega}^{+}$. This produces a $C^{1}$-function on
$\Omega^{+}$, the _cosmological time_
$\textsc{ct}:\Omega^{+}\rightarrow\mathbb{R}_{>0}$. For $\alpha>0$ we will
denote by $L_{\alpha}$ the level set of the cosmological time for the value
$\alpha$. It is a $C^{1}$ spacelike convex Cauchy surface. We refer to
[Bon05b] for details.

We will need the following basic results.

######  Lemma 2.2 ([Bon05b], Proof of Theorem 5.1, Steps 2 and 3).

Let $p\in\Omega^{+}$ and $C:={\rm conv}(p)$. Then $C$ is a future totally
convex set with compact boundary.

######  Lemma 2.3.

Let $C\subset\Omega^{+}$ be a totally convex set such that the cosmological
time is bounded away from zero on $C$. Then it is future and $\partial C$ is a
spacelike convex Cauchy surface.

###### Proof.

From Lemma 2.2 for every $p\in C$ we have $C\supseteq{\rm conv}(p)\supset
I^{+}(p)$. Hence, $C$ is a future set. Because of this, its boundary $\partial
C$ is a closed achronal topological surface [O’N83, Cor. 14.27]. Lemma 2.2
also says that $\partial{\rm conv}(p)$ is compact, thereby the cosmological
time is bounded from above on it. Thus $C$ contains all level surfaces of
sufficiently large cosmological time in the interior. Since the cosmological
time is also bounded from below on $C$, the complement to the closure of $C$
also contains a level surface of cosmological time. Hence, every inextendible
causal curve intersects $\partial C$. The total convexity of $C$ implies that
the intersection is unique. Indeed, suppose the converse that $p,q\in\partial
C$ are in causal relation. Then the line through them meets ${\rm int}(C)$ in
a precompact set, and thus does not meet $L_{\alpha}$ for sufficiently large
$\alpha$, which is a contradiction as all $L_{\alpha}$ are Cauchy surfaces.
Hence, $\partial C$ is a Cauchy surface. Thereby, $\tilde{C}$ has only
spacelike or lightlike supporting planes. Suppose that there is a lightlike
supporting plane $\Pi$ at a point $p\in\partial\tilde{C}$. Then as
$I^{+}(p)\subset\tilde{C}$, then the boundary of $C$ must contain the
lightlike ray $\Pi\cap(\partial I^{+}(p))$, which contradicts the definition
of Cauchy surface. ∎

For the next result we will make use of the spherical model of
$\mathbb{R}^{2,1}$: a geodesic map sending it injectively to an open half-
sphere of the standard sphere $\mathbb{S}^{3}$. By
$\partial_{\infty}\mathbb{R}^{2,1}$ we will denote its boundary in the
spherical model, by
$\partial^{0}_{\infty}\mathbb{R}^{2,1}\subset\partial_{\infty}\mathbb{R}^{2,1}$
we will denote the subset of endpoints of the future-directed lightlike rays,
and by $\overline{\mathbb{R}^{2,1}}$ we will denote the compactification
$\mathbb{R}^{2,1}\cup\partial_{\infty}\mathbb{R}^{2,1}$.

######  Lemma 2.4.

Let $\tau\in\mathcal{Z}^{1}(\rho)$, $p\in\tilde{\Omega}^{+}$, and $P$ be the
$\rho_{\tau}$-orbit of $p$. Then the limit set of $P$ in
$\overline{\mathbb{R}^{2,1}}$ is $\partial_{\infty}^{0}\mathbb{R}^{2,1}$.

###### Proof.

Let $L\subset\tilde{\Omega}^{+}$ be the level surface of the cosmological time
that contains $P$. For $\gamma\in\pi_{1}S$ denote the supporting plane to $L$
at $\rho_{\tau}(\gamma)p$ by $\Pi_{\gamma}$. Denote by
$\mathbb{B}^{2}\subset\mathbb{R}^{2,1}$ the unit disk in the horizontal plane
translated by the future unit vertical vector. We consider now
$\mathbb{B}^{2}$ as a model for $\mathbb{H}^{2}$ and use it as the domain for
the support function $s:\mathbb{B}^{2}\rightarrow\mathbb{R}$ of $L$. More
precisely, $s(x)$ is the oriented Lorentzian distance from the origin to the
support plane of $L$ with inward normal $(x,1)$, see e.g. [BF17] for details.
By [BF17, Proposition 3.7] there exists a continuous function
$r_{\rho,\tau}:\partial\mathbb{B}^{2}\rightarrow\mathbb{R}$ such that the
support function of every $\rho_{\tau}$-invariant future convex set admits a
continuous extension to $\partial\mathbb{B}^{2}$ by $r_{\rho,\tau}$. Denote
$\mathbb{B}^{2}\cup\partial\mathbb{B}^{2}$ by $\overline{\mathbb{B}^{2}}$, and
consider now this as the domain of the continuous extension of $s$, which we
still denote by $s$.

Let $\partial_{\infty}^{c}\mathbb{R}^{2,1}$ and
$\partial_{\infty}^{t}\mathbb{R}^{2,1}$ be the subsets of
$\partial_{\infty}\mathbb{R}^{2,1}$ corresponding to the future causal and
future timelike rays respectively. Let
$\iota:\overline{\mathbb{B}^{2}}\rightarrow\partial_{\infty}^{c}\mathbb{R}^{2,1}$
be the homeomorphism induced by the projection from the origin. Abusing the
notation, we will denote $s\circ\iota^{-1}$ still by $s$, i.e., now we
consider $s$ defined over $\partial_{\infty}^{c}\mathbb{R}^{2,1}$. The group
$\rho(\pi_{1}S)$ acts naturally on $\partial_{\infty}^{c}\mathbb{R}^{2,1}$.
Moreover, since for any Fuchsian group isomorphic to the fundamental group of
a closed surface, the limit set of its action on $\mathbb{H}^{2}$ is the whole
$\partial_{\infty}\mathbb{H}^{2}$, the action of $\rho$ on
$\partial_{\infty}^{t}\mathbb{R}^{2,1}$ associates the Gromov boundary
$\partial_{\infty}\pi_{1}S$ with $\partial_{\infty}^{0}\mathbb{R}^{2,1}$.

Let $q\in\partial_{\infty}^{0}\mathbb{R}^{2,1}$. There exists a sequence
$\gamma_{i}$ converging to $q$ as to an element of
$\partial_{\infty}\pi_{1}S$. Then for any
$x\in\partial_{\infty}^{t}\mathbb{R}^{2,1}$ the sequence $\rho(\gamma_{i})x$
converges to $q$. Hence, the sequence of planes $\Pi_{\gamma_{i}}$ converges
to the weakly supporting lightlike plane $\Pi$ to $L$ determined by $s(q)$.
Note that since $L$ is a closed subset of $\mathbb{R}^{2,1}$ and
$\rho_{\tau}(\pi_{1}S)$ acts properly discontinuously on it, the
$\rho_{\tau}$-orbit of $p$ cannot have limit points in $\mathbb{R}^{2,1}$.
Suppose that a subsequence of $\rho_{\tau}(\gamma_{i})p$ converges to
$q^{\prime}\in\partial_{\infty}\mathbb{R}^{2,1}$, $q^{\prime}\neq q$. Then
$q^{\prime}\in\partial_{\infty}\Pi$, where $\partial_{\infty}\Pi$ is the limit
set of $\Pi$ in $\partial_{\infty}{\mathbb{R}^{2,1}}$. Then $q^{\prime}$
either corresponds to a spacelike ray, or to a past lightlike ray. Note,
however, that the mentioned result of [BF17] means that an affine coordinate
system in $\mathbb{R}^{2,1}$ can be chosen so that $L$ belongs to the future
cone of the origin. This clearly means that $q^{\prime}$ must correspond to a
future causal ray. Hence, $q^{\prime}=q$, i.e. $\rho_{\tau}(\gamma_{i})p$
converge to $q$ in $\overline{\mathbb{R}^{2,1}}$.

Suppose that $\gamma_{i}\in\pi_{1}S$ is a sequence such that
$\rho_{\tau}(\gamma_{i})p$ has a limit point
$q\in\overline{\mathbb{R}^{2,1}}$. Up to a subsequence, there exists a limit
point $q^{\prime}\in\partial_{\infty}^{0}\mathbb{R}^{2,1}$ for $\gamma_{i}$,
where $\partial_{\infty}^{0}\mathbb{R}^{2,1}$ is perceived as
$\partial_{\infty}\pi_{1}S$. By the previous argument
$\rho_{\tau}(\gamma_{i})p$ must converge to $q^{\prime}$, hence
$q=q^{\prime}$. This finishes the proof. ∎

######  Definition 2.5.

We call a _future polyhedral cone_ in $\mathbb{R}^{2,1}$ the intersection of
the closed future half-spaces of a finite number of spacelike planes meeting
at a common point.

We say that a subset $C\subset\Omega^{+}$ is _convex polyhedral_ if it is
closed, totally convex, its boundary is locally modeled on future polyhedral
cones and the cosmological time is bounded away from zero on it.

We say that a point $p\in\partial C$ is a _face-point_ if it has a local model
on a future polyhedral cone with one plane, it is an _edge-point_ if it has a
local model on a future polyhedral cone with two planes, and a _vertex_
otherwise. A _face_ of $C$ is the closure of a connected component of face-
points and an _edge_ is the closure of a connected component of edge-points.

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

######  Lemma 2.6.

Every convex polyhedral subset $C\subset\Omega^{+}$ is the convex hull of its
vertices. Conversely, if $V\subset\Omega^{+}$ is a finite set, then ${\rm
conv}(V)$ is a convex polyhedral subset.

###### Proof.

Let $V$ be the set of vertices of $C$. Consider the closure $\overline{C}$ of
$\tilde{C}\subset\tilde{\Omega}^{+}\subset\mathbb{R}^{2,1}$ in
$\overline{\mathbb{R}^{2,1}}$. By Lemma 2.4,
$\overline{C}\cap\partial_{\infty}\mathbb{R}^{2,1}$ contains the set
$\partial^{c}_{\infty}\mathbb{R}^{2,1}$ of the future causal directions. On
the other hand, since we can choose the affine coordinate system so that
$\Omega^{+}$ is contained in the future cone of the origin,
$\overline{C}\cap\partial_{\infty}\mathbb{R}^{2,1}$ is contained in
$\partial^{c}_{\infty}\mathbb{R}^{2,1}$. Thus, these sets coincide. It follows
that $\overline{C}={\rm
conv}(\tilde{V}\cup\partial^{0}_{\infty}\mathbb{R}^{2,1})$. By the
Caratheodory Theorem every point of $\partial\tilde{C}$ is a convex
combination of at most three points from
$\tilde{V}\cup\partial^{0}_{\infty}\mathbb{R}^{2,1}$ (and, clearly, at least
one from $\tilde{V}$). If there is a point of
$\partial^{0}_{\infty}\mathbb{R}^{2,1}$ participating in the convex
combination, then there exists a lightlike ray in $\partial\tilde{C}$. This
contradicts to that $\partial C$ is a spacelike surface. It is easy to deduce
then that the interior points of $\tilde{C}$ are also convex combinations of
$\tilde{V}$ and then obtain the first claim.

For the second claim, denote ${\rm conv}(V)$ by $C$. Note that because the
level sets of the cosmological time are convex, the infimum of the
cosmological time on $C$ is achieved at a point of $V$. Hence, due to Lemma
2.3, the boundary $\partial C$ is a spacelike convex Cauchy surface.

Consider the full preimage $\tilde{V}$ in $\tilde{\Omega}^{+}$. Let
$\overline{C}$ be the closure of $\tilde{C}$ in $\overline{\mathbb{R}^{2,1}}$,
so $\overline{C}=\overline{{\rm conv}}(\tilde{V})$. By Lemma 2.4,
$\overline{C}={\rm conv}(\tilde{V}\cup\partial^{0}_{\infty}\mathbb{R}^{2,1})$.
By the same argument as above, $\partial\tilde{C}\subset{\rm
conv}(\tilde{V})$, i.e. $\tilde{C}$ is closed in $\mathbb{R}^{2,1}$, and $C$
is closed in $\Omega^{+}$. Note that there are finitely many edges of
$\partial\tilde{C}$ emanating from a point of $\tilde{V}$. Indeed, otherwise
once again there is a lightlike ray belonging to $\partial\tilde{C}$. It
follows that $\tilde{C}$ is the intersection of a locally finite collection of
spacelike planes, i.e., $C$ is convex polyhedral. ∎

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 $\Sigma$ be a Cauchy surface in
$\Omega^{+}$. We will need the following classical result.

######  Claim 1.

The marking isomorphism of the fundamental groups of $\pi_{1}S$ and
$\pi_{1}\Omega^{+}$ induces a homeomorphism $S\rightarrow\Sigma$ determined up
to isotopy.

###### Proof.

It is a general topological fact that the isomorphism of fundamental group
gives a homotopy equivalence between the surfaces [Hat02, Theorem 1.B.8]. In
our hyperbolic type case, any homotopy equivalence is homotopic to a
homeomorphism [FM12, Theorem 8.9], and for compact surfaces, the homotopy is
an isotopy [FM12, Theorem 1.12]. ∎

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 $S$.

######  Definition 2.7.

Let $V$ be a finite set in $S$. A _cellulation_ $\mathscr{C}$ of $(S,V)$ is a
collection of simple disjoint arcs with endpoints in $V$ that cut $S$ into
cells homeomorphic to the 2-disk. The points of $V$ are called _vertices_ of
$\mathscr{C}$, the arcs are called _edges_ and the cells are called _faces_.
Two cellulations are _equivalent_ if there exists a homeomorphism
$S\rightarrow S$ fixing $V$ and isotopic to identity that sends one
cellulation to another.

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

###  2.3. Spacetimes with marked points

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

######  Definition 2.8.

A (future) _quasi-Fuchsian flat spacetime with marked points_ is a triple
$\mathbf{p}=(\rho,\tau,f)$, where $\rho\in\mathcal{T}(S)$,
$\tau\in\mathcal{H}^{1}(\rho)$ and $f:V\rightarrow\Omega^{+}(\rho,\tau)$ is a
map, the _vertex marking map_. The space of such marked spacetimes is denoted
by $\mathcal{P}(S,V)$.

By $\tilde{V}$ we denote the full preimage of $V$ in $\tilde{S}$ equipped with
the natural $\pi_{1}S$-action. By $\widetilde{\mathcal{P}}(S,V)$ we denote the
set of triples $(\rho,\tau,\tilde{f})$, where $\rho$ is a Fuchsian
representation, $\tau\in\mathcal{Z}^{1}(\rho)$ and
$\tilde{f}:\tilde{V}\rightarrow\tilde{\Omega}^{+}(\rho,\tau)$ is an
equivariant map. This is an open subset of the space
$T\mathcal{R}(S)\times(\mathbb{R}^{2,1})^{V}$. The group $SO_{0}(2,1)$ acts on
$\mathcal{R}(S)$ freely and properly by conjugation, by extending this to
$T\mathcal{R}(S)$ and adding the usual action on $(\mathbb{R}^{2,1})^{V}$ we
get an action on $T\mathcal{R}(S)\times(\mathbb{R}^{2,1})^{V}$, and
$\widetilde{\mathcal{P}}(S,V)$ is invariant with respect to this action. Also
for each $\rho\in\mathcal{R}(S)$, the group $\mathcal{B}^{1}(\rho)$ acts
freely and properly on the fiber
$\\{\rho\\}\times\mathcal{Z}^{1}(\rho)\times(\mathbb{R}^{2,1})^{V}$, and the
intersection of this fiber with $\widetilde{\mathcal{P}}(S,V)$ is invariant
with respect to this action. Finally, for each $\rho\in\mathcal{R}(S)$,
$\tau\in\mathcal{Z}^{1}(\rho)$, the group $\rho_{\tau}(\pi_{1}S)$ acts freely
and properly discontinuously on the set
$\\{\rho\\}\times\\{\tau\\}\times(\tilde{\Omega}^{+}(\rho,\tau))^{V}$, which
is the intersection of $\widetilde{\mathcal{P}}(S,V)$ with
$\\{\rho\\}\times\\{\tau\\}\times(\mathbb{R}^{2,1})^{V}$. All these three
actions commute and the quotient by them of $\widetilde{\mathcal{P}}(S,V)$ is
exactly $\mathcal{P}(S,V)$, which is thus endowed with the structure of an
analytic manifold of dimension $12\mathbf{g}-12+3n$, where $\mathbf{g}$ 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

$\mu:\mathcal{P}(S,V)\rightarrow T\mathcal{T}(S)$ (3)

forgetting the marked points.

######  Notation 2.9.

By $\mathcal{P}_{c}(S,V)$ we denote the subset of $\mathcal{P}(S,V)$
corresponding to $f$ in a strictly convex position, i.e., no point of $f(V)$
is a convex combination of other points (note that it implies the injectivity
of $f$).

######  Lemma 2.10.

$\mathcal{P}_{c}(S,V)$ is open in $\mathcal{P}(S,V)$. In particular,
$\mathcal{P}_{c}(S,V)$ is an analytic manifold of dimension
$12\mathbf{g}-12+3n$.

###### Proof.

Suppose the converse, then there exists
$\mathbf{p}:=(\rho,\tau,f)\in\mathcal{P}_{c}(S,V)$ and a sequence
$\mathbf{p}_{i}\in\mathcal{P}(S,V)\backslash\mathcal{P}_{c}(S,V)$ converging
to $\mathbf{p}$. Lift them to a sequence
$\tilde{\mathbf{p}}_{i}\in\widetilde{\mathcal{P}}(S,V)$ converging to
$\tilde{\mathbf{p}}$. Denote the respective vertex marking maps by
$\tilde{f}_{i}$, $\tilde{f}$. Up to passing to a subsequence, by the
Carathéodory Theorem there exists $\tilde{v}\in\tilde{V}$ and a sequence of
4-tuples of points
$p^{1}_{i},p^{2}_{i},p^{3}_{i},p^{4}_{i}\in\tilde{f}_{i}(\tilde{V}\backslash\tilde{v})$
such that $\tilde{f}_{i}(\tilde{v})$ belongs to the convex hull of
$p^{1}_{i},p^{2}_{i},p^{3}_{i},p^{4}_{i}$. Up to passing to a subsequence,
$p^{1}_{i},p^{2}_{i},p^{3}_{i},p^{4}_{i}$ converge to a 4-tuple of points
$p^{1},p^{2},p^{3},p^{4}\in\big{(}\tilde{f}(\tilde{V}\backslash\tilde{v})\cup\partial^{0}_{\infty}\mathbb{R}^{2,1}\big{)}$
by Lemma 2.4, containing $\tilde{f}(\tilde{v})$ in their convex hull. This
contradicts to that $\mathbf{p}\in\mathcal{P}_{c}(S,V)$. ∎

######  Notation 2.11.

For any quasi-Fuchsian flat spacetime with marked points
$\mathbf{p}=(\rho,\tau,f)$ in $\mathcal{P}(S,V)$, we will denote by ${\rm
conv}(\mathbf{p})$ the convex hull of $f(V)$ in $\Omega^{+}(\rho,\tau)$.

###  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-point of $d$. For a finite $V\subset S$, by
$\overline{\mathcal{M}}(S,V)$ denote the set of such metrics with $V(d)\subset
V$. 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 $\mathcal{M}(S,V)$ the subset of
$\overline{\mathcal{M}}(S,V)$ such that $V(d)=V$. Finally, by
$\overline{\mathcal{M}}_{-}(S,V)$ and $\mathcal{M}_{-}(S,V)$ we denote the
subsets of $\overline{\mathcal{M}}(S,V)$, $\mathcal{M}(S,V)$ respectively
consisting of metrics with negative singular curvatures.

A metric $d$ of $\overline{\mathcal{M}}(S,V)$ 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 $\overline{\mathcal{M}}(S,V)$ valued in
$\mathbb{R}^{E}$, where $E$ is the set of edges. If $\mathbf{g}$ is the genus
of $S$ and $n=|V|$, then, as we are considering triangulations,
$|E|=6\mathbf{g}-6+3n$. Changes of triangulation provide analytic transition
maps, endowing $\overline{\mathcal{M}}(S,V)$ with a structure of
$(6\mathbf{g}-6+3n)$-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 $\mathcal{M}(S,V)$, $\mathcal{M}_{-}(S,V)$ are open subsets of
$\overline{\mathcal{M}}(S,V)$.

Take an element $\mathbf{p}\in\mathcal{P}_{c}(S,V)$. 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
2.6 the induced metric belongs to $\mathcal{M}_{-}(S,V)$. This provides a map

$\mathcal{I}:\mathcal{P}_{c}(S,V)\to\mathcal{\mathcal{M}}_{-}(S,V)$

called the _induced metric map_.

By $\overline{\mathcal{P}}_{c}(S,V)\subset\mathcal{P}(S,V)$ we denote the
subset of injective configurations in convex position, i.e., when $f$ is
injective and $f(V)\subset\partial{\rm conv}(f(V))$. Note that it differs from
the closure of $\mathcal{P}_{c}(S,V)$ in $\mathcal{P}(S,V)$ as the latter also
include the configurations when some of the marked points collapsed. The map
$\overline{\mathcal{I}}:\overline{\mathcal{P}}_{c}(S,V)\rightarrow\overline{\mathcal{M}}_{-}(S,V)$
is defined in an obvious manner, and its restriction to $\mathcal{P}_{c}(S,V)$
is $\mathcal{I}$.

If the face celluation of $\mathbf{p}\in\mathcal{P}_{c}(S,V)$ is a
triangulation, then clearly $\mathcal{I}$ is analytic around $\mathbf{p}$. Now
we analyze the regularity of $\mathcal{I}$ in a more general situation.

######  Lemma 2.12.

Let $\mathbf{p}=(\rho,\tau,f)\in\overline{\mathcal{P}}_{c}(S,V)$,
$V(\mathbf{p})\subset V$ be its set of vertices, i.e., the maximal set such
that $f(V(\mathbf{p}))$ is in a strictly convex position, and $\mathscr{C}$ be
its face cellulation on $(S,V(\mathbf{p}))$. Then there exists a neighborhood
$U$ of $\mathbf{p}$ in $\overline{\mathcal{P}}_{c}(S,V)$ such that for all
$\mathbf{p}^{\prime}\in U$ its face cellulation is a subdivision of
$\mathscr{C}$ (with vertex set between $V(\mathbf{p})$ and $V$).

###### Proof.

Let $e$ be an edge of ${\rm conv}(\mathbf{p})$ with vertices $v$ and $w$. We
need to show that for all $\mathbf{p}^{\prime}$ close enough to $\mathbf{p}$,
$e$ is an edge of the face cellulation of ${\rm conv}(\mathbf{p}^{\prime})$.
Suppose the converse. Fix a lift
$\tilde{\mathbf{p}}\in\widetilde{\mathcal{P}}(S,V)$. Then there exists a
sequence $\tilde{\mathbf{p}}_{i}$ converging to $\tilde{\mathbf{p}}$ for which
$e$ is not an edge of ${\rm conv}(\mathbf{p}_{i})$, where $\mathbf{p}_{i}$ is
the projection of $\tilde{\mathbf{p}}_{i}$. Denote the respective vertex
marking maps by $\tilde{f}_{i}$, $\tilde{f}$. Fix a lift $\tilde{e}$ of $e$ to
$\tilde{S}$ with endpoints $\tilde{v},\tilde{w}\in\tilde{V}$. For each $i$
there exists a triple of points
$p^{1}_{i},p^{2}_{i},p^{3}_{i}\in\tilde{f}_{i}(\tilde{V}\backslash\\{\tilde{v}_{i},\tilde{w}_{i}\\})$
such that the convex hull of $p^{1}_{i},p^{2}_{i},p^{3}_{i}$ intersects the
geodesic segment between $\tilde{f}_{i}(\tilde{v})$ and
$\tilde{f}_{i}(\tilde{w})$. Up to passing to a subsequence, each sequence
$p^{j}_{i}$ converges to a point
$p^{j}\in\big{(}\tilde{f}(\tilde{V}\backslash\\{\tilde{v},\tilde{w}\\})\cup\partial_{\infty}^{0}\mathbb{R}^{2,1}\big{)}$
by Lemma 2.4, and the convex hull of the points $p^{j}$ intersects the segment
between $\tilde{f}(\tilde{v})$, $\tilde{f}(\tilde{w})$. This contradicts to
that the latter is an edge of $\widetilde{\rm conv}(\mathbf{p})$. ∎

Since there are only finitely many cellulations subdividing $\mathscr{C}$,
from the definition of the topologies, it is evident that
$\overline{\mathcal{I}}$ is continuous. We go further and show

######  Lemma 2.13.

The map $\mathcal{I}$ is $C^{1}$.

The proof is reminiscent to similar proofs in [Pro22a, Pro22b].

###### Proof.

Pick $\mathbf{p}\in\mathcal{P}_{c}(S,V)$ with a lift
$\tilde{\mathbf{p}}\in\widetilde{\mathcal{P}}(S,V)$. Let $\mathscr{C}$ be the
boundary cellulation of ${\rm conv}(\mathbf{p})$. By Lemma 2.12, there exists
a small enough neighborhood $\tilde{U}$ of $\tilde{\mathbf{p}}$ such that it
projects onto a neighborhood $U$ of $\mathbf{p}$ in $\mathcal{P}_{c}(S,V)$ and
for every $\tilde{\mathbf{p}}^{\prime}\in\tilde{U}$ the edges of $\mathscr{C}$
remain to be the edges of ${\rm conv}(\mathbf{p}^{\prime})$, where
$\mathbf{p}^{\prime}$ is the projection of $\tilde{\mathbf{p}}^{\prime}$. Let
$\mathscr{T}$ be a triangulation subdividing $\mathscr{C}$. Denote
$\mathcal{I}(\mathbf{p})$ by $d$. If $e\in E(\mathscr{T})$ is an edge of
$\mathscr{C}$, then its length $l_{e}$ is an analytic function over $U$. We
need to deal with those edges of $\mathscr{T}$ that are not edges of
$\mathscr{C}$.

Choose such an edge $e$, and let $Q$ be a face of $\widetilde{\rm
conv}(\mathbf{p})$ containing a realization of $e$, $V_{Q}$ be the set of its
vertices (which are considered here as points in $\mathbb{R}^{2,1}$). We
orient $Q$ with respect to the orientation of $\widetilde{\rm
conv}(\mathbf{p})$. Pick a small enough perturbation of the position of the
points in $V_{Q}$ such that all boundary edges of $Q$ remain to be edges of
${\rm conv}(V_{Q})$, and all faces of ${\rm conv}(V_{Q})$ remain spacelike.
Let $U_{Q}\subset(\mathbb{R}^{2,1})^{V_{Q}}$ be a small neighborhood of the
initial positions of the points in $V_{Q}$, where this holds. If ${\rm
conv}(V_{Q})$ has non-empty interior, $\partial{\rm conv}(V_{Q})$ is divided
by the cycle of boundary edges of $Q$ into two domains. One of them, chosen
with the help of the orientation and called _the lower part_ , is considered
as a deformation of $Q$. Let $l_{e,Q}$ be the length of $e$ in this
realization, considered as a function over $V_{Q}$.

Provided that $\tilde{U}$ is sufficiently small, there is a smooth map
$q:\tilde{U}\rightarrow U_{Q}$ sending
$\tilde{\mathbf{p}}^{\prime}\in\tilde{U}$ to the corresponding positions of
points of $V_{Q}$. The length function $l_{e}$ over $\tilde{U}$ coincides with
$l_{e,Q}\circ q$. Hence, it is enough to show that $l_{e,Q}$ is a
$C^{1}$-function of the positions of the points in $V_{Q}$. Abusing the
notation, we denote $l_{e,Q}$ by $l_{e}$ until the end of the proof.

Let $\mathscr{T}_{1},\ldots,\mathscr{T}_{r}$ be all the triangulations of $Q$.
For every $i=1,\ldots,r$ and every configuration in $U_{Q}$ we consider the
triangulated polyhedral surface $Q_{i}$ in $\mathbb{R}^{2,1}$ determined by
the triangulation $\mathscr{T}_{i}$ and the positions of vertices. By
decreasing $U_{Q}$ if necessary, we may assume that all face triangles of all
these surfaces are spacelike, and all boundary angles (of the surfaces in the
intrinsic metric) are less than $\pi$. Then for every such surface we can
determine the length of $e$ in this surface. This determines $r$ functions
over $U_{Q}$, which we denote by $l_{e,1},\ldots,l_{e,r}$. We note that since
for every $i$ the lengths of all edges of $Q_{i}$ are analytic over $U_{Q}$,
and $l_{e,i}$ is a analytic function of these lengths, we get that $l_{e,i}$
are analytic over $U_{Q}$.

For every configuration in $U_{Q}$, the lower part of ${\rm conv}(V_{Q})$
coincides with one of $Q_{i}$. This provides a decomposition of $U_{Q}$ into
cells $U_{Q,1},\ldots U_{Q,r}$, which are analytic manifolds with corners, and
$l_{e}$ coincides with $l_{e,i}$ over $U_{Q,i}$. We need to check that the
differentials of the corresponding $l_{e,i}$ coincide on the common points of
$U_{Q,i}$. It is enough to prove this at the initial configuration.

######  Claim 2.

The differentials of all $l_{e,i}$ at the initial configuration coincide.

Let $\Pi$ be the oriented plane containing $Q$. Introduce a coordinate system
in $\mathbb{R}^{2,1}$, where $\Pi$ is a coordinate plane, which we call
_horizontal_. The last coordinate is accordingly called _vertical_. It is
clear that at the initial configuration, all the horizontal derivatives of all
$l_{e,i}$ coincide. We now show that the vertical derivatives coincide too,
and, moreover, are equal to zero.

Let $v_{1},v_{2}$ be two points in $\mathbb{R}^{2,1}$ in spacelike position.
If $v_{1}$ moves orthogonally to the segment connecting them, then it is easy
to check that the derivative of the distance between $v_{1}$ and $v_{2}$ is
zero. This implies that at the initial configuration, the vertical derivatives
of the length of any edge of $Q_{i}$ are zero. Since for every $i$, the
function $l_{e,i}$ is an analytic function of the edge-lengths of $Q_{i}$, its
vertical derivatives are zero too. This finishes the proof. ∎

##  3\. Proof of Theorem I’

###  3.1. A topological interlude

Our space $\mathcal{P}_{c}(S,V)$ 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 $V\subset S$ be a finite subset of size
$n$ and $S^{\ast V}$ be the space of injective maps $c:V\hookrightarrow S$,
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 $S^{\ast V}$. By $H_{0}(S)$ denote the group of self-homeomorphisms
$S\rightarrow S$ isotopic to identity, by $H_{0}(S,V)$ denote the subgroup of
self-homeomorphism that fix $V$ and are isotopic to identity relative to $V$,
and by $\mathcal{C}(S,V)$ we denote 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 $H_{0}(S)\rightarrow S^{\ast V}$ given by the evaluation of
$h$ at $V$, which is clearly continuous. It factors through $H_{0}(S,V)$ as a
map ${\rm ev}_{V}:\mathcal{C}(S,V)\rightarrow S^{\ast V}.$ This map is a local
homeomorphism, see e.g. [Bir74, Chapter 4.1] (note that in [Bir74, Chapter
4.1] the homeomorphisms are not required to be isotopic to identity, however,
the argument does not change). Hence, $\mathcal{C}(S,V)$ is a manifold of
dimension $2n$.

Every homeomorphism $h:S\rightarrow S$ determines a homeomorphism
$h_{V}:S^{\ast V}\rightarrow S^{\ast V}$ by $c\mapsto h\circ c$, and if $h$ is
isotopic to identity, then so is $h_{V}$. Every self-homeomophism of $S^{\ast
V}$ isotopic to 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 $\tilde{h}_{V}:\widetilde{S^{\ast
V}}\rightarrow\widetilde{S^{\ast V}}$. Fix a lift $\widehat{V}$ of $V$ to the
universal covering $\widetilde{S^{\ast V}}$. Now we define a new evaluation
map $H_{0}(S)\rightarrow\widetilde{S^{\ast V}}$ given by the evaluation of
$\tilde{h}_{V}$ at $\widehat{V}$, which is clearly continuous. For $h\in
H_{0}(S,V)$ the map $\tilde{h}_{V}$ fixes $\widehat{V}$, hence the evaluation
map factors through $H_{0}(S,V)$ as

$\widetilde{\rm ev}_{\tilde{V}}:\mathcal{C}(S,V)\rightarrow\widetilde{S^{\ast
V}}.$

######  Lemma 3.1.

The evaluation map $\widetilde{\rm ev}_{\widehat{V}}$ is a homeomorphism.

As $\widetilde{\rm ev}_{\widehat{V}}$ is a continuous map between two
manifolds of the same dimension, it is enough to prove that it is bijective.
We will show it by induction on $n$. We need to establish few auxiliary facts.
First, assume that $v\in V$ and $W:=V\backslash\\{v\\}$. When $|V|>1$ we have
the forgetful map $\theta_{V}:S^{\ast V}\rightarrow S^{\ast W}$, so for
$c:W\hookrightarrow S$ the fiber of $\theta_{V}$ over $c$ is naturally
identified with $S\backslash c(W)$. It is known that $\theta_{V}$ is a fiber
bundle (with fiber homeomorphic to $S\backslash W$), see [FH01, Theorem 1.1].
We can then extend this bundle to the universal covering.

######  Lemma 3.2.

The universal covering $\widetilde{S^{\ast V}}$ is homeomorphic to an open
ball of dimension $2n$ and there exists a fiber bundle
$\tilde{\theta}_{V}:\widetilde{S^{\ast V}}\rightarrow\widetilde{S^{\ast W}}$
extending $\theta_{V}$.

###### Proof.

We will also prove this by induction on $n$. For $n=1$, $S^{\ast V}\cong S$
and $\widetilde{S^{\ast V}}\cong\tilde{S}$, hence we have the base case for
the first statement. Consider now the pull-back of the bundle $\theta_{V}$ to
$\widetilde{S^{\ast W}}$. It has fiber $S\backslash W$. If we know that
$\widetilde{S^{\ast W}}$ is a topological ball, then the pull-back bundle is
trivial, and we can extend it further to a trivial bundle over
$\widetilde{S^{\ast W}}$ with fiber homeomorphic to $\widetilde{S\backslash
W}$. The total space of the latter bundle is a topological ball of dimension
$2n$ as it is a trivial bundle over a ball of dimension $2n-2$ with a fiber
homeomorphic to a 2-dimensional ball. Particularly, the total space is simply
connected. It also covers $S^{\ast V}$ by construction, and this covering is
the universal covering of $S^{\ast V}$. This gives the base case for the
second statement and the induction steps for both statements. ∎

For the next two statements we need yet more notation. Let $W\subset S$ be a
finite subset, possibly empty, and $h\in H_{0}(S,W)$ (which is just $H_{0}(S)$
if $W$ is empty). It then also lifts to a self-homeomorphism of
$\widetilde{S\backslash W}$, which we denote by $\tilde{h}_{\backslash W}$.
The construction in Lemma 3.2 shows that for a lift $\widehat{W}$ of $W$ to
$\widetilde{S^{\ast W}}$ the fiber of $\tilde{\theta}_{V}$ over $\widehat{W}$
is naturally identified with $\widetilde{S\backslash W}$ in the sense that for
every $h\in H_{0}(S,W)$ the restriction of $\tilde{h}_{W}$ on the fiber
coincides with $\tilde{h}_{\backslash W}$ on $\widetilde{S\backslash W}$.

######  Lemma 3.3.

Let $\tilde{p},\tilde{q}\in\widetilde{S\backslash W}$ be two points. Then
there exists $h\in H_{0}(S,W)$ such that $\tilde{h}_{\backslash W}$ sends
$\tilde{p}$ to $\tilde{q}$.

###### Proof.

Let $\tilde{p},\tilde{q}$ project to points $p,q\in S\backslash W$. Consider
an oriented path $\tilde{\psi}$ connecting $\tilde{p}$ and $\tilde{q}$ in
$\widetilde{S\backslash W}$, let $\psi$ be its projection to an oriented path
between $p$ and $q$ in ${S\backslash W}$. We make a homotopy of $\psi$ on
$S\backslash W$ so that $\psi$ becomes the concatenation of finitely many
simple closed curves $\psi_{i}$ based at $p$ and of a simple arc $\psi_{a}$
connecting $p$ and $q$.

We now take “point-pushing” homeomorphisms along every $\psi_{i}$ and along
$\psi_{a}$. For the loops $\psi_{i}$ such a map is obtained as follows. We
take a small embedded strip along $\psi_{i}$ in ${S\backslash W}$ and consider
a homeomorphism $h_{i}:S\rightarrow S$ that is identity outside the strip,
that is also identity on $\psi_{i}$, but that makes one full twist inside the
strip in the direction determined by the orientation of $\psi_{i}$. Every
$h_{i}$ is isotopic to identity so that $W$ is fixed and $p$ moves along
$\psi_{i}$ during the isotopy. For $\psi_{a}$ we consider a small embedded
disk around $\psi_{a}$ in ${S\backslash W}$ and a homeomorphism
$h_{a}:S\rightarrow S$ that is identity outside the disk and that sends $p$ to
$q$. It is also isotopic to identity by an isotopy fixing $W$ and moving $p$
along $\psi_{a}$. By $h$ we denote the composition of all $h_{i}$ and $h_{a}$.
It is clear that it is isotopic to identity by an isotopy fixing $W$ and
moving $p$ along $\psi$. Thereby $\tilde{h}_{\backslash W}$ sends $\tilde{p}$
to $\tilde{q}$. ∎

######  Lemma 3.4.

Let $h\in H_{0}(S,W)$, $v\in S\backslash W$ be a point and
$\tilde{v}\in\widetilde{S\backslash W}$ be a lift of $v$. Assume that the lift
$\tilde{h}_{\backslash W}$ fixes $\tilde{v}$. Then $h\in H_{0}(S,V)$ for
$V:=W\cup\\{v\\}$.

###### Proof.

Let $h_{t}$, $t\in[0,1]$, be a path of homeomorphisms fixing $W$ such that
$h_{0}$ is identity and $h_{1}=h$. Since
$\tilde{h}_{0}(\tilde{v})=\tilde{h}_{1}(\tilde{v})$, the loop $h_{t}(v)$,
$t\in[0,1]$, is contractible in $S\backslash W$. Let $s:[0,1]^{2}\rightarrow
S$ be a homotopy sending $s(t,0):=h_{t}(v)$ to the constant map $s(t,1):=v$ on
$S\backslash W$ so that $s(0,\tau)=s(1,\tau)=v$ for all $\tau\in[0,1]$.
Consider the space $S\times[0,1]$ and the map $H:S\times[0,1]\rightarrow S$,
$H(p,t):=h_{t}(p)$. The map $(t,\tau)\mapsto(s(t,\tau),t)$ sends $[0,1]^{2}$
to $(S\backslash W)\times[0,1]$, and the image contains $\\{v\\}\times[0,1]$
and $H^{-1}(v)$. Turns out that the paths $\\{v\\}\times[0,1]$ and $H^{-1}(v)$
are homotopic in $S\times[0,1]$ by a homotopy respecting the levels
$S\times\\{t\\}$ and fixing pointwise the set $W\times[0,1]$. Then there
exists a homeomorphism $F:S\times[0,1]\rightarrow S\times[0,1]$ respecting the
levels, fixing the levels $S\times\\{0\\}$ and $S\times\\{1\\}$ pointwise,
fixing the set $W\times[0,1]$ pointwise, and sending $\\{v\\}\times[0,1]$ to
$H^{-1}(v)$. The map $H\circ F$ constitutes the desired isotopy from $id$ to
$h$ fixing $V$. ∎

###### Proof of Lemma 3.1.

We prove the following two statements by induction: (1) for any
$X\in\widetilde{S^{\ast V}}$ there exists $h\in H_{0}(S)$ such that
$\tilde{h}_{V}(\widehat{V})=X$; and (2) for any $h\in H_{0}(S)$ such that
$\tilde{h}_{V}$ fixes $\widehat{V}$, we have $h\in H_{0}(S,V)$. It is easy to
see that this shows that $\widetilde{\rm ev}_{\widehat{V}}$ is bijective.

The base cases are trivially given by Lemmas 3.3 and 3.4. For the induction
step we employ the (trivial) fiber bundle
$\tilde{\theta}:=\tilde{\theta}_{V}:\widetilde{S^{\ast
V}}\rightarrow\widetilde{S^{\ast W}}$ from Lemma 3.2.

(1) By the induction hypothesis there exists $h^{\prime}\in H_{0}(S)$ such
that $\tilde{h}_{W}\circ\tilde{\theta}(\widehat{V})=\tilde{\theta}(X)$. Next,
by Lemma 3.3 there exists $h^{\prime\prime}\in H_{0}(S,h^{\prime}(W))$ such
that $\tilde{h}^{\prime\prime}_{\backslash h^{\prime}(W)}$ sends
$\tilde{h}^{\prime}_{V}(\widehat{V})$ to $X$ where the fiber of
$\tilde{\theta}$ containing $X$ is naturally identified with
$\widetilde{S\backslash h^{\prime}(W)}$. It follows that
$h:=h^{\prime\prime}\circ h^{\prime}$ fits.

(2) By the induction hypothesis we have $h\in H_{0}(S,W)$. Since the fiber of
$\tilde{\theta}$ containing $\widehat{V}$ is naturally identified with
$\widetilde{S\backslash W}$, Lemma 3.4 implies then that $h\in H_{0}(S,V)$. ∎

######  Remark 3.5.

We need to identify the group of deck transformations of $\widetilde{S^{\ast
V}}$, which is the _pure braid group of $S$, i.e. the fundamental group of
$S^{\ast V}$_. This is already easy to derive from the discussion above. Let
$H_{\star}(S,V)$ be the group of homeomorphisms fixing $V$ and isotopic to
identity. Then $H_{0}(S,V)$ is a normal subgroup of $H_{\star}(S,V)$. Consider
the quotient group $H_{\star}(S,V)/H_{0}(S,V)$. It clearly acts on
$\mathcal{C}(S,V)$ from the right by deck transformations, which provides a
homomorphism

$H_{\star}(S,V)/H_{0}(S,V)\rightarrow\pi_{1}S^{\ast V}~{}.$ (4)

It remains to notice that Lemma 3.3 clearly implies that this homomorphism is
surjective, while Lemma 3.4 establishes its injectivity. Note that the fact
that (4) is an isomorphism can also be deduced from a theorem of J. Birman,
see [Bir74, Theorem 4.2].

######  Remark 3.6.

Lemma 3.1 is probably already known, but we were unable to find a precise
reference. We provided a proof for completeness. Another way to prove it would
have been to start from (the adaptation of) a classical result of J. Birman
saying that

$H_{\star}(S,V)\rightarrow H_{0}(S)\xrightarrow[]{{\rm ev}}S^{\ast V}~{}$

is a fiber bundle (it is immediate to adapt the proof given in the proof of
Theorem 4.6 in [FM12]). Then Lemma 3.1 would follow from general topological
arguments.

###  3.2. The induced metric map on fibers

We have a projection map $\pi:\mathcal{P}_{c}(S,V)\to\mathcal{T}(S)$, which is
obtained from the projection map $\mu$ in (3) composed with the natural
projection from $T\mathcal{T}(S)$ onto $\mathcal{T}(S)$. For
$\rho\in\mathcal{T}(S)$ we will denote $\pi^{-1}(\rho)$ by
$\mathcal{P}_{c}(\rho,V)$. In other terms, by
$\mathcal{P}_{c}(\rho,V)\subset\mathcal{P}_{c}(S,V)$ we denote the space of
spacetimes with marked points and with linear holonomy given by $\rho$. Other
related notation from Section 2.3 restricts to this case straightforwardly.

We denote the restriction of $\mathcal{I}$ to $\mathcal{P}_{c}(\rho,V)$ by
$\mathcal{I}_{\rho}$. Our aim is to show Theorem 1.4, that implies Theorem I’.

We have the restriction of $\mu$ to $\mathcal{P}(\rho,V)$, which we denote by
$\mu_{\rho}$:

$\mu_{\rho}:\mathcal{P}(\rho,V)\to
T_{\rho}\mathcal{T}(S)=\mathcal{H}^{1}(\rho)~{}.$

We now introduce on $\mathcal{P}(\rho,V)$ a structure of fiber bundle of
special type over $\mathcal{H}^{1}(\rho)$ with fiber
$(S\times\mathbb{R}_{>0})^{V}$. We first suppose that $|V|=1$. This is a
special case, and to distinguish it we will denote $\mathcal{P}(\rho,V)$
rather by $\mathcal{P}(\rho,v)$, meaning that $V=\\{v\\}$, and denote
$\mu_{\rho}$ by $\mu_{\rho,v}$.

######  Lemma 3.7.

There exists a fiber bundle atlas $\\{(U,\phi_{U})\\}$,
$U\subset\mathcal{H}^{1}(\rho)$, for
$\mu_{\rho,v}:\mathcal{P}(\rho,v)\rightarrow\mathcal{H}^{1}(\rho)$ with

$\phi_{U}:\mu^{-1}_{\rho,v}(U)\rightarrow U\times S\times\mathbb{R}_{>0}$

such that
(1) for every $\tau\in U,r\in\mathbb{R}_{+},$ the set
$\phi_{U}^{-1}(\tau,S,r)$ is a ct-level surface in $\Omega^{+}(\rho,\tau)$;
(2) for every $\tau\in U,p\in S,$ the set
$\phi_{U}^{-1}(\tau,p,\mathbb{R}_{>0})$ is a gradient line of ct parametrized
by ct.

###### Proof.

Lift $\rho$ to an element of $\mathcal{R}(S)$. By $\tilde{\mu}_{\rho,v}$ we
denote the natural projection
$\tilde{\mu}_{\rho,v}:\tilde{\mathcal{P}}(\rho,v)\rightarrow\mathcal{Z}^{1}(\rho)$,
which is a lift of $\mu_{\rho,v}$. Let $\tilde{U}$ be an open bounded set in
$Z^{1}(\rho)$. In [Bon05b, Section 6] Bonsante showed that there exists a
continuous map

$\Phi_{\tilde{U}}:\tilde{U}\times\tilde{S}\times\mathbb{R}_{>0}\rightarrow\mathbb{R}^{2,1}$

such that for every $\tau\in\tilde{U}$ we have
$\Phi_{\tilde{U}}(\tau,\tilde{S},\mathbb{R}_{>0})=\tilde{\Omega}^{+}(\rho,\tau)$
and
(0) for every $\tau\in\tilde{U}$ the map $\Phi_{\tilde{U}}$ is
$\pi_{1}S$-equivariant, i.e. for every $\gamma\in\pi_{1}S,p\in\tilde{S}$ and
$r\in\mathbb{R}_{>0}$ we have $\Phi_{\tilde{U}}(\tau,\gamma
p,r)=\rho_{\tau}(\gamma)\Phi_{\tilde{U}}(\tau,p,r)$,
(1) for every $\tau\in\tilde{U},r\in\mathbb{R}_{>0}$ the set
$\Phi_{\tilde{U}}(\tau,\tilde{S},r)$ is a ct-level surface in
$\tilde{\Omega}^{+}(\rho,\tau)$;
(2) for every $\tau\in\tilde{U},p\in\tilde{S}$ the set
$\Phi_{\tilde{U}}(\tau,p,\mathbb{R}_{>0})$ is a gradient line of ct
parametrized by ct.

Now we consider the map
$id\times\Phi_{\tilde{U}}:\tilde{U}\times\tilde{S}\times\mathbb{R}_{+}\rightarrow\tilde{U}\times\mathbb{R}^{2,1}$.
This is a homeomorphism onto the image, which is
$\tilde{\mu}^{-1}_{\rho,v}(\tilde{U})$. So the inverses of these maps
constitute an equivariant fiber bundle atlas of the desired form for
$\tilde{\mu}_{\rho,v}$. In turn this gives the desired atlas for
$\mu_{\rho,v}:\mathcal{P}(\rho,v)\rightarrow\mathcal{H}^{1}(\rho)$. ∎

Our fiber bundle given by Lemma 3.7 has the structure group $H_{0}(S)$, where
we consider it acting on $S\times\mathbb{R}_{>0}$ by homeomorphisms on the
first factor and trivially on the second factor. Since the space
$\mathcal{H}^{1}(\rho)$ is contractible, this fiber bundle admits a
trivialization

$\mathcal{P}(\rho,v)\cong\mathcal{H}^{1}(\rho)\times
S\times\mathbb{R}_{>0}~{},$ (5)

where for every $U\subset\mathcal{H}^{1}(\rho)$, the trivialization $U\times
S\times\mathbb{R}_{>0}$ belongs to the maximal atlas for $\mu_{\rho,v}$
satisfying the conditions (1) and (2) from Lemma 3.7. For what follows we fix
such a trivialization. This endows every Cauchy surface
$\Sigma\subset\Omega^{+}(\rho,\tau)$ with a homeomorphism $\Sigma\rightarrow
S$ given by the projection to $S$ in (5).

For $|V|=n>1$ we note that after an enumeration of points of $V$ the $n$-th
power of the fiber bundle $\mu_{\rho,v}$ is naturally diffeomorphic to the
space $\mathcal{P}(\rho,V)$ with the diffeomorphism preserving the projection
to $\mathcal{H}^{1}(\rho)$. This endows
$\mu_{\rho}:\mathcal{P}(\rho,V)\rightarrow\mathcal{H}^{1}(\rho)$ with the
desired fiber bundle structure with fiber $(S\times\mathbb{R}_{>0})^{V}$. The
trivialization (5) then gives a trivialization

$\mathcal{P}(\rho,V)\cong\mathcal{H}^{1}(\rho)\times
S^{V}\times\mathbb{R}_{>0}^{V}~{}.$

Consider a subset $\mathcal{P}^{\ast}(\rho,V)\subset\mathcal{P}(\rho,V)$
consisting of all configurations such that for every distinct $v,w\in V$ the
points $f(v)$ and $f(w)$ do not belong to the same gradient line of ct. This
is clearly an open subset of $\mathcal{P}(\rho,V)$, and we have a
trivialization

$\mathcal{P}^{\ast}(\rho,V)\cong\mathcal{H}^{1}(\rho)\times S^{\ast
V}\times\mathbb{R}_{>0}^{V}~{}.$ (6)

We consider the universal cover $\mathcal{P}^{\sharp}(\rho,V)$ of
$\mathcal{P}^{\ast}(\rho,V)$, which then has a trivialization

$\mathcal{P}^{\sharp}(\rho,V)\cong\mathcal{H}^{1}(\rho)\times\widetilde{S^{\ast
V}}\times\mathbb{R}_{>0}^{V}~{}.$

Finally, we fix an evaluation homeomorphism from Section 3.1, allowing us to
identify $\widetilde{S^{\ast V}}\cong\mathcal{C}(S,V)$.

It is clear that
$\overline{\mathcal{P}}_{c}(\rho,V)\subset\mathcal{P}^{\ast}(\rho,V)$. By
$\overline{\mathcal{P}}^{\sharp}_{c}(\rho,V)$ we denote its full preimage in
the universal cover $\mathcal{P}^{\sharp}(\rho,V)$, and by
$\mathcal{P}_{c}^{\sharp}(\rho,V)$ denote the full preimage of
$\mathcal{P}_{c}(S,V)$. Let $\mathbf{p}^{\sharp}$ be an element of
$\mathcal{P}^{\sharp}_{c}(\rho,V)$ and $\mathbf{p}$ be its projection to
$\mathcal{P}_{c}(\rho,V)$. Basically, for us $\mathbf{p}^{\sharp}$ is just
$\mathbf{p}$ equipped with one additional piece of data: a homeomorphism from
$\partial{\rm conv}(\mathbf{p})$ to $(S,V)$ respecting the vertex marking map
and defined up to isotopy fixing $V$. To see this, first, observe that the
trivialization (5) endows $\partial{\rm conv}(\mathbf{p})$ by a homeomorphism
$\chi$ to $S$. Second, the choice of the evaluation map gives an element of
$\mathcal{C}(S,V)$, i.e., a class in $H_{0}(S)/H_{0}(S,V)$. If $h\in H_{0}(S)$
is a representative of this class, then $h^{-1}\circ\chi$ is a desired
homeomorphism from $\partial{\rm conv}(\mathbf{p})$ to $S$, and
$h^{-1}\circ\chi\circ f$ is the identity on $V$.

Now by $\mathcal{M}_{-}^{\sharp}(S,V)$ we denote the respective space of
metrics up to isometry isotopic to identity relative to $V$. The forgetful map
$\mathcal{M}_{-}^{\sharp}(S,V)\rightarrow\mathcal{M}_{-}(S,V)$ is a covering
map, and we will soon see that it is the universal covering. It is
straightforward from the definitions that the group of deck transformations
can be identified with the group $\pi_{1}S^{\ast V}$, which was also mentioned
in Remark 3.5.

We can define the lift of the induced metric map

$\mathcal{I}^{\sharp}_{\rho}:\mathcal{P}_{c}^{\sharp}(\rho,V)\rightarrow\mathcal{M}_{-}^{\sharp}(S,V)~{}.$

From Section 2.4 and Lemma 2.10, we know that
$\mathcal{P}_{c}^{\sharp}(\rho,V)$ and $\mathcal{M}_{-}^{\sharp}(S,V)$ are
manifolds of the same dimension $6\mathbf{g}-6+3n$. We will need more
information about their topologies, that are given by the two following
lemmas.

######  Lemma 3.8.

$\mathcal{P}_{c}^{\sharp}(\rho,V)$ is connected.

###### Proof.

Take
$\mathbf{p}_{0}^{\sharp},\mathbf{p}_{1}^{\sharp}\in\mathcal{P}_{c}^{\sharp}(\rho,V)$.
First note that $\mathcal{P}^{\sharp}(\rho,V)$ is clearly connected. We
connect $\mathbf{p}_{0}^{\sharp}$ and $\mathbf{p}_{1}^{\sharp}$ by a path
$\mathbf{p}_{t}^{\sharp}$, $t\in[0,1]$, in $\mathcal{P}^{\sharp}(\rho,V)$. Our
first goal is to deform $\mathbf{p}_{t}^{\sharp}$ to a path in
$\overline{\mathcal{P}}_{c}^{\sharp}(\rho,V)$. Denote the projection of
$\mathbf{p}_{t}^{\sharp}$ to $\mathcal{P}^{\ast}(\rho,V)$ by
$\mathbf{p}_{t}=(\rho,\tau_{t},f_{t})$.

For $v\in V$ we define the set

$O_{v}:=\\{t\in[0,1]:f_{t}(v)\in{\rm int}({\rm conv}(\mathbf{p}_{t}))\\}~{}.$

Clearly, $O_{v}$ is an open subset of $(0,1)$, hence it consists of countably
many open intervals. For every such interval $I$ with endpoints $t_{1}$ and
$t_{2}$ we have $f_{t_{1}}(v)\in\partial{\rm conv}(\mathbf{p}_{t_{1}})$ and
$f_{t_{2}}(v)\in\partial{\rm conv}(\mathbf{p}_{t_{2}})$. Now we decrease the
cosmological time of $v$ to get $f_{t}(v)\in\partial{\rm
conv}(\mathbf{p}_{t})$. This will deform our path $\mathbf{p}_{t}$ to another
continuous path, and we lift it to a deformation of the path
$\mathbf{p}_{t}^{\sharp}$. We can do this for every $v\in V$ and every such
interval $I$. After all the modifications $\mathbf{p}_{t}^{\sharp}$ is a path
in $\overline{\mathcal{P}}_{c}^{\sharp}(\rho,V)$. When $f_{t}(v)$ is not a
vertex of ${\rm conv}(\mathbf{p}_{t})$, it remains again to decrease slightly
the cosmological time of $f_{t}(v)$. After this for all $t$ the sets
$f_{t}(V)$ get to strictly convex positions, thus $\mathbf{p}_{t}^{\sharp}$
becomes a path in $\mathcal{P}_{c}^{\sharp}(\rho,V)$. ∎

######  Lemma 3.9.

$\mathcal{M}_{-}^{\sharp}(S,V)$ is connected and simply connected.

###### Proof.

By $\mathcal{T}(S,V)$ we denote the Teichmüller space with marked points,
i.e., the space of conformal structures on $S$ up to conformal maps isotopic
to identity relative to $V$. It is classical that $\mathcal{T}(S,V)$ is
homeomorphic to an open ball of dimension $6\mathbf{g}-6+2n$. By $X^{V}$ we
denote a subset of $\mathbb{R}_{<0}^{V}$ of tuples of negative numbers with
sum $2-2\mathbf{g}.$ Consider the trivial bundle $\mathcal{T}(S,V)\times
X^{V}\times\mathbb{R}_{>0}$. To $d\in\mathcal{M}_{-}^{\sharp}(S,V)$ we
associate the following data: the underlying conformal structure, as an
element of $\mathcal{T}(S,V)$, the tuple of singular curvatures and the area.
This gives a map
$\mathcal{M}_{-}^{\sharp}(S,V)\rightarrow\mathcal{T}(S,V)\times
X^{V}\times\mathbb{R}_{>0}$, which is clearly continuous. The work of Troyanov
[Tro86] implies that it is injective and surjective, hence, from the
invariance of the domain, it is a homeomorphism.

We remark that another similar proof but using discrete conformality instead
of conformality follows from the work [GLSW18].

∎

The following result will be proven in Section 3.3. In a generic case when the
face decomposition is a triangulation, it is Theorem 1.3 of Iskhakov.

######  Proposition 3.10.

The differential $d\mathcal{I}_{\rho}$ is non-degenerate.

In Section 3.4 we will prove the following compactness result.

######  Proposition 3.11.

Let $\mathbf{p}_{i}=(\rho,\tau_{i},f_{i})$ be a sequence in
$\mathcal{P}_{c}(\rho,V)$ such that the induced metrics
$d_{i}\in\mathcal{\mathcal{M}}_{-}(S,V)$ converge to a metric
$d\in\mathcal{\mathcal{M}}_{-}(S,V)$. Then, up to a subsequence,
$\mathbf{p}_{i}$ converge to $\mathbf{p}\in\mathcal{P}_{c}(\rho,V)$.

Putting all these results together, we can now give the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

The spaces $\mathcal{P}_{c}(\rho,V)$ and $\mathcal{\mathcal{M}}_{-}(S,V)$ are
manifolds of the same dimension. From Lemma 2.13, Proposition 3.10 and the
Inverse Function Theorem, $\mathcal{I}_{\rho}$ is a local
$C^{1}$-diffeomorphism. Due to Lemma 3.11, $\mathcal{I}_{\rho}$ is proper.
Hence the image of $\mathcal{I}_{\rho}$ is closed and open so
$\mathcal{I}_{\rho}$ is surjective, as by Lemma 3.9, the target is connected.
Then $\mathcal{I}_{\rho}$ is a covering map. Consider now a commutative
diagram

${\mathcal{P}_{c}^{\sharp}(\rho,V)}$${\mathcal{\mathcal{M}}_{-}^{\sharp}(S,V)}$${\mathcal{P}_{c}(\rho,V)}$${\mathcal{\mathcal{M}}_{-}(S,V)}$$\scriptstyle{\mathcal{I}^{\sharp}_{\rho}}$$\scriptstyle{\mathcal{I}_{\rho}}$

We know that the vertical arrows are covering maps, and so is the lower
horizontal arrow. Since all spaces are connected, it follows that
$\mathcal{I}^{\sharp}_{\rho}$ is also a covering. By Lemma 3.8, the domain is
connected and by Lemma 3.9, the target is simply connected, thus
$\mathcal{I}^{\sharp}_{\rho}$ is a homeomorphism. The groups of deck
transformations of the vertical coverings are both naturally isomorphic to the
group $\pi_{1}S^{\ast V}$, see Remark 3.5, and the map
$\mathcal{I}^{\sharp}_{\rho}$ is clearly equivariant with respect to it. As it
induces an isomorphism of the deck transformation groups, it follows that
$\mathcal{I}_{\rho}$ is also a homeomorphism.

∎

###  3.3. Infinitesimal rigidity

In this section we prove Proposition 3.10.

Let $\mathbf{p}=(\rho,\tau,f)\in\mathcal{P}_{c}(\rho,V)$ and
$\dot{\mathbf{p}}\in T_{\mathbf{p}}\mathcal{P}_{c}(\rho,V)$ be such that
$d\mathcal{I}_{\rho}(\dot{\mathbf{p}})=0$. Denote $\mathcal{I}(\mathbf{p})$ by
$d$ and denote a hyperbolic metric on $S$ with holonomy $\rho$ by $h$. Choose
a triangulation $\mathscr{T}$ subdividing the face cellulation of
$\mathbf{p}$. Pick a small enough neighborhood $U$ of $\mathbf{p}$ and for
every $\mathbf{p}^{\prime}=(\rho,\tau^{\prime},f^{\prime})\in U$ consider the
polyhedral surface in $\Omega(\rho,\tau^{\prime})$ with the combinatorics
given by $\mathscr{T}$. Provided that $U$ is small enough, such a surface is
an embedded Cauchy surface, though possibly non-convex. Its edge-lengths and
oriented dihedral angles are analytic functions over $U$. By Claim 2 in the
proof of Lemma 2.13, $\dot{\mathbf{p}}$ induces zero derivatives of all edge-
lengths. We will prove that the induced variations of the (oriented) dihedral
angles of $\mathscr{T}$ are zero. One can see that this implies that
$\dot{\mathbf{p}}$ is zero.

Let $\mathscr{G}$ be the trivalent dual graph to $\mathscr{T}$. The
dualization (i.e. the image by the Gauss map of $\partial{\rm
conv}(\mathbf{p})$) realizes $\mathscr{G}$ as a geodesic graph on $(S,h)$
although some edges, which correspond to dihedral angles equal to $\pi$, have
zero length. From now and until the end of this section we will work in
$(S,h)$. We will change now our notation and until the end of the section by
$V$, $E$ and $F$ we will denote the sets of vertices, edges and faces of
$\mathscr{G}$ rather than of $\mathscr{T}$. Note that every connected
component of the subgraph of the edges of zero length is a tree. We consider
all the edges oriented arbitrarily. By corner we will mean an angle of a face
of $\mathscr{G}$, more exactly, we will use the word _corner_ to denote it as
a combinatorial object, and the word _angle_ to denote its angle-measure. By
$Q$ we denote the set of corners of $\mathscr{G}$.

Consider a pullback $\tilde{\mathscr{G}}$ of $\mathscr{G}$ to
$\tilde{S}\cong\mathbb{H}^{2}\subset\mathbb{R}^{2,1}$. By
$\tilde{V},\tilde{E}$ and $\tilde{Q}$ we denote its sets of vertices, edges
and corners respectively. For an edge $\tilde{e}\in\tilde{E}$ we will abuse
the notation and will denote its unit oriented normal (with respect to the
chosen orientation of the edge and the orientation of the surface), considered
as a point in $\mathbb{R}^{2,1}$, also by $\tilde{e}$. Note that, because the
angles between edges are known, points of $\mathbb{R}^{2,1}$ defined by edges
of zero length are also well-defined. We will use the similar convention for
the vertices $\tilde{v}\in\tilde{V}$, denoting by $\tilde{v}$ also the
respective point in $\mathbb{H}^{2}\subset\mathbb{R}^{2,1}$. For a corner
$\tilde{q}$ by $\alpha_{\tilde{q}}$ we denote its angle. We have all
$\alpha_{\tilde{q}}\in(0,\pi)$. Particularly, for any two adjacent edges, the
(non-oriented) lines determined by them are distinct. For an edge $\tilde{e}$
by $l_{\tilde{e}}$ we denote its length.

The deformation $\dot{\mathbf{p}}$ induces infinitesimal deformations
$\dot{\tilde{e}}$ on all edges $\tilde{e}\in\tilde{E}$. Here we consider the
latter as points in $\mathbb{R}^{2,1}$, and we consider $\dot{\tilde{e}}$ as
vectors in $\mathbb{R}^{2,1}$. These deformations are equivariant with respect
to $\rho$. Because $d\mathcal{I}_{\rho}(\dot{\mathbf{p}})=0$, the induced
deformations $\dot{\alpha}$ of all corner angles are zero. Our aim is to show
that then the induced deformations of all lengths are zero too. The remainder
of the present section is aimed to prove a slight generalization of Iskhakov’s
Theorem 1.3.

######  Remark 3.12.

In the case when $\mathscr{G}$ does not have edges of zero length, Theorem 1.3
was basically proven by Iskhakov in [Isk00]. We need to make two remarks.
First, we need to modify Iskhakov’s proof to allow edges of zero length.
Second, unfortunately it seems that there is a mistake in Iskhakov’s proof.
Iskhakov claims that his result holds also when the face angles can be bigger
than $\pi$. The first step of his proof is to eliminate enough edges to leave
only one face, and then he gives a proof only for this case. Because we can do
the elimination in such a way that any given edge can be kept, the one-face
case implies the general case. In the bottom of p.56 in [Isk00] Iskhakov
writes that his coefficients $\lambda$ are positive numbers, which is crucial
for him to finish the proof. However, this is wrong exactly when there are
non-convex angles, so Iskhakov’s proof does not seem to work in the non-convex
case. This also makes the reduction to the one-face case invalid, as it is
unclear how to guarantee that the resulting single face is convex, especially
with the task to keep any given edge. However, it does not seem hard to fix
Iskhakov’s proof for convex initial cellulations. Because of these remarks,
and because Iskhakov’s proof remains formally unpublished, we decided to write
an exposition of his proof. Our strategy is slightly different from
Iskhakov’s, as we do not make a reduction to the one-face case, but all the
main ideas remain his.

Recall that our edges are oriented. For a vertex $\tilde{v}\in\tilde{V}$ and
an adjacent edge $\tilde{e}\in\tilde{E}$ we define the number
$\tilde{\sigma}_{\tilde{v}\tilde{e}}$ equal to $1$ if $\tilde{v}$ is the
source of $\tilde{e}$ and equal to $-1$ if $\tilde{v}$ is the sink of
$\tilde{e}$. Hence, if $\tilde{e}$ is adjacent to $\tilde{v}$ and $\tilde{w}$,
then
$\tilde{\sigma}_{\tilde{w}\tilde{e}}=-\tilde{\sigma}_{\tilde{v}\tilde{e}}$.
Every corner $\tilde{q}\in\tilde{Q}$, considered with clockwise orientation,
can be denoted by an ordered pair of edges, which we denote
$\tilde{e}_{\tilde{q}-}$ and $\tilde{e}_{\tilde{q}+}$. By
$\tilde{\lambda}_{\tilde{v}\tilde{e}}$ we denote the number
$\tan(\frac{\alpha_{\tilde{q}_{1}}}{2})+\tan(\frac{\alpha_{\tilde{q}_{2}}}{2})$,
where $\tilde{q}_{1}$ and $\tilde{q}_{2}$ are the two corners determined by
the edge $\tilde{e}$ at the vertex $\tilde{v}$. Note that by convexity, the
corner angles are less than $\pi$, which implies that the $\tilde{\lambda}$
are positive.

For every vertex, edge and corner of $\mathscr{G}$ we choose an arbitrary lift
to $\tilde{S}$ and, abusing the notation, use these lifts to identify $V,E$
and $Q$ as subsets of $\tilde{V},\tilde{E}$ and $\tilde{Q}$ respectively. We
have the following system of equations:

$(\mathcal{S})\;\;\begin{cases}\langle\dot{e},e\rangle=0&\forall e\in E\\\
\langle\tilde{e}_{q+},\dot{\tilde{e}}_{q-}\rangle+\langle\dot{\tilde{e}}_{q+},\tilde{e}_{q-}\rangle=0&\forall
q\in Q\\\
\langle\sum_{\tilde{e}\in\tilde{E}_{v}}\tilde{\lambda}_{v\tilde{e}}\tilde{\sigma}_{v\tilde{e}}\dot{\tilde{e}},v\rangle=0&\forall
v\in V\end{cases}$

Here $\langle\cdot,\cdot\rangle$ is the Minkowski bilinear form, and the first
type of equations accounts for the fact that $e$ remains to be unit under the
deformation. The second type is responsible for that the angles of the corners
do not change. The third type comes from the fact that around a vertex $v$,
the lines containing elements of $\tilde{E}_{v}$ (the set of edges of
$\tilde{\mathscr{G}}$ adjacent to $v$) remain concurrent. Indeed, it is easy
to compute that since the oriented normals are coplanar Euclidean unit
vectors, we have
$\sum_{\tilde{e}\in\tilde{E}_{v}}\tilde{\lambda}_{v\tilde{e}}\tilde{\sigma}_{v\tilde{e}}{\tilde{e}}=0$
(see, e.g., [Isk00, Lemma A.2]). We want to prove the following.

######  Lemma 3.13.

The zero vector is the only solution of $(\mathcal{S})$.

Note that the proof below does not assume that the graph is trivalent,
moreover this generality in needed for the case of the presence of edges of
zero length.

System $(\mathcal{S})$ is a linear homogeneous system of $|E|+|Q|+|V|$
equations in variables $\dot{e}$ for $e\in E$. Indeed, every $\dot{\tilde{e}}$
for $\tilde{e}\in\tilde{E}$ is obtained from some $\dot{e}$, where $e\in E$,
by the application of an isometry of $\mathbb{R}^{2,1}$, given by $\rho$,
hence $\dot{\tilde{e}}$ is a linear combination of coefficients of $\dot{e}$.
As every edge is adjacent to 4 corners and every corner is adjacent to 2
edges, we actually have $3|E|+|V|$ parameters.

Denote the matrix of the system $(\mathcal{S})$ by $\mathcal{R}$ and consider
the adjoint matrix $\mathcal{R}^{*}$. It is the matrix of a linear operator
from $\mathbb{R}^{E\cup Q\cup V}$ to $\mathbb{R}^{3E}$. Consider an element of
its kernel. We denote its components by $a_{e},e\in E$, $b_{q},q\in Q$, and
$c_{v},v\in V$. Every edge $e$ determines a vector equation on these numbers.

Figure 2. A four-legged picture of the four corners around the edge $e$.

For every edge $e\in E$ we consider a “four-legged picture” (see Figure 2).
Denote the vertices of $e$ by $\tilde{v}_{+}$ and $\tilde{v}_{-}$. Note that
we do not make an assumption on the orientation of $e$ with respect to the
vertices. Denote the edges forming corners with $e$ at $\tilde{v}_{+}$ by
$\tilde{e}_{++}$ and $\tilde{e}_{+-}$ clockwise, and forming corners at
$\tilde{v}_{-}$ by $\tilde{e}_{-+}$, $\tilde{e}_{--}$ counterclockwise. For
simplicity, we denote the numbers corresponding to this picture by $b_{++}$,
$b_{+-}$, $b_{-+}$, $b_{--}$, $c_{+}$, $c_{-}$, $\tilde{\lambda}_{+}$,
$\tilde{\lambda}_{-}$, $\tilde{\sigma}_{+}$, $\tilde{\sigma}_{-}$
respectively. By noting that the equations in $(\mathcal{S})$ are invariant
for hyperbolic isometries, we get the following equation from
$\mathcal{R}^{*}$

$a_{e}e+b_{++}\tilde{e}_{++}+b_{+-}\tilde{e}_{+-}+b_{-+}\tilde{e}_{-+}+b_{--}\tilde{e}_{--}+c_{+}\tilde{\lambda}_{+}\tilde{\sigma}_{+}\tilde{v}_{+}+c_{-}\tilde{\lambda}_{-}\tilde{\sigma}_{-}\tilde{v}_{-}=0~{}.$
(7)

Denote also
$\tilde{\sigma}_{++}:=\tilde{\sigma}_{\tilde{v}_{+}\tilde{e}_{++}}$,
$\tilde{\sigma}_{+-}:=\tilde{\sigma}_{\tilde{v}_{+}\tilde{e}_{+-}}$, and
denote the angles of the respective corners by $\alpha_{++}$ and
$\alpha_{+-}$. Note that

$\begin{split}&\langle\tilde{v}_{-},\tilde{\sigma}_{++}\tilde{e}_{++}\rangle=-\sin(\alpha_{++})\sinh(\ell_{e})~{},\\\
&\langle\tilde{v}_{-},\tilde{\sigma}_{+-}\tilde{e}_{+-}\rangle=\sin(\alpha_{+-})\sinh(\ell_{e})~{},\end{split}$

where $\ell_{e}$ is the hyperbolic length of $e$. For example for the first
line, we consider an orthogonal projection of $\tilde{v}_{-}$ onto the line
directed by $\tilde{e}_{++}$ in the hyperbolic plane, and use formula for
hyperbolic right-angled triangle, and then use the relation between Minkowski
scalar product and the oriented distance from a point to a geodesic in the
hyperbolic plane, see also [Isk00].

Thus, when we take the Minkowski product of Equation (7) and
$-\tilde{\sigma}_{+}\tilde{v}_{-}$, it turns to

$\sinh(\ell_{e})\left(b_{++}\tilde{\sigma}_{++}\tilde{\sigma}_{+}\sin(\alpha_{++})-b_{+-}\tilde{\sigma}_{+-}\tilde{\sigma}_{+}\sin(\alpha_{+-})\right)+c_{+}\tilde{\lambda}_{+}\cosh(\ell_{e})-c_{-}\tilde{\lambda}_{-}=0~{}.$
(8)

At this point, we need to treat separately the presence of edges of zero
length. Actually, the case without edges of zero length is included in the
case with edges of zero length, but we provide a proof for completeness,
because it is more straightforward, and also because this is the argument the
closest to [Isk00].

###### Proof of Lemma 3.13 when all edges have positive length..

In this case, Equation (8) can be transformed to

$b_{++}\tilde{\sigma}_{++}\tilde{\sigma}_{+}\sin(\alpha_{++})-b_{+-}\tilde{\sigma}_{+-}\tilde{\sigma}_{+}\sin(\alpha_{+-})+\frac{c_{+}\tilde{\lambda}_{+}\cosh(\ell_{e})-c_{-}\tilde{\lambda}_{-}}{\sinh(\ell_{e})}=0~{}.$
(9)

This expression depends only on the choice of the image of $\tilde{v}_{+}$ in
$V$ and on $e$. Now for $v\in V$ by $E_{v}$ we denote the set of edges of
$\mathscr{G}$ adjacent to $v$. We need to add one technical peculiarity: if
$e\in E$ is a loop in $\mathscr{G}$, then we consider two copies of it in
$E_{v}$. We then mark these copies by the orientation of $e$ at $v$. For $e\in
E_{v}$, define

$\lambda_{ve}=\tilde{\lambda}_{\tilde{v}\tilde{e}}~{},\;\sigma_{ve}=\tilde{\sigma}_{\tilde{v}\tilde{e}}$
(10)

for adjacent lifts $\tilde{v}$ of $v$ and $\tilde{e}$ of $e$. As here $e\in
E_{v}$, it can be one of two copies of a loop and thus a loop contributes two
times. By $E_{vw}$ we denote the set of edges adjacent, as edges of
$\mathscr{G}$, to both $v,w\in V$. Fixing $v\in V$ and summing the Equations
(9) over all edges incident to $v$ we see that the components $b$ get
eliminated and we obtain a system of equations on the components $c$ only:

$c_{v}\sum_{e\in
E_{v}}\lambda_{ve}\frac{\cosh(\ell_{e})}{\sinh(\ell_{e})}-\sum_{w\in
V}c_{w}\sum_{e\in E_{vw}}\lambda_{we}\frac{1}{\sinh(\ell_{e})}=0\,\forall v\in
V~{}.$ (11)

The matrix for this system is a $(|V|\times|V|)$-matrix, which we denote by
$\mathcal{X}$. It has positive coefficients on the diagonal, non-positive
coefficients in other entries, and has a positive sum of the entries in each
column. Indeed, for the column corresponding to a vertex $v$ the sum of the
coefficients is

$\sum_{e\in E_{v}}\lambda_{ve}\frac{\cosh(\ell_{e})-1}{\sinh(\ell_{e})}>0~{}.$
(12)

Then $\mathcal{X}$ is an instance of what is called a strictly diagonally
dominant matrix. It is easy to check that it has a non-zero determinant, see
e.g. [Isk00, Appendix B]. Thereby, all the components $c_{v}$ are zero.

This implies that Equation (8) turns to

$b_{++}\tilde{\sigma}_{++}\sin(\alpha_{++})=b_{+-}\tilde{\sigma}_{+-}\sin(\alpha_{+-})~{}.$
(13)

For a given vertex $v\in V$ and any corner $q$ incident to $v$ in
$\mathscr{G}$, Equation (13) says that the expression
$b_{q}\sigma_{q+}\sigma_{q-}\sin(\alpha_{q})$ is independent on $q$, where the
meaning of the notation $\sigma_{q\pm}$ is obvious. Hence, there exists a real
number $t_{v}$ such that
$b_{q}=\frac{t_{v}}{\sigma_{q+}\sigma_{q-}\sin(\alpha_{q})}$. The components
$a_{e}$ are expressed in terms of $t_{v}$ from Equation (7). Thus, the nullity
of $\mathcal{R}^{*}$ is at most $|V|$, but it is also at least $|V|$, hence it
is equal to $|V|$. Then the rank of $\mathcal{R}^{*}$, as well as the rank of
$\mathcal{R}$, is $3|E|$, thereby the nullity of $\mathcal{R}$ is zero. ∎

###### Proof of Lemma 3.13 in the general case..

By $E_{0}\subset E$ we denote the subset of edges of zero length. First, note
that if $e\in E_{0}$, then Equation (8) changes to

$c_{+}\tilde{\lambda}_{+}-c_{-}\tilde{\lambda}_{-}=0~{}.$ (14)

We still sum it together with the other equations for each vertex as for (11),
and obtain the matrix $\mathcal{X}$. Namely, due to (14), the equations
defining $\mathcal{X}$ are, $\forall v\in V$, using notation from (10),

$c_{v}\left(\sum_{e\in E_{v}\backslash
E_{0}}\lambda_{ve}\frac{\cosh(\ell_{e})}{\sinh(\ell_{e})}+\sum_{e\in E_{v}\cap
E_{0}}\lambda_{ve}\right)-\sum_{w\in V}c_{w}\left(\sum_{e\in E_{vw}\backslash
E_{0}}\lambda_{we}\frac{1}{\sinh(\ell_{e})}+\sum_{e\in E_{vw}\cap
E_{0}}\lambda_{we}\right)=0~{}.$ (15)

The sum of the coefficients in a column is now equal to

$s_{v}:=\sum_{e\in E_{v}\backslash
E_{0}}\lambda_{ve}\frac{\cosh(\ell_{e})-1}{\sinh(\ell_{e})}~{},$ (16)

which is zero when $v$ is adjacent only to edges of zero length, then we
cannot directly conclude that $\mathcal{X}$ has non-zero determinant. However,
$\mathcal{X}$ is still non-degenerate. Indeed, consider a non-zero vector
$c\in\mathbb{R}^{V}$, and let $c_{v}$ be such that $|c_{v}|$ is maximal among
the coefficients of $c$. Without loss of generality, let us suppose that
$c_{v}$ is positive. Let $x_{vw}$ be the coefficients of $\mathcal{X}^{T}$.
There are two cases.

  * •

If there exists an edge $e$ adjacent to $v$ and with positive length, then
$s_{v}>0$. On the other hand, $\left(\mathcal{X}^{T}c\right)_{v}\geq
c_{v}s_{v}>0$, i.e. $c$ is not in the kernel of $\mathcal{X}^{T}$.

  * •

Suppose now that all edges adjacent to $v$ have zero length, i.e. $s_{v}=0$
i.e. $x_{vv}=-\sum_{w\not=v}x_{vw}$ and suppose that $c$ is in the kernel of
$\mathcal{X}^{T}$, i.e. $x_{vv}c_{v}=-\sum_{w\not=v}x_{vw}c_{w}$. This is
impossible if $c_{v}>c_{w}$ when $x_{vw}\not=0$, then for $w\in V$ adjacent to
$v$, we have $|c_{v}|=|c_{w}|$. As the graph of $\mathscr{G}$ is connected,
and as there are edges of positive length, we are back to the previous case.

In turn, the kernel of $\mathcal{X}^{T}$ is trivial, so is the kernel of
$\mathcal{X}$.

Now we have to study the coefficients $b_{q}$. A difficulty is that we do not
have Equations (13) for edges of zero length.

Consider the restricted matrix $\mathcal{R}_{r}$ obtained from $\mathcal{R}$
be deleting all the equations of the vertex-type. Define $\bar{E}:=E\backslash
E_{0}$. By $\bar{V}$ we denote the set of visible vertices, i.e., the vertices
of the quotient of $\mathscr{G}$ obtained by contracting all edges of zero
length. Similarly, by $\bar{Q}$ denote the set of visible corners. Consider
the matrix $\bar{\mathcal{R}}_{r}$ coming from the system of equations

$\begin{cases}\langle\dot{e},e\rangle=0&\forall e\in\bar{E}\\\
\langle\tilde{e}_{q+},\dot{\tilde{e}}_{q-}\rangle+\langle\dot{\tilde{e}}_{q+},\tilde{e}_{q-}\rangle=0&\forall
q\in\bar{Q}\end{cases}~{}.$

By the argument from the proof of the non-zero edge length case,
$\dim(\ker(\bar{\mathcal{R}}_{r})\cap\mathbb{R}^{\bar{E}})\leq|\bar{V}|$. Due
to Lemma 3.14, all rows of $\bar{\mathcal{R}}_{r}$ are linear combinations of
the rows of $\mathcal{R}_{r}$, see Figure 3. Hence, also

$\dim(\ker(\mathcal{R}_{r})\cap\mathbb{R}^{\bar{E}})\leq|\bar{V}|~{}.$

Figure 3. On the left picture are two corners defining rows in
$\mathcal{R}_{r}$, the doted edge between the two vertices having zero length.
On the right is the corresponding corner defining a row in
$\bar{\mathcal{R}}_{r}$. Lemma 3.14 says that those rows are linearly
dependent.

We claim that $\dim(\ker(\mathcal{R}_{r}))\leq|\bar{V}|+|E_{0}|$. Indeed,
order all edges in $E_{0}$ as $e_{1},\ldots,e_{k}$ such that every $e_{i}$ is
adjacent to some $e\in\bar{E}_{i-1}:=\bar{E}\cup\\{e_{1},\ldots,e_{i-1}\\}$,
and pick a corner $q_{i}$ adjacent to $e_{i}$ and to such $e$. Consider the
matrix $\bar{\mathcal{R}}_{r,1}$ obtained from $\bar{\mathcal{R}}_{r}$ by
adding the equations

$\begin{split}&\langle\dot{e}_{1},e_{1}\rangle=0~{},\\\
&\langle\tilde{e}_{q_{1}+},\dot{\tilde{e}}_{q_{1}-}\rangle+\langle\dot{\tilde{e}}_{q_{1}+},\tilde{e}_{q_{1}-}\rangle=0~{}.\end{split}$

Clearly, the corresponding two rows are linearly independent with each other
and with all the previous rows: the entries corresponding to $\dot{e}_{1}$
constitute two linearly independent vectors, while for the previous rows those
are zero-vectors. Hence, we have

$\dim(\ker(\bar{\mathcal{R}}_{r,1})\cap\mathbb{R}^{\bar{E}_{1}})\leq|\bar{V}|+1~{}.$

By continuing similarly, we obtain matrices $\bar{\mathcal{R}}_{r,i}$,
$i=1,\ldots,k$, and in the end we get
$\dim(\ker(\bar{\mathcal{R}}_{r,k}))\leq|\bar{V}|+|E_{0}|~{}.$ But all the
rows of $\bar{\mathcal{R}}_{r,k}$ are linear combinations of the rows of
$\mathcal{R}_{r}$. Hence,

$\dim(\ker(\mathcal{R}_{r}))\leq|\bar{V}|+|E_{0}|=|V|~{}.$

It follows that the nullity of $\mathcal{R}$ is zero and the proof of Lemma
3.13 is finished. ∎

During the proof of Lemma 3.13, we used the following fact.

######  Lemma 3.14.

Let $e_{1}$, $e_{2}$, $e_{3}$ be three pairwise distinct hyperbolic lines
intersecting at a single point and $\dot{e}_{1}$, $\dot{e}_{2}$, $\dot{e}_{3}$
be their infinitesimal deformations inducing zero-deformations on the angle
between $e_{1}$ and $e_{2}$ and on the angle between $e_{2}$ and $e_{3}$. Then
the induced deformation on the angle between $e_{1}$ and $e_{3}$ is also zero.

Note that here the infinitesimal deformations are not assumed to preserve the
concurrence of the lines.

###### Proof.

We orient the lines arbitrarily and use the same notation $e_{i}$ for their
unit oriented normals in $\mathbb{R}^{2,1}$. Then the deformations
$\dot{e}_{i}$ also can be considered as vectors in $\mathbb{R}^{2,1}$. Since
the lines intersect at a point, for some
$\lambda_{1},\lambda_{2},\lambda_{3}\neq 0$ we have

$\lambda_{1}e_{1}+\lambda_{2}e_{2}+\lambda_{3}e_{3}=0~{}.$

Then

$\begin{split}\langle\dot{e}_{1},e_{3}\rangle=\langle\dot{e}_{1},-\frac{\lambda_{2}}{\lambda_{3}}e_{2}\rangle&=\frac{\lambda_{2}}{\lambda_{3}}\langle\dot{e}_{2},e_{1}\rangle=\frac{\lambda_{2}}{\lambda_{3}}\langle\dot{e}_{2},-\frac{\lambda_{3}}{\lambda_{1}}e_{3}\rangle=\\\
&=\frac{\lambda_{2}}{\lambda_{1}}\langle\dot{e}_{3},e_{2}\rangle=\frac{\lambda_{2}}{\lambda_{1}}\langle\dot{e}_{3},-\frac{\lambda_{1}}{\lambda_{2}}e_{1}\rangle=-\langle\dot{e}_{3},e_{1}\rangle~{}.\end{split}$

∎

###  3.4. Compactness

The aim of this section is to prove Proposition 3.11. We first prove the
subconvergence of the spacetimes in Section 3.4.1, then the subconvergence of
the marked points in Section 3.4.2.

####  3.4.1. Convergence of ambient spacetimes

The aim of this subsection is to prove Lemma 3.18, which gives the
subconvergence of the classes of cocycles when induced distances converge.

######  Lemma 3.15.

Let $\Pi_{0},\Pi_{1},\Pi_{2}\subset\mathbb{R}^{2,1}$ be three pairwise
distinct spacelike planes passing through a common point; $\psi$ be the
intersection line of $\Pi_{1}$ and $\Pi_{2}$; $C$ be the future-convex wedge
formed by $\Pi_{1}$ and $\Pi_{2}$; and $\chi$ be the intersection curve of
$\partial C$ and $\Pi_{0}$. By $\alpha_{+}$ denote the intrinsic angle in
$\partial C$ of $\chi$ at its intersection point with $\psi$ in the future
side from $\Pi_{0}$. Then $\alpha_{+}\geq\pi$.

###### Proof.

By applying a Lorentzian isometry, we can assume that the intersection point
of all three planes is the origin and that the wedge $C$ is symmetric with
respect to the $x_{0}$-axis. This particularly means that the line $\psi$ is
horizontal. Denote the two rays that constitute $\chi$ by
$\chi_{1}\subset\Pi_{1}$ and $\chi_{2}\subset\Pi_{2}$. By
$\chi^{\prime}_{1}\subset\Pi_{2}$ denote the ray obtained from $\chi_{1}$ by
the symmetry in $\mathbb{R}^{2,1}$ with respect to the $x_{0}$-axis. Note that
by symmetry $\chi_{1}$ and $\chi_{1}^{\prime}$ constitute an intrinsic
geodesic in $\partial C$. On the other hand, one can see that
$\chi_{1}^{\prime}$ belongs to the future side of all spacelike planes
containing the ray $\chi_{1}$, particularly of $\Pi_{0}$. Therefore, the
intrinsic angle between $\chi_{1}$ and $\chi_{2}$ in $\partial C$ that is in
the future of $\Pi_{0}$ is $\geq\pi$. ∎

######  Lemma 3.16.

Let $\Pi_{1},\Pi_{2}\subset\mathbb{R}^{2,1}$ be two non-parallel spacelike
planes; $\psi$ be the intersection line of $\Pi_{1}$ and $\Pi_{2}$; $C$ be the
future-convex wedge formed by $\Pi_{1}$ and $\Pi_{2}$; $\Pi_{0}$ be a timelike
plane intersecting $\psi$ transversely; and $\chi$ be the intersection curve
of $\partial C$ and $\Pi_{0}$. Assume that the origin $o$ is the intersection
point of all three planes and that a normal vector $n$ to $\Pi_{0}$ belongs to
$C$. By $\alpha_{+}$ denote the intrinsic angle in $\partial C$ of $\chi$ at
its intersection point with $\psi$ in the opposite side to $n$ with respect to
$\Pi_{0}$. Then $\alpha_{+}\geq\pi$.

###### Proof.

We consider the configuration and the notation as in the proof above. Denote
the horizontal plane by $\Pi$ and the future-directed unit vertical vector by
$n_{v}$. For any timelike plane, if it has a normal vector in the future of
$\Pi$, then it separates this vector from $n_{v}$. Hence, in our situation $n$
and $n_{v}$ cannot belong to the same open halfspace with respect to
$\Pi_{0}$. But $n_{v}$ is in the convex hull of $\chi_{1}$ and
$\chi^{\prime}_{1}$. Thus, since $\Pi_{0}$ contains $\chi_{1}$, $n$ and
$\chi^{\prime}_{1}$ cannot belong to the same open halfspace with respect to
$\Pi_{0}$, which implies the statement of the lemma. ∎

The proofs of the next two lemmas use the ideas of Bonsante from his proof of
Proposition 7.6 in [Bon05b].

######  Lemma 3.17.

Let $C\subset\mathbb{R}^{2,1}$ be a spacelike future convex Cauchy polyhedral
set. Fix an affine system of coordinates such that the origin $o$ is in
$\partial C$ and the horizontal plane $\Pi$ through the origin is a supporting
plane to $C$. Let $\phi:\Pi\rightarrow\mathbb{R}$ be the convex function
defining $\partial C$. Let $p\in\partial C$ and $\psi:[0,T]\rightarrow\partial
C$ be an intrinsic geodesic segment connecting $o$ with $p$. Let
$\psi_{h}:[0,T]\rightarrow\Pi$ be the orthogonal projection of $\psi$ to
$\Pi$. Then $\phi\circ\psi_{h}$ and $\|\psi_{h}\|$ are non-decreasing
functions.

In this section, $\|\cdot\|$ is the Minkowski norm, that gives the Euclidean
norm over $\Pi$.

###### Proof.

The intrinsic metric on $\partial C$ is CAT(0), which implies that a geodesic
segment between any two points is unique. Note also that $\psi$ is a
piecewise-linear map. For $a>0$ define
$D_{a}:=\\{x=(x_{0},x_{1},x_{2})\in\partial C:x_{0}\leq a\\}$. Lemma 3.15
implies that $D_{a}$ is convex in the intrinsic geometry of $\partial C$.
Indeed, it implies directly that at every kink point of $\partial D_{a}$, that
is the intersection of a horizontal plane with an edge of $\partial C$, the
intrinsic angle is $\leq\pi$. But for the kink points that come from the
vertices of $\partial C$ the same follows by continuity. From the convexity of
$D_{a}$ it is easy to see that $\phi\circ\psi_{h}$ is non-decreasing. Indeed,
suppose the converse. Then some $t\in[0,T]$ belongs to a segment where
$\phi\circ\psi_{h}$ is decreasing. Then the intrinsic geodesic from $o$ to
$\psi(t)$ does not completely belong to $D_{\phi\circ\psi_{h}(t)}$. This
contradicts to the convexity of $D_{\phi\circ\psi_{h}(t)}$.

For the second claim, pick $q\in\Pi$ and define the set
$D_{q}:=\\{x\in\partial C:\langle x,q\rangle\leq\|q\|\\}$. We claim that it is
also convex. Indeed, consider the timelike plane

$\Pi_{q}:=\\{x\in\mathbb{R}^{2,1}:\langle x,q\rangle=\|q\|\\}~{}.$

Because $C$ is future-convex and $\Pi$ is a supporting plane to $C$ at $o$,
for every $p\in\partial D_{q}$ the ray from $p$ orthogonal to $\Pi_{q}$,
pointing to the halfspace containing $o$, does not point to the exterior of
$C$. Thereby, for any two supporting planes $\Pi_{1}$ and $\Pi_{2}$ to $C$
that form an edge of $C$ intersecting $\partial D_{q}$, Lemma 3.16 shows that
the intrinsic angle at the corresponding kink point of $\partial D_{q}$ is
$\leq\pi$. The same holds by continuity for all kink points of $\partial
D_{q}$. It follows that $D_{q}$ is convex.

Suppose now that $\|\psi_{h}\|$ is somewhere decreasing. Then some $t\in[0,T]$
belongs to a segment where $\|\psi_{h}\|$ is decreasing. Then the geodesic
from $o$ to $\psi(t)$ does not completely belong to $D_{\psi_{h}(t)}$, which
contradicts to the convexity of the latter. ∎

######  Lemma 3.18.

Let $\mathbf{p}_{i}=(\rho,\tau_{i},f_{i})$ be a sequence in
$\mathcal{P}_{c}(\rho,V)$ such that the induced metrics
$\mathcal{I}(\mathbf{p}_{i})=d_{i}\in\mathcal{\mathcal{M}}_{-}(S,V)$ converge
to a metric $d\in\mathcal{\mathcal{M}}_{-}(S,V)$. Then, up to a subsequence,
the cohomology classes $\tau_{i}\in\mathcal{H}^{1}(\rho)$ converge.

###### Proof.

Fix $v_{0}\in\tilde{V}$ and choose a sequence representative of the classes of
cocycles, which we will still denote by $\tau_{i}$, such that for the
corresponding maps
$\tilde{f}_{i}:\tilde{V}\rightarrow\tilde{\Omega}^{+}(\rho,\tau_{i})$ we have
$\tilde{f}_{i}(v_{0})=o$, where $o$ is the origin, and the horizontal plane
$\Pi$ through $o$ is a supporting plane to all $\widetilde{\rm
conv}(\mathbf{p}_{i})$. Up to passing to a subsequence, we may assume that
$\tau_{i}$ converge to a map
$\overline{\tau}:\pi_{1}S\rightarrow\overline{\mathbb{R}^{2,1}}$. Pick a basis
$\Xi$ for $\pi_{1}S$. If for all $\gamma\in\Xi$ we have
$\overline{\tau}(\gamma)\in\mathbb{R}^{2,1}$, then $\overline{\tau}$ is a
cocycle, determining the desired $\tau\in\mathcal{H}^{1}(\rho)$.

Suppose that for some $\gamma\in\Xi$ we have
$\overline{\tau}(\gamma)\in\partial_{\infty}\mathbb{R}^{2,1}$. The rest of the
proof is devoted to finding a contradiction to this.

Let $K\subset\mathbb{R}^{2,1}$ be the Euclidean unit ball centered at $o$.
Consider a sequence of positive numbers $\lambda_{i}$ such that for all
$\gamma\in\Xi$ we have $\lambda_{i}\tau_{i}(\gamma)\in K$ and for some
$\gamma_{i}\in\Xi$ we have $\lambda_{i}\tau_{i}(\gamma_{i})\in\partial K$.
Then the sequence $\lambda_{i}$ converges to zero and, up to passing to a
subsequence, $\lambda_{i}\tau_{i}$ converge to a cocycle
$\tau_{\lambda}\in\mathcal{Z}^{1}(\rho)$ such that for some $\gamma_{0}\in\Xi$
we have $\tau_{\lambda}(\gamma_{0})\in\partial K$, particularly
$\tau_{\lambda}(\gamma_{0})\neq 0$.

By $\overline{C}_{i}$ we denote the closures of $C_{i}:=\widetilde{\rm
conv}(\lambda_{i}\mathbf{p}_{i})$ in $\overline{\mathbb{R}^{2,1}}$. Up to
passing to a subsequence, the sets $\overline{C}_{i}$ converge in the
Hausdorff sense to a closed convex set
$\overline{C}\subset\overline{\mathbb{R}^{2,1}}$ (here we mean the Hausdorff
convergence with respect to the topology of $\overline{\mathbb{R}^{2,1}}$),
and the maps $\lambda_{i}\tilde{f}_{i}$ converge to a map
$\tilde{f}:\tilde{V}\rightarrow\partial\overline{C}$. Because for all $i$ we
have $o\in\partial C_{i}$ and
$\partial^{0}_{\infty}\mathbb{R}^{2,1}\subset\partial\overline{C}_{i}$, the
set $C:=\overline{C}\cap\mathbb{R}^{2,1}$ is non-empty, and is a future convex
set. Note that $\Pi$ is a supporting plane to $C$, which implies that for all
$\gamma\in\pi_{1}S$, the vector $\tau_{\lambda}(\gamma)$ is spacelike or zero.
Indeed, if for some $\gamma\in\pi_{1}S$ this is wrong, then
$\tau_{\lambda}(\gamma^{-1})$ does not belong to the future side of $\Pi$, as
for all $i$ we have $\tau_{i}(\gamma^{-1})=-\rho(\gamma)\tau_{i}(\gamma)$.
Because $C$ is future convex, $\partial C$ is a graph over $\Pi$. Because
$\lambda_{i}$ converge to zero, the metrics $\lambda_{i}d_{i}$ on $\tilde{S}$
also converge to zero.

By $\phi_{i},\phi:\Pi\rightarrow\mathbb{R}$ we will denote the convex
functions defining $\partial C_{i}$, $\partial C$. Let $\psi_{i}$ be the
geodesic segment in $\partial C_{i}$ connecting $o$ with
$\lambda_{i}\tilde{f}_{i}(\gamma_{0}v_{0})$. By definition, this sequence of
points converges to the point
$\tilde{f}(\gamma_{0}v_{0})=\tau_{\lambda}(\gamma_{0})\in\partial K$. Let
$\psi_{h,i}$ be the orthogonal projection of $\psi_{i}$ onto $\Pi$, and let
$\ell_{i}$ be its length. Note that since $\tilde{f}(\gamma_{0}v_{0})$ is in
$\partial K$ and is spacelike, $\ell_{i}$ is bounded away from zero. We remark
that either the sequence $\ell_{i}$ is bounded, or there is a subsequence that
goes to infinity. We will see below that any of these cases leads to the
wanted contradiction.

Suppose first that $\ell_{i}$ are bounded. Then, up to passing to a
subsequence, $\ell_{i}$ converge to a positive number $\ell$. Abusing
notation, we will denote by $\psi_{h,i}$ and $\psi_{i}$ the parametrization
proportional to arc-length over $[0,\ell]$ of the respective paths, with $0$
sent to $o$. By Arzelà–Ascoli, the $\psi_{h,i}$ converge to a path
$\psi_{h}:[0,\ell]\rightarrow\Pi$, with length $\ell_{0}\leq\ell$. By
$\psi:[0,\ell]\rightarrow\partial C$ we denote its lift to $\partial C$, which
is also a Lipschitz path. Since the metrics $\lambda_{i}d_{i}$ converge to
zero, the lengths of the paths $\psi_{i}$ converge to zero. We have

$\lim_{i\rightarrow+\infty}\int_{0}^{\ell}\|\psi_{i}^{\prime}(t)\|dt=0~{}.$

Therefore, $\lim_{i\rightarrow\infty}\|\psi_{i}^{\prime}(t)\|=0$ for almost
all $t\in[0,\ell]$. By Lemma 3.17, the numbers
$(\phi_{i}\circ\psi_{h,i})^{\prime}(t)$ are non-negative where defined (which
is almost everywhere). As by definition,
$\|\psi_{h,i}^{\prime}(t)\|=\frac{\ell_{i}}{\ell}$ almost everywhere, and as
$\|\psi_{i}^{\prime}(t)\|^{2}=\|\psi_{h,i}^{\prime}(t)\|^{2}-((\phi_{i}\circ\psi_{h,i})^{\prime}(t))^{2}$
almost everywhere, it follows that almost everywhere,
$\lim_{i\rightarrow\infty}(\phi_{i}\circ\psi_{h,i})^{\prime}(t)=1$. By the
Dominated Convergence Theorem, $\phi_{i}\circ\psi_{h,i}(t)$ converges then to
$t$. But it also converges to $\phi\circ\psi_{h}(t)$, so the latter is equal
to $t$. On the other hand,

$\|\psi_{h}(t)\|\leq{\rm len}(\psi_{h}|_{[0,t]})\leq\liminf_{i}{\rm
len}(\psi_{h,i}|_{[0,t]})=\lim_{i}t\frac{\ell_{i}}{\ell}=t~{}.$

We see that for every $t$ the Minkowski vector $\psi(t)$ is causal.
Particularly so is
$\psi(\ell)=\tilde{f}(\gamma_{0}v_{0})=\tau_{\lambda}(\gamma_{0})$. This is a
contradiction as $\tau_{\lambda}(\gamma_{0})$ must be spacelike.

Suppose now that up to passing to a subsequence, $\ell_{i}\to\infty$. Note
however that the endpoint of $\psi_{i}$ is in $K$, and by Lemma 3.17, for an
arc-length parametrization, $\|\psi_{h,i}\|$ are non-decreasing for every $i$.
Hence, the images of $\psi_{h,i}$ belong to the compact set $K\cap\Pi$. Then,
up to a subsequence again, the maps $\psi_{h,i}$ converge uniformly on compact
sets to a Lipschitz map $\psi_{h}:[0,+\infty)\rightarrow K\cap\Pi$. More
precisely, let us parametrize all the $\psi_{h,i}$ by arc-lengths. Up to a
finite number of elements, we assume that all lengths are greater than 2.
Using Arzelà–Ascoli we choose a subsequence of $\psi_{h,i}$ such that
$\psi_{h,i}|_{[0,2]}$ converges to a 1-Lipschitz path $\psi_{h}|_{[0,2]}$.
Starting with this subsequence, we reproduce the same procedure, with $3$
instead of $2$, and the limiting path $\psi_{h}|_{[0,3]}$ coincides with
$\psi_{h}|_{[0,2]}$ over $[0,2]$ by uniqueness of the limit. We repeat this
countably many times to define $\psi_{h}$ over $[0,+\infty)$.

Denote its lift to $\partial C$ by $\psi:[0,+\infty)\rightarrow\partial C$.
Similarly to the previous case we get $\phi\circ\psi_{h}(t)=t$. Since
$\psi_{h}(t)$ belongs to $K$, we have $\|\psi_{h}(t)\|\leq 1$ for all $t$.
Therefore, for some $t$ the Minkowski vector $\psi(t)\in\partial C$ is
timelike. This contradicts to the fact that $C$ is a future convex set.

At the end of the day, such a sequence $(\ell_{i})_{i}$ cannot exist, and we
have arrived at the wanted contradiction. ∎

####  3.4.2. Convergence of marked points

######  Lemma 3.19.

For every $\alpha>0$ and every compact set $U\subset\mathcal{H}^{1}(\rho)$
there exists a constant $\beta\geq\alpha$ such that for every convex Cauchy
surface $\Sigma\subset\Omega^{+}(\rho,\tau)$, $\tau\in U$, if the infimum of
the cosmological time over $\Sigma$ is at most $\alpha$, then its supremum is
at most $\beta$.

###### Proof.

This lemma in the case when $U$ is reduced to a single point is proved in
[BF17, Corollary 3.19]. Now let us consider the general case, and suppose that
the result is false. Hence, there exist $\alpha>0$ and a sequence
$(\tau_{i})_{i}$ of cocycles converging to a cocycle $\tau$ and a sequence of
convex Cauchy surfaces $(\Sigma_{i})_{i}$ with the infimum of cosmological
time bounded above by $\alpha$ and the supremum of the cosmological time going
to $\infty$. We have that $\tilde{\Omega}^{+}(\rho,\tau_{i})$ converge to
$\tilde{\Omega}^{+}(\rho,\tau)$, as well as the respective level sets of the
cosmological time, see [BF20, Lemma 3.27] or Propositions 6.2 and 6.4 in
[Bon05b]. (It is worth noting the relation between support functions and
extremal values of the cosmological time, cf. [BF17, Lemma 3.10].) This
implies that for any sequence of points
$\tilde{p}_{i}\in\tilde{\Omega}^{+}(\rho,\tau_{i})$ converging to
$\tilde{p}\in\tilde{\Omega}^{+}(\rho,\tau)$ we have
$\textsc{ct}_{i}(\tilde{p}_{i})$ converge to $\textsc{ct}(\tilde{p})$.

Let $p_{i}\in\Sigma_{i}\cap L_{\alpha}$. As the convex hull $P_{i}$ of $p_{i}$
is in the future of $\Sigma_{i}$, the cosmological time of $\partial P_{i}$
also goes to infinity. As $L_{\alpha}$ is compact, $(p_{i})_{i}$ subconverges
to a point $p$ in $L_{\alpha}$. From Lemma 2.12, for all sufficiently large
$i$ the boundary cellulation of ${\rm conv}(p_{i})$ is a subdivision of the
boundary cellulation of ${\rm conv}(p)$. From this it is easy to see that
$\partial\widetilde{\rm conv}(p_{i})$ converges to $\partial\widetilde{\rm
conv}(p)$. This and the remark above on the behavior of the cosmological time
imply that ct is unbounded from above on $\partial{\rm conv}(p)$. But by Lemma
2.6, the latter is a convex Cauchy surface, particularly compact, which shows
the contradiction. ∎

Consider a level surface of the cosmological time. It is a
$C^{1}$-submanifold, hence the induced metric is induced by a Riemannian
structure. Its universal covering can be put in the future cone of the origin
of Minkowski space, so that it is locally a graph of a function over the
hyperboloid. We may locally approximate this graph by smooth strictly convex
ones, whose induced metric has negative sectional curvature. Then, following
the proof of Proposition 3.12 in [FS19], the induced distances locally