# Hyperbolic 3-manifolds with boundary of polyhedral type

Roman Prosanov This research was funded by the Austrian Science Fund (FWF)
ESPRIT grant ESP-12-N.

###### Abstract

Let $M$ be a compact orientable 3-manifold with hyperbolizable interior and
non-empty boundary such that all boundary components have genii at least 2. We
study an Alexandrov-Weyl-type problem for convex hyperbolic cone-metrics on
$\partial M$. We consider a class of hyperbolic metrics on M with convex
boundary, which we call _bent_ metrics, and which naturally generalize
hyperbolic metrics on $M$ with convex polyhedral boundary. We show that for
each convex hyperbolic cone-metric $d$ on $\partial M$, with few simple
exceptions, there exists a bent metric on $M$ such that the induced intrinsic
metric on $\partial M$ is $d$. Next, we prove that if a bent realization is
what we call controllably polyhedral, then it is unique up to isotopy. We
exhibit a large subclass of hyperbolic cone-metrics on $\partial M,$ called
_balanced_ , which is open and dense among all convex hyperbolic cone-metrics
in the sense of Lipschitz topology, and for which we show that their bent
realizations are controllably polyhedral. We additionally prove that any
convex realization of a convex hyperbolic cone-metric on $\partial M$ is bent.

Finally, we deduce that there exists an open subset of the space of convex
cocompact metrics on the interior of $M$, including all metrics with
polyhedral convex cores, such that the metrics in this subset are (1) globally
rigid with respect to the induced intrinsic metrics on the boundaries of their
convex cores; (2) infinitesimally rigid with respect to their bending
laminations. This gives partial progress towards conjectures of W. Thurston.

##  1 Introduction

###  1.1 Motivation

Let $P$ be a convex polyhedron in one of the model constant-curvature
3-spaces, $\mathbb{E}^{3}$, $\mathbb{S}^{3}$ or $\mathbb{H}^{3}$. We will say
_model spaces_ for short, and will assume that polyhedra are compact. The
boundary of $P$ is the topological 2-sphere, and it is not hard to see that
the induced intrinsic metric on the boundary is what is called a convex cone-
metric.

######  Definition 1.1.

Let $S$ be a surface. We say that $d$ is a _Euclidean cone-metric_ (resp.
_spherical_ or _hyperbolic_) on $S$ if it is locally isometric to the
Euclidean plane (resp. to the standard sphere or the hyperbolic plane) except
a discrete set of points where it is locally isometric to a Euclidean cone
(resp. spherical or hyperbolic) of total angle $\neq 2\pi$. The exceptional
points are called _vertices_ of $d$. We say that $d$ is _convex_ if all cone
angles are $<2\pi$.

We note that we allow the set of vertices to be empty, in which case $d$ is a
smooth metric of constant curvature $0$ or $\pm 1$.

In [1] Alexandrov proved

######  Theorem 1.2.

Every Euclidean convex cone-metric on the 2-sphere can be realized on the
boundary of a convex polyhedron $P\subset\mathbb{E}^{3}$, which is unique up
to isometry.

Later Alexandrov extended Theorem 1.2 to the other model spaces, see [2] and
Alexandrov’s books [3, 4]. The existence part of Theorem 1.2 is a polyhedral
counterpart to the renowned _Weyl problem_ , which asks if a smooth metric of
positive curvature on the 2-sphere can always be realized as the induced
intrinsic metric on the boundary of a convex body in $\mathbb{E}^{3}$. It was
resolved positively by Nirenberg [66] employing the methods of partial
differential equations (preceded by a proof in the analytic class due to Lewy
[51]), and independently by Alexandrov [3] and Pogorelov [69] as an outcome of
their general theory of (not necessarily smooth) convex surfaces in
$\mathbb{E}^{3}$. The methods of Alexandrov and Pogorelov also cover the
realizations in $\mathbb{S}^{3}$ and $\mathbb{H}^{3}$. The uniqueness part of
Theorem 1.2 states that convex polyhedra in $\mathbb{E}^{3}$ are rigid. This
extends the classical Legendre–Cauchy theorem on the rigidity of polyhedra
with the same combinatorics, and is parallel to the rigidity of smooth convex
bodies in $\mathbb{E}^{3}$ established by Herglotz [42] and preceded by a
proof in the analytic class due to Cohn-Vossen [27].

In 70s Thurston rocked the field of low-dimensional topology by formulating
his geometrization conjecture (whose proof was finalized by Perelman), showing
many its spectacular corollaries and providing non-trivial evidences in its
favor. One of the main parts of the conjecture highlighted the unexpected
ubiquity and diversity of hyperbolic manifolds in dimension 3. This attracted
many geometers to a deeper study of hyperbolic 3-manifolds and their
properties. Some attention was paid to hyperbolic 3-manifolds with convex
boundary. In particular, Labourie proved in [49] an analogue of the Weyl
problem for compact hyperbolic 3-manifolds with boundary. For the rest of the
paper, let $M$ be an oriented smooth compact 3-manifold with non-empty
boundary, distinct from the solid torus and from the 3-ball, admitting a
hyperbolic metric with convex boundary, and let $N$ be the interior of $M$.
The result of Labourie is

######  Theorem 1.3.

For every smooth Riemannian metric $d$ on $\partial M$ of Gaussian curvature
$>-1$ there exists a hyperbolic metric $g$ on $M$ such that the boundary is
convex and the induced intrinsic metric is isometric to $d$.

Thanks to the foundational work of Mostow [64] it is known that the geometry
of closed hyperbolic manifolds in dimension at least 3 is in some sense
determined by their topology. In dimension 3 this was strengthened by
Gabai–Meyerhoff–Thurston [39] to the following result

######  Theorem 1.4.

Every two hyperbolic metrics on a closed 3-manifold are isotopic.

Theorem 1.4 together with the rigidity of smooth convex bodies in
$\mathbb{H}^{3}$ make it reasonable to conjecture that the geometry of compact
hyperbolic 3-manifolds with non-empty smooth convex boundary should be
dictated by their topology and by the intrinsic geometry of the boundary. This
was confirmed by Schlenker in [79] provided that the boundary is strictly
convex, which means that the shape operator of the boundary is everywhere
positive-definite.

######  Theorem 1.5.

Let $g_{1}$, $g_{2}$ be two hyperbolic metrics on $M$ such that the boundary
is smooth, strictly convex and the induced intrinsic boundary metrics are
isotopic. Then $g_{1}$ and $g_{2}$ are isotopic.

We note that in [79] Schlenker also gave another prove of Theorem 1.3 and
considered the toy case of the solid torus.

The results on convex bodies in the model spaces go in parallel in the smooth
and polyhedral realms, with the proofs in the latter case normally simpler
than the proofs in the former. Curiously enough, there was no polyhedral
counterparts of Theorems 1.3 and 1.5, and the main purpose of the present
paper is to fill this gap. One of the main difficulties is that convex
hyperbolic cone-metrics on $\partial M$ admit convex realizations that are not
polyhedral and that are somewhat more difficult to handle. Another difficulty
is that the core proof-ideas of Theorems 1.3 and 1.5 in [49] and [79] are
quite specific to the smooth setting, and do not admit direct polyhedral
generalizations. In the same time, standard approaches to the rigidity of
polyhedra in the model spaces do not have direct generalizations to non-
trivial 3-manifolds with polyhedral boundary. Thus, although our general
framework (_the continuity method_) is similar to the one employed by
Schlenker in [79], in our implementation of all main technical steps we have
to choose a different route to follow.

###  1.2 Our results

Let us specify what we mean by a hyperbolic metric on $M$ with (convex)
polyhedral boundary.

######  Definition 1.6.

We say that a hyperbolic metric $g$ on $M$ is _polyhedral_ if $\partial M$ is
locally convex and each point $p\in\partial M$ has a neighborhood isometric to
$U\cap P$, where $P\subset\mathbb{H}^{3}$ is a convex polyhedron in
$\mathbb{H}^{3}$ and $U\subset\mathbb{H}^{3}$ is an open domain intersecting
$\partial P$.

Now we generalize this definition.

######  Definition 1.7.

We say that a hyperbolic metric $g$ on $M$ is _bent_ if $\partial M$ is
locally convex and each point $p\in\partial M$, except finitely many, belongs
to the relative interior of a geodesic segment. The exceptional points are
called _vertices_ of $g$.

Here and further by a geodesic segment we mean a geodesic segment of $g$. If
we need to consider a geodesic segment in $\partial M$ with respect to the
induced intrinsic metric, possibly bent in $(M,g)$, we will be referring to it
as to an _intrinsic geodesic segment_. It is easy to see that a polyhedral
hyperbolic metric on $M$ is bent. Moreover, one can notice that in the model
spaces a closed convex surface satisfying the bent condition is actually a
polyhedron. This is no longer true for compact hyperbolic 3-manifolds with
boundary. Well-known examples of non-polyhedral bent hyperbolic metrics
without vertices are given by the convex cores of so-called convex cocompact
hyperbolic 3-manifolds, which we define now.

######  Definition 1.8.

A subset $C$ of a complete Riemannian manifold is called _totally convex_ if
it is closed and contains every geodesic segment between any two points of
$C$.

######  Definition 1.9.

A hyperbolic metric $\overline{g}$ on $N$ is called _convex cocompact_ if it
is complete and $(N,\overline{g})$ contains a compact totally convex subset.

It is not hard to see that $M$ admits a hyperbolic metric with convex boundary
if and only if its interior $N$ admits a convex cocompact metric. Moreover,
for every hyperbolic metric $g$ on $M$ with convex boundary there exists a
unique convex cocompact metric $\overline{g}$ on $N$ and a unique isometric
embedding $(M,g)\hookrightarrow(N,\overline{g})$, where the uniqueness is in
some sense, which we will clarify in Section 2.4. We will mostly use the
notation $\overline{g}$ to denote the metric on $N$ obtained from $g$ by this
construction, but sometimes we will just write $\overline{g}$ to denote a
convex cocompact metric on $N$ in the absence of metric $g$. Our restrictions
on $M$ imply that $M$ is irreducible and atoroidal. Since $M$ is distinct from
the solid torus and the 3-ball, this particularly means that all components of
$\partial M$ have genii at least 2. Thurston’s hyperbolization theorem implies
that the converse holds: every compact 3-manifold satisfying these topological
conditions admits a hyperbolic metric with convex boundary, see [55, 46]. In
particular, there are plenty of such 3-manifolds.

######  Definition 1.10.

The _convex core_ $C(\overline{g})$ of a convex cocompact hyperbolic metric
$\overline{g}$ on $N$ is the inclusion-minimal closed totally convex subset of
$(N,\overline{g})$.

We will recall in Section 2.4 that the convex core is well-defined and, except
few degenerate cases, is homeomorphic to $M$. It was shown by Thurston in [88]
that the induced intrinsic metric on $\partial C(\overline{g})$ is hyperbolic,
but $\partial C(\overline{g})$ may be bent in $(N,\overline{g})$ along an
irrational geodesic lamination, in which case it is neither smooth nor
polyhedral. However, $\partial C(\overline{g})$ satisfies our bent condition.
In Section 3.1.2 we will sketch an argument showing the existence of bent
metrics with vertices. We will also prove in that section that the induced
intrinsic metric on the boundary of a bent hyperbolic 3-manifold is a convex
hyperbolic cone-metric.

We need to introduce two important subclasses of polyhedral hyperbolic metrics
on $M$.

######  Definition 1.11.

We say that a polyhedral hyperbolic metric $g$ on $M$ is _strictly polyhedral_
if $\partial M$ is at positive distance from $C(\overline{g})$. We say that a
polyhedral hyperbolic metric $g$ on $M$ is _controllably polyhedral_ if
$\partial M\cap C(\overline{g})$ is empty or consists only of simple closed
geodesics.

In particular, every strictly polyhedral metric is controllably polyhedral. We
also need to distinguish a special class of metrics on $\partial M$.

######  Definition 1.12.

We say that a convex hyperbolic cone-metric $d$ on $\partial M$ is _proto-
Fuchsian_ if $M$ is homeomorphic to $S\times[-1,1]$ for a surface $S$ and
either (1) $S$ is closed, $d$ does not have vertices and the restrictions of
$d$ to the components of $\partial M$ are isotopic as metrics on $S$; or (2)
$S$ is compact with non-empty boundary and $(\partial M,d)$ is the double of a
hyperbolic metric on $S$ with convex piecewise-geodesic boundary and with no
cone-points in the interior of $S$. Otherwise, we say that $d$ is _non-proto-
Fuchsian_.

Note that in case (2) $M$ is a handlebody. Now we can formulate the main
result of our paper.

######  Theorem A.

For each non-proto-Fuchsian convex hyperbolic cone-metric $d$ on $\partial M$
there exists a bent metric $g$ on $M$ such that the induced intrinsic metric
on $\partial M$ is isometric to $d$. If either (1) $g$ is controllably
polyhedral, or (2) $g$ is polyhedral and does not have vertices, then $g$ is
unique up to isotopy (among bent metrics on $M$).

Note that in the smooth case the condition that the Gaussian curvature of a
metric $d$ is $>-1$ implies that a convex realization of $d$ is at positive
distance from the convex core. Hence, Theorem A includes polyhedral
counterparts to Theorems 1.3 and 1.5. When $d$ is proto-Fuchsian, every
realization of $d$ degenerates to a surface, and it is not possible to realize
$d$ by a hyperbolic metric on $M$ with convex boundary. We complement Theorem
A by another result

######  Theorem B.

Let $g$ be a hyperbolic metric on $M$ such that the boundary is convex and the
induced intrinsic metric $d$ on $\partial M$ is a hyperbolic cone-metric. Then
$g$ is bent.

Our approach to Theorem A implies a special corollary, which we present now.
By $\mathcal{CH}(N)$ we denote the space of convex cocompact hyperbolic
metrics on $N$ up to isotopy, endowed with a natural topology, which we define
in Section 2.4. As we mentioned, the induced intrinsic metric on the boundary
of the convex core of a convex cocompact metric is hyperbolic. This defines a
map $\mathcal{I}:\mathcal{CH}(N)\rightarrow\mathcal{T}(\partial M)$, where
$\mathcal{T}(\partial M)$ is the Teichmüller space of $\partial M$. It follows
from the work of Labourie [49] and from an approximation argument that this
map is surjective. (The present paper provides an alternative proof of the
surjectivity of this map as a particular case of Theorem A.) There is another
object called _the bending lamination_ (some authors say _pleating
lamination_) of the convex core measuring how much the boundary is bent, see
Section 3.1.1 for details. (In the polyhedral case this is just the data of
the exterior dihedral angles.) This defines a map
$\mathcal{I}^{*}:\mathcal{CH}(N)\rightarrow\mathcal{ML}(\partial M)$, where
$\mathcal{ML}(\partial M)$ is the space of _measured geodesic laminations_ on
$\partial M$. In this case the map is not surjective, and its image is
described in the works [17, 50] of Bonahon–Otal and Lecuire. Note that the
space $\mathcal{ML}(\partial M)$ lacks a natural differential structure, but
has a natural PL-structure.

Thurston conjectured that $\mathcal{I}$ is a homeomorphism and
$\mathcal{I}^{*}$ is a homeomorphism onto the image outside the Fuchsian
locus. The injectivity of these maps remains an open problem, though it was
shown in [17] that if $\overline{g}$ has a non-degenerate polyhedral convex
core, then $\mathcal{I}^{*}(\overline{g})$ has a single
$\mathcal{I}^{*}$-preimage. By the work [14] of Bonahon, the map $\mathcal{I}$
is $C^{1}$ and the map $\mathcal{I}^{*}$ is tangentiable, which is a
relaxation of the notion of differentiability to the PL-setting. We show

######  Corollary C.

Let $\overline{g}\in\mathcal{CH}(N)$ have non-degenerate polyhedral convex
core. Then there exists a neighborhood $U$ of $\overline{g}$ in
$\mathcal{CH}(N)$ such that
(1) the restriction of $\mathcal{I}$ to $U$ is a $C^{1}$-diffeomorphism onto
the image. Moreover, for each $\overline{g}^{\prime}\in U$ the metric
$\mathcal{I}(\overline{g}^{\prime})$ has a single $\mathcal{I}$-preimage in
$\mathcal{CH}(N)$.
(2) For each $\overline{g}^{\prime}\in U$ the tangent map of $\mathcal{I}^{*}$
is an isomorphism at $\overline{g}^{\prime}$.

The key point here is the infinitesimal rigidity of $\mathcal{I}$ at
$\overline{g}$. Together with the inverse function theorem it implies the
first claim of Part (1). We will show that local rigidity implies global
rigidity, obtaining the second claim of Part (1). The infinitesimal rigidity
problems for $\mathcal{I}$ and for $\mathcal{I}^{*}$ are equivalent as implied
by the works of Bonahon [14, 13]. Note that the rigidity part of [17] does not
imply infinitesimal or local injectivity of $\mathcal{I}$ or $\mathcal{I}^{*}$
at $\overline{g}$.

A convex realization of a convex hyperbolic cone-metric might be also obtained
from Theorem 1.3 together with an approximation argument, though some non-
trivial work is required to exclude possible degenerations. In the case when
$M$ is homeomorphic to $S\times[-1,1]$ this approach was implemented by
Slutskiy in [82]. Theorem B says that such a realization is necessarily bent,
leading to an alternative proof of the existence part of Theorem A. The most
non-trivial contribution of the present paper is the proof of the uniqueness
part of Theorem A. The proof of Theorem B is rather simple and follows the
ideas of Olovyanishnikov, who similarly proved that a convex realization of a
Euclidean cone-metric on the 2-sphere in the Euclidean 3-space is a polyhedron
[67]. We provide Theorem B to complement our main result Theorem A.

As we already mentioned, our proof of Theorem A is obtained from a
generalization of the classical continuity method popularized by Alexandrov.
Let $V\subset\partial M$ be a finite set, possibly empty. By
$\mathcal{D}_{c}(\partial M,V)$ we denote the space of convex hyperbolic cone-
metrics on $\partial M$ with the vertex set $V$ up to isotopy. By
$\mathcal{P}_{b}(M,V)$ we denote the space of bent hyperbolic metrics on $M$
with the vertex set $V$ up to isotopy. (There are some technical restrictions
on the isotopies, see details in Sections 2.2 and 3.1.2.) We will endow these
spaces with some natural smooth topologies making them finite-dimensional
manifolds of the same dimension. There is the realization map

$\mathcal{I}_{V}:\mathcal{P}_{b}(M,V)\rightarrow\mathcal{D}_{c}(\partial M,V)$

sending a bent metric on $M$ to the induced intrinsic metric on the boundary.

We will frequently abuse the notation and write $g\in\mathcal{P}_{b}(M,V)$ or
$d\in\mathcal{D}_{c}(\partial M,V)$ meaning that $g$ or $d$ are metrics with
their isotopy classes belonging to the spaces $\mathcal{P}_{b}(M,V)$ and
$\mathcal{D}_{c}(\partial M,V)$. The uniqueness part of Theorem A follows from
the following intermediate steps.

######  Proposition 1.13.

If $g\in\mathcal{P}_{b}(M,V)$ is locally rigid, i.e., the map
$\mathcal{I}_{V}$ is a local homeomorphism around $g$, then $g$ is globally
rigid, i.e., for every other $g^{\prime}\in\mathcal{P}_{b}(M,V)$ such that
$\mathcal{I}_{V}(g)=\mathcal{I}_{V}(g^{\prime})$, the metrics $g$ and
$g^{\prime}$ are isotopic.

######  Proposition 1.14.

Controllably polyhedral metrics on $M$ and polyhedral metrics without vertices
are locally rigid.

We state these propositions separately because we expect that the methods of
the present paper can be generalized to show that there are other classes of
bent metrics satisfying the premise of Proposition 1.13. The next case to
handle should be the case of general polyhedral metrics. It might be possible
to generalize the techniques of [14] to show that $\mathcal{I}_{V}$ is
everywhere $C^{1}$. With some additional information on the differential, our
approach will imply that all polyhedral metrics are rigid. It is natural to
expect that $\mathcal{I}_{V}$ is a $C^{1}$-diffeomorphism, but in the greatest
generality this currently seems to be out of range. Proposition 1.13 also
implies the second claim of Part (1) of Corollary C.

Proposition 1.14 for controllably polyhedral metrics is proved in three steps:
we show that the subset of controllably polyhedral metrics is open in
$\mathcal{P}_{b}(M,V)$, that the map $\mathcal{I}_{V}$ is differentiable over
this subset, and that the differential is non-degenerate. We remark that two
most technically complicated places in our paper are the proofs of the latter
two claims occupying Sections 3.4 and 4.

Contrary to the smooth case, in our setting it is not immediately obvious if
there are many convex hyperbolic cone-metrics on $\partial M$ admitting
controllably polyhedral realizations. We will, however, exhibit a large
subclass of such metrics, which we call _balanced_. The actual definition is
slightly technical, and we postpone it to Section 2.2.1, but the heuristic is
that a cone-metric is balanced if the set of vertices is sufficiently dense.
We have

######  Proposition 1.15.

The subset of balanced metrics is open and dense in the space of convex
hyperbolic cone-metrics on a closed surface $S$ with respect to the Lipschitz
topology.

######  Proposition 1.16.

Every convex realization of a convex balanced hyperbolic cone-metric on
$\partial M$ is controllably polyhedral.

###  1.3 Related work

Here we review some related work. The only previously known case of the
rigidity part of Theorem A was established by Fillastre [33], who considered
the case of so-called _Fuchsian manifolds_ with convex boundary. These are
homeomorphic to ${S\times[-1,1]}$, where $S$ is a closed surface of genus
greater than one, and contain an embedded totally geodesic surface isotopic to
$S\times\\{0\\}$. Fillastre also proved that for every convex hyperbolic cone-
metric on $S$ with vertices there exists a Fuchsian manifold with convex
boundary (which is necessarily strictly polyhedral) realizing it. In [34]
Fillastre went further to allow so-called ideal and hyperideal vertices. There
is a similar work of Fillastre–Izmestiev [35] on the case when the manifold is
homeomorphic to $T\times(-1,1]$, where $T$ is the 2-torus, but this does not
fit to our restrictions on $M$.

Similar results were obtained for the model Lorentzian spaces of constant
curvature. In the case of the _de Sitter 3-space_ $d\mathbb{S}^{3}$ (of
constant curvature 1) there is a well-known duality between points in
$d\mathbb{S}^{3}$ and planes in $\mathbb{H}^{3}$, which allows to consider the
corresponding questions as “dual” to questions in $\mathbb{H}^{3}$. The
cornerstone is the work [76] of Rivin–Hodgson, where they gave a description
of the dual metrics on convex polyhedra in $\mathbb{H}^{3}$. It builds on a
previous work of Andreev [6] characterizing hyperbolic polyhedra with acute
dihedral angles. It was further generalized by Rivin [75] to _ideal_ (non-
compact) hyperbolic polyhedra. This is of special interest as it allowed to
resolve a long-standing question of Steiner to describe the combinatorial
types of polyhedra admitting Euclidean realizations with all vertices on the
standard sphere. The work [76] was generalized to the smooth setting by
Schlenker [77], and was recently generalized by the author to the setting of
strictly polyhedral hyperbolic manifolds [72]. The case of strictly polyhedral
manifolds with ideal vertices was previously considered by Schlenker in an
unpublished manuscript [78]. In [34] Fillastre examined similar questions for
Fuchsian Lorentzian spacetimes, and in [36] Fillastre–Izmestiev dealt with the
case of dual metrics on $T\times(-1,1]$. Quite recently Smith [84] resolved
the smooth Weyl problem for _Minkowski spacetimes_. In [87] Tamburelli solved
the realization part for _anti-de Sitter spacetimes_.

Up to now we were mentioning in parallel the results on convex realizations in
the smooth and in the polyhedral cases. It is natural to expect that there
should be some common generalization of these settings. This is the topic of
the Alexandrov–Pogorelov theory of general convex surfaces. Consider convex
surfaces in the Euclidean 3-space $\mathbb{E}^{3}$. Alexandrov managed to give
a complete intrinsic description of the induced intrinsic metrics on them.
These are so-called _CBB(0) metrics_ in modern terminology, whose definition
is slightly technical and we refer to the textbooks [4, 22]. This notion saw a
huge development in last decades due to the connection with degenerations of
Riemannian manifolds. With the help of an approximation argument, Alexandrov
showed in [1] that every CBB(0) metric on the 2-sphere is realized as the
induced intrinsic metric on the boundary of a convex body. The respective
rigidity question was solved by Pogorelov [69] and is known as notoriously
difficult, another approach was given by Volkov [90]. In recent years there
was a streamline of works transferring the general Alexandrov realization
theorem to various settings, see [37, 38, 48, 82]. Of special relevance is the
above-mentioned work of Slutskiy [82], where he considered general convex
realization results for $M$ homeomorphic to $S\times[-1,1]$. However, there
are no generalizations of the Pogorelov rigidity theorem, except that to the
other model spaces (due to Pogorelov himself, see [69], and to Milka [61]),
and except the recent work of the author [70] on the case of Fuchsian
manifolds.

Speaking of the injectivity problems for the maps $\mathcal{I}$ and
$\mathcal{I}^{*}$, in the work [16] Bonahon considered manifolds $N$
homeomorphic to $S\times(-1,1)$ for a closed surface $S$ of higher genus and
showed that $\mathcal{I}^{*}$ is a homeomorphism onto the image near the so-
called _Fuchsian locus_ of $\mathcal{CH}(N)$. In the work [81] Series proved
the conjecture on $\mathcal{I}^{*}$ for manifolds homeomorphic to
$S\times(-1,1)$, where $S$ is a once-punctured torus (the latter manifolds
have cusps and are outside of the scope of this paper). As the infinitesimal
rigidity problems for $\mathcal{I}$ and $\mathcal{I}^{*}$ are equivalent,
these works also imply the corresponding infinitesimal and local rigidity
results for $\mathcal{I}$. We also remark that bending laminations can be
considered as degenerate cases of the dual metrics, which we mentioned just
above.

The questions whether the maps $\mathcal{I}$ and $\mathcal{I}^{*}$ are
homeomorphisms onto the images are very similar to the known parametrization
of the space $\mathcal{CH}(N)$. Every convex cocompact hyperbolic metric on
$N$ determines a conformal structure “at infinity” on $\partial M$, defining
another map $\mathcal{CH}(N)\rightarrow\mathcal{T}(\partial M)$ to the
Teichmüller space $\mathcal{T}(\partial M)$. The famous Bers _double
uniformization_ theorem [9] asserts that this map is a homeomorphism in the
case of $N$ homeomorphic to $S\times(-1,1)$. It was extended to the case of
arbitrary $N$ admitting a convex cocompact hyperbolic metric (and, more
generally, a geometrically finite hyperbolic metric) in the collective works
of Ahlfors, Bers, Kra, Marden, Maskit and Sullivan. See, e.g., the discussion
in [55, Chapter 5]. A long string of efforts was devoted to generalize it to
the case of all complete hyperbolic metrics on $N$ (where we need other type
of invariants at infinity, called _ending laminations_) culminated in the
works [62, 20]. Recently Chen and Schlenker [26] explored the case when on
each boundary component one prescribes either a conformal structure at
infinity or a (smooth) induced metric/dual metric.

A recognized trouble with the continuity method is that while it gives a proof
of the existence of a convex realization, it provides no recipe how to
construct it. The first approach to overcome it in the case of Theorem 1.2 was
undertook by Volkov in [91] by a variational method, and another approach was
proposed in [10] by Bobenko–Izmestiev. It is interesting to ask if any kind of
constructive approach can be given to the realization part of Theorem A. In
the work [70] the author built on this approach to show the rigidity of
Fuchsian manifolds with general convex boundary. Hence, it can be helpful also
on the way to generalize Pogorelov’s rigidity to other hyperbolic manifolds,
and to prove the injectivity of $\mathcal{I}$ and $\mathcal{I}^{*}$ for non-
polyhedral convex cores.

The key infinitesimal rigidity statement in our paper is obtained with the
help of the works on the rigidity of hyperbolic cone-3-manifolds, see Section
2.3 for the definition. There was a long string of results showing their
rigidity in various settings, see [43, 58, 63, 94, 95]. We specifically employ
the work [95] of Weiss. The rigidity of hyperbolic cone-3-manifolds has
interesting applications to some other geometry problems, see [11, 21, 44, 71]
for examples.

Acknowledgments. We thank François Fillastre, Ivan Izmestiev and Jean-Marc
Schlenker for many valuable discussions and thank Michael Eichmair for his
support.

##  2 Preliminaries

For a background in hyperbolic geometry we refer to [8, 55, 56]. We will
mostly rely either on the hyperboloid model or on the Klein model of the
hyperbolic space. We will frequently use the hyperbolic version of the well-
known Busemann–Feller lemma [60]

######  Lemma 2.1.

Let $C\subset\mathbb{H}^{3}$ be a closed convex set with non-empty interior,
$\tau\subset(\mathbb{H}^{3}\backslash{\rm int}(C))$ be a rectifiable curve and
$\tau^{\prime}\subset\partial C$ be its nearest-point projection on $C$. Then
$\tau^{\prime}$ is rectifiable and its length is at most the length of $\tau$.

We mentioned in Section 1.3 that the space of oriented planes of
$\mathbb{H}^{3}$ can be identified with _the de Sitter space_
$d\mathbb{S}^{3}$, the model Lorentzian space of curvature 1. Vice versa,
points of $\mathbb{H}^{3}$ correspond to space-like planes in
$d\mathbb{S}^{3}$. Sometimes we will be using some simple facts on this
duality. We refer to [76, 77, 72] for more details.

###  2.1 Trigonometry

In this subsection we review some elementary facts from hyperbolic
trigonometry that we will use in the remainder.

By a trapezoid we mean a convex hyperbolic quadrilateral
$p_{1}p_{2}q_{2}q_{1}$ such that
${\angle}p_{1}q_{1}q_{2}={\angle}p_{2}q_{2}q_{1}=\pi/2$. Take a trapezoid
$p_{1}p_{2}q_{2}q_{1}$ and denote the lengths $p_{1}q_{1}$, $p_{2}q_{2}$,
$p_{1}p_{2}$, $q_{1}q_{2}$ by $a$, $b$, $c$, $\gamma$ and the angles
${\angle}q_{1}p_{1}p_{2}$, ${\angle}q_{2}p_{2}p_{1}$ by $\beta$ and $\alpha$
respectively. In what follows we will be using the analogues of the cosine and
sine laws for trapezoids. The proofs are straightforward following, e.g., the
approach of Thurston in [88, Chapter 2.6].

######  Lemma 2.2 (The cosine laws).

For a trapezoid we have

$\cos(\alpha)=\frac{\sinh(b)\cosh(c)-\sinh(a)}{\cosh(b)\sinh(c)},$
$\cosh(\gamma)=\frac{\sinh(a)\sinh(b)+\cosh(c)}{\cosh(a)\cosh(b)}.$

######  Lemma 2.3 (The sine law).

For a trapezoid we have

$\frac{\sin(\alpha)}{\cosh(a)}=\frac{\sin(\beta)}{\cosh(b)}=\frac{\sinh(\gamma)}{\sinh(c)}.$

Next, we recall few standard formulas on right-angled hyperbolic pentagons and
hexagons. Proofs can be found in [23, Chapter 2.3].

######  Lemma 2.4.

Consider a right-angled hyperbolic pentagon with side lengths $a$, $b$,
$\alpha$, $c$, $\beta$ clockwise. Then we have

$\cosh(c)=\sinh(a)\sinh(b),$ $\cosh(c)=\coth(\alpha)\coth(\beta).$

######  Lemma 2.5.

Consider a right-angled hyperbolic hexagon with side lengths $a$, $\gamma$,
$b$, $\alpha$, $c$, $\beta$ clockwise. Then we have

$\cosh(\alpha)=\frac{\cosh(b)\cosh(c)+\cosh(a)}{\sinh(a)\sinh(b)}.$

The next lemma, due to Kubota [47], is a hyperbolic analogue of the well-known
Ptolemy theorem.

######  Lemma 2.6.

Consider a hyperbolic quadrilateral inscribed in a circle with side lengths
$a$, $b$, $c$ and $d$ clockwise and diagonal lengths $x$ and $y$. Then

$\sinh\left(\frac{a}{2}\right)\sinh\left(\frac{c}{2}\right)+\sinh\left(\frac{b}{2}\right)\sinh\left(\frac{d}{2}\right)=\sinh\left(\frac{x}{2}\right)\sinh\left(\frac{y}{2}\right).$

Last, we will need a geometric estimate on hyperbolic pairs of pants following
from

######  Lemma 2.7.

Let $H=p_{1}p_{2}p_{3}p_{4}p_{5}p_{6}\subset\mathbb{H}^{2}$ be a convex right-
angled hexagon with sides $a,\gamma,b,\alpha,c,\beta$ ($p_{1}p_{2}=a$ and then
cyclically). Assume that for some $\Delta>0$ we have $a,b,c\geq\Delta$. Then
there exist a number $\delta=\delta(\Delta)>0$ and a point $p\in H$ such that

$d_{\mathbb{H}^{2}}(p,\partial H)\geq\delta.$

###### Proof.

Take any $\delta_{1}>0$. First suppose that $\alpha\geq\delta_{1}$. It is easy
to see that there exists $\delta_{2}=\delta_{2}(\delta_{1},\Delta)>0$ such
that since $b\geq\Delta$ and $\alpha\geq\delta_{1}$, the inradius of the
right-angled triangle $p_{3}p_{4}p_{5}$ is at least $\delta_{2}$. We can take
the incenter of $p_{3}p_{4}p_{5}$ as $p$.

The same procedure applies if $\beta$ or $\gamma$ are at least $\delta_{1}$.
Hence, it remains to suppose that all $\alpha,\beta,\gamma\leq\delta_{1}$.

Let $q$ be the intersection of the altitudes and let the foot of the altitude
divide $p_{1}p_{2}$ in the segments of lengths $a_{1},a_{2}$ respectively. We
define similarly $\gamma_{1},\gamma_{2}$, $b_{1},b_{2}$ and so on (in the
cyclic order). By $h_{a}$ denote $d_{\mathbb{H}^{2}}(q,p_{1}p_{2})$. Then

$\tanh(h_{a})=\cosh(a_{1})\tanh(\beta_{2}).$

From Lemma 2.4 we can also express

$\tanh(\beta_{2})=\frac{\coth(\gamma)}{\cosh(a)}.$

Substitute this and get

$\tanh(h_{a})=\frac{\cosh(a_{1})\cosh(\gamma)}{\cosh(a)\sinh(\gamma)}=\frac{\cosh(a_{1})\cosh(\gamma)\sinh(\beta)}{\cosh(\beta)\cosh(\gamma)+\cosh(\alpha)}=$
$=\frac{\sinh(a_{1})\cosh(a_{1})\cosh(\gamma)\sinh(\beta)}{\sinh(a_{1})(\cosh(\beta)\cosh(\gamma)+\cosh(\alpha))}=\frac{\coth(a_{1})\cosh(\gamma){\cosh(\alpha_{1})}}{\cosh(\beta)\cosh(\gamma)+\cosh(\alpha)}.$

Here we used Lemma 2.4 and Lemma 2.5. It is not hard to see that under the
condition $\alpha,\beta,\gamma\leq\delta_{1}$ the last expression is bounded
away from zero. Hence, for some $\delta_{3}=\delta_{3}(\delta_{1})>0$ we have
$h_{a}\geq\delta_{3}$.

Let $q_{a}$ be the foot of the altitude to $p_{1}p_{2}$. Either $a_{1}$ or
$a_{2}$ is $\geq\Delta/2$. Assume that $a_{1}\geq\Delta/2$. Consider the
right-angled triangle $qq_{a}p_{1}$. There exists
$\delta_{4}=\delta_{4}(\delta_{1},\Delta)>0$ such that since
$a_{1}\geq\Delta/2$ and $h_{a}\geq\delta_{3}$, the inradius of $qq_{a}p_{1}$
is at least $\delta_{4}$. In this case we can take the incenter of
$qq_{a}p_{1}$ as $p$.

Hence, we can set $\delta$ as the minimum of $\delta_{2}$ and $\delta_{4}$. ∎

######  Corollary 2.8.

There exists a continuous function
$\delta=\delta(\Delta):\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that if
$P$ is a hyperbolic pair of pants with the length of each boundary component
at least $\Delta>0$, then for some $p\in P$ we have $d(p,\partial P)>\delta$,
where $d$ is the metric on $P$.

Until the end of the paper we assume that we fixed a function $\delta$ from
Corollary 2.8.

###  2.2 Cone-metrics on surfaces

Let $S$ be a closed surface and $V\subset S$ be a finite set of marked points.

######  Definition 2.9.

A _triangulation_ $\mathcal{T}$ of $(S,V)$ is a collection of simple disjoint
arcs with endpoints in $V$ that cut $S$ into triangles. Here a triangle is a
topological 2-disk with three marked points at the boundary (possibly
coinciding as points of $V$). We call two triangulations of $(S,V)$
_equivalent_ if they are isotopic on $S$. Here if

$\\{f_{t}:S\rightarrow S,~{}~{}~{}t\in[0,1],~{}~{}~{}f_{0}=id\\}$

is an isotopy, then for each $v\in V$ we require $f_{0}(v)=f_{1}(v)$, but we
do not require $f_{t}(v)=v$ for $t\in(0,1)$.

We denote the set of edges of $\mathcal{T}$ be $E(\mathcal{T})$. Now we review
some facts about hyperbolic cone-metrics on $S$. In what follows we will be
saying just “cone-metric” always meaning “hyperbolic cone-metric”. We denote
the set of cone-points of $d$ by $V(d)$. We say that $d$ is a cone-metric on
$(S,V)$ if $d$ is a cone-metric on $S$ and $V\supseteq V(d)$. For $v\in V$ we
denote its cone-angle by $\kappa_{v}$ (which is equal to $2\pi$ if $v\notin
V(d)$). We call two hyperbolic cone-metrics on $(S,V)$ _equivalent_ if they
are isometric by an isometry fixing $V$ and isotopic to identity. Here the
isotopy does not have to fix $V$.

######  Definition 2.10.

A _geodesic triangulation_ of $(S,V,d)$ is a triangulation of $(S,V)$ such
that all edges are geodesics in $d$.

######  Definition 2.11.

Let $\mathcal{T}$ be a triangulation of $(S,V)$ and $d$ be a cone-metric on
$(S,V)$. We say that $\mathcal{T}$ is _realized_ by $d$ if there is a geodesic
triangulation of $(S,V,d)$ equivalent to $\mathcal{T}$.

It is not hard to notice that for every cone-metric $d$ on $(S,V)$ there
exists a geodesic triangulation of $(S,V,d)$. Of special importance are
_Delaunay triangulations_. Generically a Delaunay triangulation of $(S,V,d)$
is unique, but in some cases it is not and we need to consider a more general
_Delaunay decomposition_.

######  Definition 2.12.

Let $d$ be a cone-metric on $(S,V)$. A _Delaunay decomposition_ of $(S,V,d)$
is a collection of simple disjoint geodesic arcs with endpoints in $V$ that
cut $S$ into cells isometric to compact convex hyperbolic polygons such that
each of the polygons can be inscribed in a circle and if we develop any two
adjacent cells to $\mathbb{H}^{2}$, then the closed disk bounded by the
circumcircle of each polygon does not contain vertices of the other polygon
except the endpoints of the common edge.

Note that here a priori by circle we mean either a metric circle in
$\mathbb{H}^{2}$, or a horocycle, or a hypercycle. However, it is known that
actually the circumcircle of each cell in the Delaunay decomposition of a
closed surface is a metric circle, see [41, Theorem 14] and [71, Remark 3.7].
Another well-known fact is

######  Theorem 2.13 ([57], Section 4).

A Delaunay decomposition of $(S,V,d)$ exists and is unique.

######  Definition 2.14.

A _Delaunay triangulation_ of $(S,V,d)$ is a triangulation subdividing the
Delaunay decomposition.

In particular, there are finitely many Delaunay triangulations, and any two
can be connected by a finite sequence of flips via Delaunay triangulations.

For a map $f:(S_{1},d_{1})\rightarrow(S_{2},d_{2})$ between two compact metric
spaces we define its _distortion_ ${\rm dist}(f)$ as

${\rm dist}(f):=\sup_{p,q\in S_{1},~{}p\neq
q}\left|\ln\frac{d_{2}(f(p),f(q))}{d_{1}(p,q)}\right|.$

We say that a sequence of metrics $\\{d_{i}\\}$ on a surface $S$ converges to
a metric $d$ _in the strong Lipschitz sense_ if the identity maps
$(S,d_{i})\rightarrow(S,d)$ have distortions $\varepsilon_{i}\rightarrow 0$.
By $\mathfrak{D}(S,V)$ we denote the space of hyperbolic cone-metrics on
$(S,V)$ endowed with the topology of the strong Lipschitz convergence. By
$\mathcal{D}(S,V)$ we denote the space of equivalence classes of hyperbolic
cone-metrics. For a triangulation $\mathcal{T}$ of $(S,V)$ by
$\mathcal{D}(S,\mathcal{T})\subset\mathcal{D}(S,V)$ we denote the subset of
classes of metrics realizing $\mathcal{T}$. It is evident that if $d$ realizes
$\mathcal{T}$, then $d$ is determined up to isotopy by $\mathcal{T}$ and by
the edge lengths. Hence, the set $\mathcal{D}(S,\mathcal{T})$ can be
considered as an open polyhedron in $\mathbb{R}^{E(\mathcal{T})}_{>0}$ defined
by strict triangle inequalities. The charts $\mathcal{D}(S,\mathcal{T})$ endow
$\mathcal{D}(S,V)$ with the structure of a smooth manifold of dimension
$3(n-k)$ where $n:=|V|$ and $k:=\chi(S)$. It is not hard to check that this
topology coincides with the topology induced from $\mathfrak{D}(S,V)$.
Moreover, $\mathfrak{D}(S,V)$ is homeomorphic to the product of
$\mathcal{D}(S,V)$ with the space of homeomorphisms $S\rightarrow S$ fixing
each $v\in V$ and isotopic to identity. In particular, any continuous map to
$\mathcal{D}(S,V)$ can be lifted to a continuous map to $\mathfrak{D}(S,V)$.
We will not introduce a special notation for equivalence classes of metrics,
so $d$ could sometimes mean a metric on $S$, and sometimes could mean an
equivalence class of metrics, but the meaning should be clear each time from
the context.

It is known from the works [89] of Troyanov and [41] of Gu–Guo–Luo–Sun–Wu that
$\mathcal{D}(S,V)$ is diffeomorphic to the product of $\mathbb{R}^{n}$ with
the Teichmüller space $\mathcal{T}(S,V)$ of $(S,V)$. This implies

######  Theorem 2.15.

The space $\mathcal{D}(S,V)$ is diffeomorphic to $\mathbb{R}^{3(n-k)}$.

By $\overline{\mathfrak{D}}_{c}(S,V)$ and $\overline{\mathcal{D}}_{c}(S,V)$ we
denote the subsets of $\mathfrak{D}_{c}(S,V)$ and $\mathcal{D}_{c}(S,V)$
consisting of convex cone-metrics and their equivalence classes. By
$\mathfrak{D}_{c}(S,V)$ and $\mathcal{D}_{c}(S,V)$ we denote the subsets of
$\mathfrak{D}_{c}(S,V)$ and $\mathcal{D}_{c}(S,V)$ consisting of convex cone-
metrics with $V=V(d)$ and their equivalence classes. It is easy to see that
$\mathfrak{D}_{c}(S,V)$ and $\mathcal{D}_{c}(S,V)$ are open in
$\mathfrak{D}_{c}(S,V)$ and $\mathcal{D}_{c}(S,V)$ and
$\overline{\mathfrak{D}}_{c}(S,V)$ and $\overline{\mathcal{D}}_{c}(S,V)$ are
their closures.

We need now the following statement

######  Lemma 2.16.

Let $d\in\mathcal{D}(S,V)$, $\mathcal{T}$ be its Delaunay triangulation and
$t\geq 0$ be a number. For each edge $e\in E(\mathcal{T})$ of length $l_{e}$
in $d$ we set a new length $l_{e,t}$ given by

$\sinh\left(\frac{l_{e,t}}{2}\right):=e^{t}\sinh\left(\frac{l_{e}}{2}\right).$

Then the new edge-lengths define a metric $d_{t}\in\mathcal{D}(S,V)$ with the
same Delaunay decomposition as $d$ and for every $v\in V$ and $t>0$ we have
$\kappa_{v}(d_{t})<\kappa_{v}(d)$.

###### Proof.

Let us first show that $d_{t}$ is well-defined, i.e., that the obtained edge-
lengths satisfy the triangle inequalities for each triangle of $\mathcal{T}$.
Consider the function

$f(x):=2\operatorname{arcsinh}(e^{t/2}\sinh(x/2)).$

Its derivatives are given by

$f^{\prime}(x)=\frac{e^{t/2}\cosh(x/2)}{\sqrt{e^{t}\sinh^{2}(x/2)+1}},$
$f^{\prime\prime}(x)=\frac{e^{t/2}\sinh(x/2)(1-e^{t})}{2(e^{t}\sinh^{2}(x/2)+1)^{3/2}}.$

Hence, for $t>0$ and $x>0$ it is strictly increasing and concave. Since
$f(0)=0$, $f$ is subadditive under these conditions. Hence, if three edge-
lengths of $\mathcal{T}$ in $d$ satisfy the triangle inequalities, the
modified lengths satisfy them too.

It was shown in [41, Proposition 10] that a triangulation $\mathcal{T}$ is
Delaunay for $d$ if and only if for every two adjacent triangles with the
length of the common edge $l_{0}$ and with the other edge-lengths $l_{1}$ and
$l_{2}$ in one triangle and $l^{\prime}_{0}$, $l^{\prime}_{1}$ in another, we
have

$\frac{\sinh^{2}(l_{1}/2)+\sinh^{2}(l_{2}/2)-\sinh^{2}(l_{0}/2)}{\sinh(l_{1}/2)\sinh(l_{2}/2)}+\frac{\sinh^{2}(l^{\prime}_{1}/2)+\sinh^{2}(l^{\prime}_{2}/2)-\sinh^{2}(l_{0}/2)}{\sinh(l^{\prime}_{1}/2)\sinh(l^{\prime}_{2}/2)}\geq
0.$

This implies that $\mathcal{T}$ remains Delaunay in all metrics $d_{t}$.
Moreover, Lemma 2.6 implies that metrics $d_{t}$ are independent on the choice
of a Delaunay triangulation $\mathcal{T}$.

It remains to show that for all $t$ and all $v$ we have
$\partial\kappa_{v}(d_{t})/\partial t<0$. This was basically done in [71,
Lemma 5.3.iii]. It is enough to consider $t=0$. Let
$\overline{t}\in\mathbb{R}^{V}$ be a vector close to zero. Define a length
$l_{e,\overline{t}}$ for $e\in E(\mathcal{T})$ with endpoints $v$ and $w$ by

$\sinh\left(\frac{l_{e,\overline{t}}}{2}\right):=\exp\left(\frac{\overline{t}_{v}+\overline{t}_{w}}{2}\right)\sinh\left(\frac{l_{e}}{2}\right).$

If $\overline{t}$ is sufficiently close to zero, then all triangle
inequalities still hold for the new lengths and define a cone-metric on
$(S,V)$, which we denote by $d_{\overline{t}}$. For $v,w\in V$ we denote
$\partial\kappa_{v}(d_{\overline{t}})/\partial t_{w}$ by $X_{vw}$. Lemma
5.3.iii in [71] states that for each $v$ we have $\sum_{w\in V}X_{vw}<0$,
which implies the desired claim. Note that [71] uses a slightly different
setting, but the connection with the present context is explained in [71,
Section 4.3]. ∎

We employ Lemma 2.16 to prove the following

######  Lemma 2.17.

Let $D\subset\mathcal{D}(S,V)$ be a compactum. There exists an isotopy
$\iota_{t}:D\hookrightarrow\mathcal{D}(S,V)$, $t\in[0,1]$, of the inclusion
map $\iota_{0}:D\hookrightarrow\mathcal{D}(S,V)$ such that
$\iota_{1}(D)\subset\mathcal{D}_{c}(S,V)$ and for each $d\in D$ the cone-angle
$\kappa_{v}$ of each $v\in V$ strictly decreases along the respective path
$\\{\iota_{t}(d):t\in[0,1]\\}$. In particular, if
$D\subset\overline{\mathcal{D}}_{c}(S,V)$, then
$\iota_{t}(D)\subset\mathcal{D}_{c}(S,V)$ for all $t>0$.

###### Proof.

For each $d\in D$ choose a Delaunay triangulation $\mathcal{T}$ and define
$\iota_{t}(d)$ by setting for each edge $e\in E(\mathcal{T})$ and for each
$t>0$

$\sinh\left(\frac{l_{e}(\iota_{t}(d))}{2}\right):=e^{t}\sinh\left(\frac{l_{e}(d)}{2}\right),$

where $l_{e}(d)$ is the length of $e$ in metric $d$. Lemma 2.16 shows that
this indeed defines a class of metrics in $\mathcal{D}(S,\mathcal{T})$ and
that for each $v\in V$ the cone-angle $\kappa_{v}$ strictly decreases. Lemma
2.6 implies that $\iota_{t}(d)$ is independent on the choice of a Delaunay
triangulation. This means that $\iota_{t}$ is a continuous homotopy. Thus, for
some $t_{0}>0$ we obtain $\iota_{t_{0}}(D)\subset\mathcal{D}_{c}(S,V)$. It
only remains to reparametrize the obtained path. If
$D\subset\overline{\mathcal{D}}_{c}(S,V)$, then
$\iota_{t}(D)\subset\mathcal{D}_{c}(S,V)$ for all $t>0$ as the cone-angles
strictly decrease. ∎

By $\mathbb{S}^{1}_{l}$ we denote the metric circle of length $l$. A _closed
$\varepsilon$-quasi-geodesic_ is the image of $S^{1}_{l}$ under an
$\varepsilon$-quasi-isometry, i.e., of a map
$\mathbb{S}^{1}_{l}\rightarrow(S,d)$ with distortion at most $\varepsilon$.
The following consequence of the Arzelá–Ascoli theorem was shown in [76,
Theorem 6.1]:

######  Lemma 2.18.

Let $\\{d_{i}\\}$ be a sequence of metrics on $S$ converging in the strong
Lipschitz sense to a metric $d$. Let $\\{\psi_{i}\subset S\\}$ be closed
$\varepsilon_{i}$-quasi-geodesics in $d_{i}$ with $\varepsilon_{i}\rightarrow
0$ and the lengths $l_{i}$ of $\psi_{i}$ converging to $l>0$. Then up to
extracting a subsequence, $\psi_{i}$ converge to a closed geodesic
$\psi\subset S$ in $d$ of length $l$.

We recall the following well-known notion

######  Definition 2.19.

Let $d$ be a metric on $S$. The _systole_ $\operatorname{sys}(d)$ of $d$ is
the infimum of lengths of non-contractible loops of $S$.

Lemma 2.18 implies that the infimum is achieved.

######  Lemma 2.20.

The function $\operatorname{sys}(.)$ is continuous over $\mathfrak{D}(S,V)$.

###### Proof.

Let $\\{d_{i}\\}$ be a sequence of metrics on $S$ converging in the strong
Lipschitz sense to a metric $d$ and $\\{\psi_{i}\subset S\\}$ be a sequence of
systolic loops. Lemma 2.18 shows that up to taking a subsequence, they
converge to a loop $\psi\subset S$ of length
$\liminf_{i\rightarrow\infty}\operatorname{sys}(d_{i})$. Obviously since
$\psi_{i}$ are non-contractible, so is $\psi$. Hence

$\liminf_{i\rightarrow\infty}\operatorname{sys}(d_{i})\geq\operatorname{sys}(d).$

On the other hand, if $\psi^{\prime}$ is a systolic loop of $d$, then for a
sequence $\varepsilon_{i}\rightarrow 0$ the lengths of $\psi^{\prime}$ in
$d_{i}$ is $\operatorname{sys}(d)+\varepsilon_{i}$. Thus,

$\limsup_{i\rightarrow\infty}\operatorname{sys}(d_{i})\leq\operatorname{sys}(d).$

∎

It is evident that $\operatorname{sys}(.)$ is constant over the isotopy class
of $d$, hence determines a function over $\mathcal{D}_{c}(S,V)$, which we
continue to denote by $\operatorname{sys}(.)$. Lemma 2.20 implies

######  Corollary 2.21.

The function $\operatorname{sys}(.)$ is continuous over $\mathcal{D}(S,V)$.

####  2.2.1 Balanced cone-metrics

######  Definition 2.22.

Let $S$ be a closed surface, $V\subset S$ be a finite set and $d$ be a metric
on $S$. The _sparsity_ $\operatorname{sp}(V,d)$ of $V$ in $d$ is

$\operatorname{sp}(V,d):=\sup_{p\in S}d(p,V).$

For a cone-metric $d$ we define its _cone-sparsity_
$\operatorname{cosp}(d):=\operatorname{sp}(V(d),d)$.

Due to compactness, it is easy to see that there exist $p\in S$, $v\in V$ such
that $d(p,v)=\operatorname{sp}(V,d)$.

######  Lemma 2.23.

The function $cosp$ is upper semicontinuous over $\mathfrak{D}(S,V)$.

###### Proof.

Let $\\{d_{i}\\}$ be a sequence of cone-metrics on $(S,V)$ converging in the
strong Lipschitz sense to a metric $d$ and $\\{p_{i}\subset S\\}$ be a
sequence of points realizing $\operatorname{cosp}(d_{i})$. Up to taking a
subsequence they converge to a point $p\in S$. Since for each $v\in V$ the
function $\kappa_{v}$ is continuous over $\mathfrak{D}(S,V)$, if $v\notin
V(d_{i})$ for infinitely many $d_{i}$, then $v\notin V(d)$. Hence,

$d(p,V(d))\geq\limsup_{i\rightarrow\infty}d(p_{i},V(d_{i})),$
$\operatorname{cosp}(d)\geq\limsup_{i\rightarrow\infty}\operatorname{cosp}(d_{i}).$

∎

It is evident that $\operatorname{cosp}(.)$ is constant over the isotopy class
of $d$, hence it determines a function over $\mathcal{D}_{c}(S,V)$, which we
continue to denote by $\operatorname{cosp}(.)$. Lemma 2.23 implies

######  Corollary 2.24.

The function $\operatorname{cosp}(.)$ is upper-semicontinuous over
$\mathcal{D}(S,V)$.

######  Definition 2.25.

A cone-metric $d$ on a surface $S$ is called _balanced_ if

$\operatorname{cosp}(d)<\delta(\operatorname{sys}(d))$

where $\delta$ is the function from Corollary 2.8.

By ${\mathfrak{D}}_{bc}(S,V)$ and ${\mathcal{D}}_{bc}(S,V)$ we denote the
subsets of $\mathfrak{D}(S,V)$ and $\mathcal{D}(S,V)$ consisting of balanced
convex cone-metrics and their equivalence classes. Lemma 2.20 and Lemma 2.23
imply

######  Lemma 2.26.

The sets ${\mathfrak{D}}_{bc}(S,V)$ and ${\mathcal{D}}_{bc}(S,V)$ are open.

For two metrics $d_{1}$ and $d_{2}$ on $S$ _the Lipschitz distance_ between
$d_{1}$ and $d_{2}$ is defined as the infimum of the distortions of all
homeomorphisms $(S,d_{1})\rightarrow(S,d_{2})$. We say that a sequence of
metrics $\\{d_{i}\\}$ on a surface $S$ converges to a metric $d$ _in the
Lipschitz sense_ if the Lipschitz distance between them tends to zero. The
obtained Lipschitz topology is one of the most common to endow the space of
all metrics on $S$. Now we are going to prove Proposition 1.15 stating that
the set of convex balanced cone-metrics is open and dense in the space of all
convex cone-metrics on $S$ equipped with the Lipschitz topology. It is evident
that it is dense, so it remains only to show that it is open. It follows from

######  Lemma 2.27.

The function $cosp$ is upper-semicontinuous on the space of all cone-metrics
on $S$ in the Lipschitz topology.

###### Proof.

The proof is almost identical to the proof of Lemma 2.23, but we need to
establish one additional fact. Let $\\{d_{i}\\}$ be a sequence of hyperbolic
cone-metrics converging to a hyperbolic cone-metric $d$ in the Lipschitz
sense.

######  Claim 2.27.1.

For each $v\in V(d)$ there exists a sequence $\\{v_{i}\in V(d_{i})\\}$
converging to $v$.

Indeed, otherwise there exists $v\in V(d)$, a small neighborhood $U$ of $v$
not containing other point of $V(d)$, a domain
$U^{\prime}\subset\mathbb{H}^{2}$ and a sequence of homeomorphisms
$f_{i}:U\rightarrow U^{\prime}$ with distortions converging to zero. Let
$\sigma$ be a circle of small radius $r>0$ around $v$ in $U$. Then
$f_{i}(\sigma)$ converges in the Hausdorff sense to a circle of radius $r$
around $f(u)$ in $U^{\prime}$. Hence, their lengths must converge. On the
other hand, their lengths are clearly different.

The rest of the proof is straightforward. ∎

######  Definition 2.28.

For a hyperbolic cone-metric $d$ and $V\supseteq V(d)$ we call $V$ _balancing_
$d$ if

$\operatorname{sp}(V,d)<\delta(\operatorname{sys}(d))$

where $\delta$ is the function from Corollary 2.8.

It is evident that $\operatorname{sp}(V,.)$ is constant over the classes in
$\mathcal{D}(S,V)$. By $\overline{\mathfrak{D}}_{bc}(S,V)$ and
$\overline{\mathcal{D}}_{bc}(S,V)$ we denote the subsets of
$\mathfrak{D}_{c}(S,V)$ and $\mathcal{D}_{c}(S,V)$ consisting of metrics
(classes of metrics), for which $V$ is balancing. Note that in contrast with
$\overline{\mathcal{D}}_{c}(S,V)$, which is the closure of
${\mathcal{D}}_{c}(S,V)$, $\overline{\mathcal{D}}_{bc}(S,V)$ is not the
closure of ${\mathcal{D}}_{bc}(S,V)$ in ${\mathcal{D}}(S,V)$ as we do not add
metric classes with $\operatorname{sp}(V,d)=\delta(\operatorname{sys}(d))$. In
particular, $\overline{\mathcal{D}}_{bc}(S,V)$ is open in
$\overline{\mathcal{D}}_{c}(S,V)$ endowed with the subspace topology. The same
is true about $\overline{\mathfrak{D}}_{bc}(S,V)$.

Balanced convex cone-metrics play a special role in our paper. Of particular
importance are the following “weak” connectivity and simple-connectivity
properties of $\mathcal{D}_{c}(S,V)$ through balanced convex cone-metrics.

######  Lemma 2.29.

For every $d_{0},~{}d_{1}\in\mathcal{D}_{c}(S,V)$ there exist $W\supseteq V$
and a continuous path
$\alpha:[0,1]\rightarrow\overline{\mathcal{D}}_{bc}(S,W)$ such that
$\alpha(0)$, $\alpha(1)$ are lifts of $d_{0}$, $d_{1}$, and for all
$t\in(0,1)$ we have $\alpha(t)\in\mathcal{D}_{bc}(S,W)$.

###### Proof.

Let $\alpha^{\prime}:[0,1]\rightarrow\mathcal{D}(S,V)$ be a path connecting
$d_{0}$ and $d_{1}$ in $\mathcal{D}(S,V)$. Lemma 2.17 shows that it can be
transformed to a path
$\alpha^{\prime\prime}:[0,1]\rightarrow\mathcal{D}_{c}(S,V)$ connecting
$d_{0}$ and $d_{1}$: just do the respective homotopy and use as
$\alpha^{\prime\prime}$ the concatenation of the upper and the horizontal
sides of the homotopy square. Lift $\alpha^{\prime\prime}$ to a path
$\mathfrak{a}^{\prime\prime}:[0,1]\rightarrow\mathfrak{D}_{c}(S,V)$. By
compactness of the image of $\mathfrak{a}^{\prime\prime}$, we can choose a
sufficiently dense set $W\subset S$ such that $W$ balances
$\mathfrak{a}^{\prime\prime}(t)$ for all $t$. This projects to a path
$\alpha^{\prime\prime\prime}:[0,1]\rightarrow\overline{\mathcal{D}}_{bc}(S,W)$.
Applying once again Lemma 2.17 we modify it to the desired path $\alpha$ with
the property $\alpha(t)\in\mathcal{D}_{bc}(S,W)$ for all $t\in(0,1)$. ∎

######  Lemma 2.30.

Let $\beta:S^{1}\rightarrow\overline{\mathcal{D}}_{c}(S,V)$ be a continuous
loop. Then there exist $W\supseteq V$ and a lift
$\alpha:S^{1}\rightarrow\overline{\mathcal{D}}_{c}(S,W)$ of $\beta$ such that
$\alpha(S^{1})\subset\overline{\mathcal{D}}_{bc}(S,W)$ and $\alpha$ is
contractible in $\overline{\mathcal{D}}_{bc}(S,W)$.

###### Proof.

By $\overline{D}^{2}$ we denote the closed topological 2-disk with the
boundary identified with $S^{1}$. Let
$\xi^{\prime}:\overline{D}^{2}\rightarrow\mathcal{D}(S,V)$ be a map
contracting $\beta$ in $\mathcal{D}(S,V)$. Lemma 2.17 allows us to obtain from
$\xi^{\prime}$ a map
$\xi^{\prime\prime}:\overline{D}^{2}\rightarrow\overline{\mathcal{D}}_{c}(S,V)$
contracting $\beta$. Lift it to a map
$\mathfrak{x}^{\prime\prime}:\overline{D}^{2}\rightarrow\mathfrak{D}(S,V)$. By
compactness of the image of $\mathfrak{x}^{\prime\prime}$, we can choose a
sufficiently dense set $W\subset S$ such that $W$ balances
$\mathfrak{x}^{\prime\prime}(z)$ for all $z\in\overline{D}^{2}$. This projects
to the desired map
$\xi:\overline{D}^{2}\rightarrow\overline{\mathcal{D}}_{bc}(S,W)$ such that
$\xi|_{S^{1}}$ is a lift of $\beta$. ∎

###  2.3 Hyperbolic cone-3-manifolds

A hyperbolic cone-3-manifold is a 3-dimensional generalization of a hyperbolic
cone-surface. We do not require that much theory about the set of cone-metrics
on a given 3-manifold as we do about cone-metrics on surfaces, so we present
the former theory in a simplified form.

######  Definition 2.31.

_A hyperbolic cone-3-manifold_ is a pair $(K,\Sigma)$, where $K$ is a
3-manifold, possibly with boundary, equipped with an intrinsic metric and
$\Sigma$ is an embedded geodesic graph such that

  * •

every interior point belonging to $K\backslash\Sigma$ has a neighborhood
isometric to a domain in $\mathbb{H}^{3}$;

  * •

every boundary point belonging to $K\backslash\Sigma$ has a neighborhood
isometric to a domain of a boundary point in a half-space of $\mathbb{H}^{3}$;

  * •

every interior point belonging to $\Sigma$ has a neighborhood isometric to a
cone of curvature $-1$ over a 2-dimensional spherical cone-surface
homeomorphic to the 2-sphere, but not isometric to the standard sphere;

  * •

every boundary point belonging to $\Sigma$ has a neighborhood isometric to a
cone of curvature $-1$ over a 2-dimensional spherical cone-surface with
boundary, homeomorphic to the 2-disk, but not isometric to a half of the
standard sphere.

We refer to [28, 43, 58, 94] for an introduction to the theory of closed
hyperbolic cone-3-manifolds. If a point of $\Sigma$ has a neighborhood
isometric to a cone over the sphere with two conical points (in the case of an
interior point), or to a spherical lune (in the case of a boundary point), we
call it an _edge point_ , otherwise we call it a _vertex_.

The main source of cone-3-manifolds are gluings of hyperbolic polyhedra along
isometric 2-dimensional faces. We say that a cone-3-manifold $K$ is
_triangulated_ if $\mathcal{T}$ is its geodesic triangulation such that all
vertices of $\mathcal{T}$ belong to $\Sigma$ and $\Sigma$ is included in the
1-skeleton of $\mathcal{T}$. We note that we do not necessarily require that
all vertices of $\mathcal{T}$ are vertices of $\Sigma$ or that all edges of
$\mathcal{T}$ belong to $\Sigma$. A triangulated hyperbolic cone-3-manifold
$(K,\Sigma,\mathcal{T})$ is uniquely determined by the topology of
$(K,\mathcal{T})$ and by the edge lengths of $\mathcal{T}$. Let
$(K,\mathcal{T})$ be a triangulated 3-manifold and $E(\mathcal{T})$ be the set
of edges. There is a domain
$\mathcal{C}(K,\mathcal{T})\subset\mathbb{R}^{E(\mathcal{T})}$ corresponding
to the condition that every $l\in\mathcal{C}(K,\mathcal{T})$ turns every
tetrahedron of $\mathcal{T}$ to a non-degenerate hyperbolic tetrahedron.
Hence, we can consider $\mathcal{C}(K,\mathcal{T})$ as the space of hyperbolic
cone-structures on $(K,\mathcal{T})$. Note, however, that the singular locus
of such structures is variable, though it stays belonging to the union of the
edges of $\mathcal{T}$. For $e\in E(\mathcal{T})$ and
$l\in\mathcal{C}(K,\mathcal{T})$ by $\nu_{e}$ we denote the total dihedral
angle of $e$ in the cone-structure determined by $l$. This is an analytic
function over $\mathcal{C}(K,\mathcal{T})$.

Our proof of the infinitesimal rigidity is based on a comparison of
infinitesimal variations of $l\in\mathcal{C}(K,\mathcal{T})$ with the standard
approach to the infinitesimal variations of hyperbolic cone-structures via the
bundle of infinitesimal isometries developed in [43, 94, 95]. This occupies
Section 4.

In what follows we will need triangulations with few special properties

######  Definition 2.32.

Let $\mathcal{T}$ be a triangulation of a hyperbolic cone-3-manifold
$(K,\Sigma)$, and $L:=K\backslash\Sigma$. The triangulation $\mathcal{T}$ is
called _large_ if for every tetrahedron $\widetilde{T}$ of $\mathcal{T}$
lifted to $\widetilde{L}$ and every $\gamma\in\pi_{1}(L)$, $\gamma\neq e$, the
tetrahedron $\gamma\widetilde{T}$ does not share with $\widetilde{T}$ any
2-face, or any edge coming from $e\in E(\mathcal{T})$ that is not in $\Sigma$.
It is called _super-large_ if $\gamma\widetilde{T}$ never shares a vertex with
$\widetilde{T}$.

###  2.4 Convex cocompact hyperbolic 3-manifolds

This section largely follows Section 2.3 in [72]. Here we repeat the basic
definitions and give a brief summary of the basic facts that we will use, but
for additional explanations we refer to [72, Section 2.3].

Recall that $M$ is an oriented smooth compact 3-manifold with non-empty
boundary such that its interior $N$ admits a cocompact hyperbolic metric, and
that $M$ is distinct from the solid torus. Let $g$ be a hyperbolic metric on
$M$ such that $\partial M$ is locally convex (which implies that it is
globally convex). It is well-known that

######  Lemma 2.33.

There exists a unique up to isotopy convex cocompact metric $\overline{g}$ on
$N$ such that there is a (unique) isometric embedding
$(M,g)\hookrightarrow(N,\overline{g})$ such that the complement to the image
is a neighborhood of $\partial M$ intersected with $N$, homeomorphic to
$\partial M\times(0,1)$, and the induced map $\pi_{1}(\partial
M)\rightarrow\pi_{1}(\partial M)$ is identity.

For a proof look the beginning of [72, Section 2.3]. Recall that we are going
to use the notation $\overline{g}$ to denote the metric on $N$ obtained from
$g$ by Lemma 2.33, but sometimes we will just write $\overline{g}$ to denote a
convex cocompact metric on $N$ in the absence of metric $g$. For a convex
cocompact hyperbolic metric $\overline{g}$ we denote the hyperbolic manifold
$(N,\overline{g})$ by $N(\overline{g})$. We denote the image of $(M,g)$ in
$N(\overline{g})$ under the embedding from Lemma 2.33 by $M(g)\subset
N(\overline{g})$. By $\partial_{\infty}N(\overline{g})$ we denote the boundary
at infinity of $N(\overline{g})$, i.e., $\partial M$ equipped with the
respective conformal structure.

By $G$ we denote the group $\rm{Iso}^{+}(\mathbb{H}^{3})$ of orientation-
preserving isometries of $\mathbb{H}^{3}$, which we identify with
$PSL(2,\mathbb{C})$. We will frequently assume that some developing map
$\widetilde{N}(\overline{g})\rightarrow\mathbb{H}^{3}$ is fixed, so we
identify the universal cover $\widetilde{N}(\overline{g})$ with
$\mathbb{H}^{3}$ and $\widetilde{M}(g)$ with a convex subset of
$\mathbb{H}^{3}$. By $\rho_{\overline{g}}:\pi_{1}(M)\rightarrow G$ we denote
the corresponding holonomy representation. By
$\Lambda(\rho_{\overline{g}})=\Lambda(\overline{g})$ we denote its limit set
and by $\widetilde{C}(\overline{g})$ we denote its convex hull. The latter is
$\rho_{\overline{g}}$-invariant and projects to the _convex core_
$C(\overline{g})\subset N(\overline{g})$, an alternative definition of which
was given in the introduction as Definition 1.10.

We recall that $\overline{g}$ and $\rho_{\overline{g}}$ are called _Fuchsian_
if $\Lambda(\overline{g})$ is contained in a circle. In the Fuchsian case
$C(\overline{g})$ is 2-dimensional, and we will frequently have to say
something exceptional about Fuchsian metrics. We need to distinguish the
subcase when $\Lambda(\overline{g})$ is a circle itself, so $\overline{g}$ is
called _Fuchsian of the first kind_. It is called _Fuchsian of the second
kind_ in the remaining case when $\Lambda(\overline{g})$ is a proper subset of
a circle. In the former case $M$ is the oriented interval bundle over a closed
surface and $C(\overline{g})$ is a totally geodesic surface. In the latter
case $M$ is the oriented interval bundle over a compact surface with non-empty
boundary, i.e., a handlebody and $C(\overline{g})$ is a totally geodesic
surface with non-empty geodesic relative boundary. A well-known observation of
Thurston is

######  Lemma 2.34.

Let $\overline{g}$ be non-Fuchsian. The induced intrinsic metric on $\partial
C(\overline{g})$ is hyperbolic.

See, e.g., [31] and [83] for a detailed proof. Thus, a connected component of
$\partial C(\overline{g})$ is an embedded hyperbolic surface, which is,
however, non-geodesic as it may be bent along a geodesic lamination.

By $\mathcal{CH}(N)$ we denote the space of convex cocompact hyperbolc metrics
on $N$ up to isotopy. As before, we will not introduce a special notation for
an equivalence class of metrics, and will write $\overline{g}$ sometimes for a
precise metric, and sometimes for an equivalence class. By
$\mathcal{T}(\partial M)$ we denote the Teichmüller space of $\partial M$. The
following version of the generalized Ahlfors–Bers theorem holds

######  Theorem 2.35.

The map $\mathcal{AB}:\mathcal{CH}(N)\rightarrow\mathcal{T}(\partial M)$
sending $\overline{g}$ to the conformal structure at infinity of
$\partial_{\infty}N(\overline{g})$ is a bijection.

We did not find an exact reference with this formulation of this result, but
in [72, Section 2.3] we explain how to derive it from the existing literature.
We endow $\mathcal{CH}(N)$ with the smooth topology of $\mathcal{T}(\partial
M)$. In order to introduce in what follows the topology on the space of bent
metrics on $M$ we need to recall few other objects from [72, Section 2.3]. By
$\mathcal{CH}_{h}(N)$ we denote the space of convex cocompact hyperbolic
metrics on $N$ up to homotopy. By $MCG_{h}(M)$ we denote the group of isotopy
classes of self-diffeomorphisms of $M$ homotopic to identity. We recal that in
the case of incompressible boundary Waldhausen [92] showed that $MCG_{h}(M)$
is trivial, while the case of compressible boundary is considered in the
monograph [59] of McCullough and Miller. The group $MCG_{h}(M)$ acts freely
and properly discontinuously on $\mathcal{CH}(N)$ and $\mathcal{T}(\partial
M)$. The quotient of the former is exactly $\mathcal{CH}_{h}(N)$, the quotient
of the latter is denoted by $\mathcal{T}_{h}(\partial M,M)$. The map
$\mathcal{AB}$ commutes with the action of $MCG_{h}(M)$, hence
$\mathcal{CH}_{h}(N)$ and $\mathcal{T}_{h}(\partial M,M)$ are diffeomorphic.
Note, however, that there is another more standard way to introduce the
topology on $\mathcal{CH}_{h}(N)$. By $\mathcal{R}(\pi_{1}(M),G)$ we denote
the space of representations of $\pi_{1}(M)$ in $G$ endowed with the topology
of a complex algebraic variety coming from the complex algebraic group
structure on $G$, and by $\mathcal{X}(\pi_{1}(M),G)$ we denote the quotient of
$\mathcal{R}(\pi_{1}(M),G)$ by the conjugation action of $G$. The developing
maps allow us to embed $\mathcal{CH}_{h}(N)$ into $\mathcal{X}(\pi_{1}(M),G)$.
The global topology of $\mathcal{X}(\pi_{1}(M),G)$ is not well-behaved, but
results of Marden [54] imply that the image of $\mathcal{CH}_{h}(N)$ belongs
to the smooth part of $\mathcal{X}(\pi_{1}(M),G)$, so this embedding endows
$\mathcal{CH}_{h}(N)$ with the topology of a complex manifold. It is another
part of the Ahlfors–Bers theorem that these topologies on
$\mathcal{CH}_{h}(N)$ coincide. See [72, Section 2.3] for more explanations.

We will use the following result on the behavior of limit sets under the
convergence in $\mathcal{R}(\pi_{1}(M),G)$.

######  Theorem 2.36.

Let $\\{\rho_{i}:\pi_{1}(M)\rightarrow G\\}$ be a sequence converging in
$\mathcal{R}(\pi_{1}(M),G)$ to $\rho:\pi_{1}(M)\rightarrow G$ such that all
$\rho_{i}$ and $\rho$ are holonomy maps of convex cocompact hyperbolic metrics
on $N$. Then $\Lambda(\rho_{i})$ converge to $\Lambda(\rho)$ in the Hausdorff
sense.

See Proposition 4.5.3 in [55] or Theorem E in [5].

##  3 Hyperbolic manifolds with boundary of polyhedral type

###  3.1 Polyhedral and bent manifolds

We start to study bent metrics on $M$. Denote the set of vertices of a bent
metric $g$ by $V(g)$. If $p\in\partial M$ belongs to the relative interiors of
two geodesic segments with distinct tangents at $p$, then the convexity
implies that $p$ has a neighborhood isometric to a neighborhood of a point in
the boundary of a half-space in $\mathbb{H}^{3}$. Then we say that $p$ is
_regular_ in $g$. A connected component of the set of regular points is called
a _face_ of $g$. If $p$ is not a vertex and not regular, it is called an
_edge-point_ in $g$. Let it belong to the relative interior of a geodesic
segment $\psi$. There is a neighborhood of $p$ in the relative interior of
$\psi$ consisting only of edge-points. A maximal geodesic segment consisting
of edge-points is called an _edge_ of $g$. It is straightforward that the
edges are disjoint and simple. We can also speak about bent surfaces in
hyperbolic 3-manifolds and about their vertices, edges and faces.

We will frequently consider $\widetilde{M}(g)\subset\mathbb{H}^{3}$ and we
denote its set of vertices, i.e. the full preimage of $V(g)$ in
$\widetilde{M}(g)$, by $\widetilde{V}(g)$. We will use all our notions
interchangeably between $M(g)$ and $\widetilde{M}(g)$. We note that in
$\partial\widetilde{M}(g)$ every edge extends from both sides either to a
point from $\widetilde{V}(g)$, or to a point in
$\partial_{\infty}\mathbb{H}^{3}$. In the latter case it necessarily belongs
to $\Lambda(\overline{g})$ as otherwise the distance to
$\widetilde{C}(\overline{g})$ would grow to infinity along this edge, which
contradicts to the compactness of $M$. This provides a classification of the
edges of $g$: either a lift of an edge has two endpoints in
$\widetilde{V}(g)$, or two endpoints in $\Lambda(\overline{g})$, or it is
mixed. We will show below that the third case is impossible. In the second
case note that by definition such an edge also belongs to $\partial
C(\overline{g})$.

We recall that for any non-Fuchsian convex cocompact metric $\overline{g}$ on
$N$ a homeomorphism between $C(\overline{g})$ and $M$ provides an example of a
bent metric on $M$ without vertices. We devote the next subsection to recall
the basic facts about the boundary structure of $C(\overline{g})$.

####  3.1.1 Boundary geometry of convex cores

Let $\overline{g}$ be a convex cocompact hyperbolic metric on $N$. Due to
Lemma 2.34, the induced intrinsic metric on $\partial C(\overline{g})$ is
hyperbolic. The union of edges of $\partial C(\overline{g})$ constitutes what
is called a _geodesic lamination_.

######  Definition 3.1.

Let $(S,d)$ be a closed hyperbolic surface. A _geodesic lamination_ $\lambda$
is a closed subset of $S$ foliated by simple disjoint geodesics, which we will
call _edges_ of $\lambda$.

We refer to [12, 25, 24, 68] for various aspects of the theory of geodesic
laminations. We remark that we step away from the conventional terminology in
favor of coherence as the edges of a geodesic lamination are commonly known in
the literature as _leaves_.

A geodesic lamination is called _minimal_ if it does not contain any proper
geodesic sub-laminations. A simple closed geodesic provides an example of a
minimal geodesic lamination. Any other minimal geodesic lamination is called
_irrational_. Every minimal irrational lamination consists of uncountably many
edges. A general geodesic lamination is called _irrational_ if it contains a
minimal irrational sub-lamination, and is called _rational_ otherwise. For
examples of irrational laminations we refer to [12, Chapter 1.3] or [15].

######  Theorem 3.2 ([24], Theorem I.4.2.8).

A geodesic lamination $\lambda$ on a closed hyperbolic surface consists of
finitely many minimal sub-laminations and finitely many open geodesics,
isolated in $\lambda$, spiralling onto a minimal sub-lamination from both
ends.

We will call connected components of $S\backslash\lambda$ _faces_ of
$\lambda$. A metric completion of a face is a hyperbolic surface of finite
type with piecewise-geodesic boundary, whose components may be either closed
or open geodesics. Every end of an open boundary-geodesic is asymptotic to
another such end forming what we call a _spike_. An edge of $\lambda$ is
called _boundary_ if it is a boundary component of a face. Note that even
though an irrational lamination contains uncountably many leaves, only
finitely many of them are boundary. A lamination is called _maximal_ if the
completion of each face is isometric to an ideal triangle.

Identify the universal cover $(\widetilde{S},d)$ with $\mathbb{H}^{2}$ and let
$\widetilde{\lambda}$ be the full preimage of a minimal irrational lamination
$\lambda$. Then every geodesic $\widetilde{\psi}$ of $\widetilde{\lambda}$ is
a limit of other geodesics of $\widetilde{\lambda}$. If $\widetilde{\psi}$ is
a lift of a boundary edge, then it is a limit from only one side.

######  Definition 3.3.

Let $\lambda$ be a geodesic lamination on $(S,d)$. A _transverse arc_ to
$\lambda$ is a simple regular arc transverse to each edge of $\lambda$ with
endpoints in faces. A _transverse measure_ $\mu$ on $\lambda$ is a locally
finite Borel measure $\mu_{\tau}$ on each transverse arc $\tau$ such that

(1) for a transverse sub-arc $\tau^{\prime}\subset\tau$ the measure
$\mu_{\tau^{\prime}}$ is the restriction of $\mu_{\tau}$;

(2) the support of $\mu_{\tau}$ is $\tau\cap\lambda$;

(3) the measure is invariant through isotopies of $S$ preserving $\lambda$.

The pair $(\lambda,\mu)$ is called a _measured (geodesic) lamination_.

Note that in particular $\mu_{\tau}(\tau)=0$ if and only if
$\tau\cap\lambda=\emptyset$. We will denote $\mu_{\tau}(\tau)$ just by
$\mu(\tau)$. We will sometimes abuse the notation and denote the measured
lamination $(\lambda,\mu)$ just by $\mu$. We can speak about the measure of
each geodesic of $\lambda$ meaning the measure of any point belonging to this
geodesic on any transverse arc intersecting this geodesic. Not every geodesic
lamination can support a transverse measure. A useful criterion is

######  Lemma 3.4 ([56], Proposition 8.3.8).

Let $\lambda$ be a geodesic lamination supporting a transverse measure and
$\widetilde{\lambda}$ be its full preimage in $\mathbb{H}^{2}$. Every point
$q\in\partial\mathbb{H}^{2}$ can be adjacent to at most two geodesics from
$\widetilde{\lambda}$, and if it is adjacent to two, then both of them are
lifts of boundary edges of $\lambda$ belonging to a minimal irrational sub-
lamination.

Theorem 3.2 and Lemma 3.4 imply (see a detailed proof also, e.g., in [68,
Corollary 1.7.3])

######  Lemma 3.5.

The support of a measured lamination consists of finitely many minimal sub-
laminations.

By identifying the universal cover $\widetilde{S}$ with $\mathbb{H}^{2}$ a
measured lamination on $S$ gives rise to a $\pi_{1}(S)$-invariant measure on
the space of lines in $\mathbb{H}^{2}$. This allows to consider measured
laminations from a metric independent point of view. By $\mathcal{ML}(S)$ we
denote the space of measured laminations on $S$ equipped with the topology of
*-weak convergence of measures. There is a natural PL-structure on the space
$\mathcal{ML}(S)$ induced by so-called _train tracks_ on $S$. See the details
in [12, 68]. A well-known result is

######  Theorem 3.6 ([12, 68]).

The space $\mathcal{ML}(S)$ is PL-homeomorphic to $\mathbb{R}^{-3\chi(S)}$.

Now we return to our setting when $\overline{g}$ is a convex cocompact
hyperbolic metric on $N$. The set of edges of $\partial C(\overline{g})$
clearly forms a geodesic lamination, which we denote $\lambda_{\overline{g}}$.
There is a transverse measure $\mu_{\overline{g}}$ measuring how $\partial
C(\overline{g})$ is bent in $N(\overline{g})$. A careful construction is given
in [31]. We sketch now another construction of $\mu_{\overline{g}}$.

Recall that oriented planes in $\mathbb{H}^{3}$ correspond to point in the de
Sitter space $d\mathbb{S}^{3}$. Let $\tau$ be a transverse arc and
$\widetilde{\tau}$ be its lift to $\partial\widetilde{C}(\overline{g})$.
Consider the set of all pairs $(p,\Pi)$ where $p\in\widetilde{\tau}$ and $\Pi$
is a supporting plane to $\widetilde{C}(\overline{g})$ at $p$, which we
consider oriented outwards $\widetilde{C}(\overline{g})$. Denote the
projection of this set to the second coordinate by $\tau^{*}\subset
d\mathbb{S}^{3}$. It is not hard to see that $\tau^{*}$ is an arc in
$d\mathbb{S}^{3}$, i.e., the image of a closed interval. One can show that
$\tau^{*}$ is rectifiable. Define $\mu_{\overline{g}}(\tau)$ as the length of
$\tau^{*}$.

The pair $(\lambda_{\overline{g}},\mu_{\overline{g}})$ is called _the bending
lamination_ of $\overline{g}$. It is possible to extend the notion of geodesic
laminations to cone-surfaces (a step towards that was taken, e.g., in [18])
and the notion of bending lamination to bent surfaces. This would give
alternative proofs to the basic facts about bent surfaces in the next section.
We, however, chose to take a simpler, more geometric route. We will need the
following easy statements.

######  Lemma 3.7.

Let $\psi\subset\lambda_{\overline{g}}$ be a geodesic belonging to an
irrational component, and
$\widetilde{\psi}\subset\partial\widetilde{C}(\overline{g})$ be its lift. Then
there is a unique supporting plane to $\widetilde{C}(\overline{g})$ at
$\widetilde{\psi}$.

###### Proof.

The geodesic $\psi$ is open, hence, there is an accumulation point of $\psi$
in $\partial C(\overline{g})$. Thereby, there exists a transverse arc
intersecting $\psi$ infinitely many times. Thus, the measure of $\psi$ is
zero. If there are two distinct support planes to
$\widetilde{C}(\overline{g})$ at $\widetilde{\psi}$, the measure of $\psi$ is
at least the exterior dihedral angle between these planes, which is non-zero.
∎

######  Lemma 3.8.

Let
$\widetilde{\psi}_{1},\widetilde{\psi}_{2}\subset\partial\widetilde{C}(\overline{g})$
be two edges of the same component of $\partial\widetilde{C}(\overline{g})$
ending in the same point $q\in\Lambda(\overline{g})$. Then
$\widetilde{\psi}_{1}$ and $\widetilde{\psi}_{2}$ belong to a minimal
irrational sub-lamination of $\lambda_{\overline{g}}$, and there exists a face
of $\partial\widetilde{C}(\overline{g})$ incident to both of them.

###### Proof.

Pick $p\in\widetilde{\psi}_{1}$. We claim that the intrinsic distance in
$\partial\widetilde{C}(\overline{g})$ from $p$ to $\widetilde{\psi}_{2}$ tends
to zero as we move $p$ towards $q$. Indeed, there exists a finite sequence
$\Pi_{1},\ldots,\Pi_{r}$ of supporting planes to $\widetilde{C}(\overline{g})$
such that $\Pi_{1}$ is tangent to $\widetilde{\psi}_{1}$, $\Pi_{r}$ is tangent
to $\widetilde{\psi}_{2}$ and every two subsequent planes intersect. This
follows from [31, Lemma II.1.8.3]. By the Busemann-Feller lemma, the intrinsic
distance from $p$ to $\widetilde{\psi}_{2}$ in
$\partial\widetilde{C}(\overline{g})$ is at least the intrinsic distance from
$p$ to $\widetilde{\psi}_{2}$ in the union of pieces of these support planes
forming a convex surface $C$. Take the horosphere at $q$ passing through $p$
and consider in $C$ the curve obtained in the intersection. The length of this
curve from $p$ to the intersection with $\widetilde{\psi}_{2}$ tends to zero
as we move $p$ towards $q$, which implies our claim.

Develop the universal cover of this component of
$\partial\widetilde{C}(\overline{g})$ to $\mathbb{H}^{2}$. (We remark that as
the boundary of $M$ could be compressible, the boundary components of
$\widetilde{M}$ could be infinitely connected.) Due to our claim, there are
lifts of $\widetilde{\psi}_{1}$ and $\widetilde{\psi}_{2}$ ending in the same
point $q^{\prime}\in\partial_{\infty}\mathbb{H}^{2}$. Due to Lemma 3.4, they
belong to a minimal irrational sub-lamination of $\lambda_{\overline{g}}$ and
there are no other edges of our lifted lamination ending in $q^{\prime}$.
Hence, in the universal cover there is a face between our lifts of
$\widetilde{\psi}_{1}$ and $\widetilde{\psi}_{2}$, which comes from a face of
$\partial\widetilde{C}(\overline{g})$ adjacent to $\widetilde{\psi}_{1}$ and
$\widetilde{\psi}_{2}$. ∎

####  3.1.2 Boundary geometry of bent manifolds

Let $g$ be a bent metric on $M$. We will be frequently considering the closed
set $\partial M(g)\cap\partial C(\overline{g})$. Let $p\in\partial
M(g)\cap\partial C(\overline{g})$. As $\partial C(\overline{g})$ does not have
vertices, $p$ belongs to the relative interior of a segment
$\psi\subset\partial C(\overline{g})$. It follows from convexity that
$\psi\subset\partial M(g)$. Hence, $\partial M(g)\cap\partial C(\overline{g})$
is the union of some edges and faces of $\partial C(\overline{g})$. Note
however that such an edge or a face of $\partial C(\overline{g})$ is not
necessarily an edge or a face of $\partial M(g)$. The following lemma is
rather trivial

######  Lemma 3.9.

Let $g$ be a bent metric on $M$ with at least one vertex. Then $M(g)$ is the
closed convex hull of its vertices in $N(\overline{g})$ and all vertices are
at positive distance from $C(\overline{g})$. Conversely, let $\overline{g}$ be
a convex cocompact hyperbolic metric on $N$ and $V\subset N(\overline{g})$ be
a finite set. Then $\overline{\rm conv}(V)$ produces a bent metric on $M$
except the case when $\overline{g}$ is Fuchsian and $V$ belongs to the
complete totally geodesic surface containing $C(\overline{g})$.

Here $\overline{\rm conv}(V)$ is _the closed convex hull_ of $V$, i.e., the
inclusion-minimal closed totally convex subset of $N(\overline{g})$ containing
$V$. We will also apply the notion on closed convex hull to subsets of
$\mathbb{H}^{3}$.

###### Proof.

Consider $\widetilde{M}(g)\subset\mathbb{H}^{3}$. A lift of an edge of $g$ is
a geodesic segment in $\mathbb{H}^{3}$ that ends either at $\widetilde{V}(g)$
or at $\Lambda(\overline{g})$. A lift of a maximal geodesic segment of $M(g)$
passing through a regular point ends either at $\widetilde{V}(g)$ or at
$\Lambda(\overline{g})$ or at an edge-point. Thus, $\widetilde{M}(g)={\rm
conv}(\widetilde{V}(g)\cup\Lambda(\overline{g}))$. But each point of
$\Lambda(\overline{g})$ is an accumulation point of the orbit of every vertex.
We get $\widetilde{M}(g)=\overline{\rm conv}(\widetilde{V}(g))$ and
$M(g)=\overline{\rm conv}(V(g))$.

If $p\in(\partial\widetilde{M}(g)\cap\partial\widetilde{C}(\overline{g}))$,
then it lies in the relative interior of a geodesic segment
$\psi\subset\partial\widetilde{C}(\overline{g})$. Then
$\psi\subset\partial\widetilde{M}(g)$ and $p$ is not a vertex of $g$. Thus,
all vertices of $g$ are at positive distance from $C(\overline{g})$.

For the last claim, it is easy to see that if $V\subset N(\overline{g})$ is a
finite point-set, then $\overline{\rm conv}(V)$ is homeomorphic to $M$ except
the case when $\overline{g}$ is Fuchsian and $V$ belongs to the complete
totally geodesic surface containing $C(\overline{g})$. We need to show that
the boundary is bent. Denote $\overline{\rm conv}(V)$ by $M(g)$ and consider
$\widetilde{M}(g)\subset\mathbb{H}^{3}$. The Caratheodory theorem (which we
can apply to $\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$ due to the
Klein model) says that each $p\in\widetilde{M}(g)$ is in the convex hull of at
most four points from $\widetilde{V}(g)\cup\Lambda(\overline{g})$. If a point
belongs to the convex hull of four of these points, but no less, then it
belongs to the interior of the convex hull. If follows that every point in
$\partial\widetilde{M}(g)$ belongs to the relative interior of a geodesic
segment except possibly points of $\widetilde{V}$. This finishes the proof. ∎

We will also make use of the following easy observation

######  Lemma 3.10.

For every $v\in V(g)$ there exists a neighborhood of $v$ in $\partial M$ that
does not contain edges of $g$ except those that are incident to $v$.

###### Proof.

Let $B$ be a small embedded geodesic ball around $v$ in $N(\overline{g})$ that
does not contain other vertices. If there is a sequence of edges not incident
to $v$ that converge to $v$, then their intersections with $B$ are disjoint
geodesic segments with the endpoints in $\partial B$. Then they converge to a
geodesic segment in $M(g)\cap B$ containing $v$ in its relative interior.
Thus, $v$ is not a vertex. ∎

We now describe the intrinsic geometry of bent metrics.

######  Lemma 3.11.

If $g$ is a bent metric on $M$, then the induced intrinsic metric $d$ on
$\partial M$ is a convex hyperbolic cone-metric.

This relies on the following proposition, which strengthens the observation of
Thurston that the induced intrinsic metric on the boundary of a convex core is
intrinsically hyperbolic.

######  Proposition 3.12 ([83]).

Let $X$ be a closed subset of $\mathbb{H}^{3}$. If ${\rm conv}(X)$ has non-
empty interior, then $\partial{\rm conv}(X)\backslash X$ is locally isometric
to $\mathbb{H}^{2}$.

###### Proof of Lemma 3.11.

Pick a point $p\in\partial M\backslash V(g)$ and lift it to
$\widetilde{p}\in\partial\widetilde{M}(g)\subset\mathbb{H}^{3}$. Consider a
small open ball $B$ centered at $\widetilde{p}$ that contains no vertices and
does not intersect the components of $\partial\widetilde{M}(g)$ except the one
containing $\widetilde{p}$. Set $X=\partial B\cap\partial\widetilde{M}(g)$.
Either $X$ is a great circle and $\partial\widetilde{M}(g)\cap B$ is a planar
disk or ${\rm conv}(X)$ has non-empty interior, $\partial\widetilde{M}(g)\cap
B$ is a connected component of $\partial{\rm conv}(X)\backslash X$ and we can
apply Proposition 3.12.

We proved that outside the vertices $\partial M$ is locally isometric to the
hyperbolic plane. Now pick $v\in V(g)$ and take a small simply connected
neighborhood $U$ of $v$ in $\partial M$ that does not intersect neither edges
nor faces not incident to $v$, which is guaranteed by Lemma 3.10. Then each
point of $U\backslash v$ lies on a segment emanating from $v$. Consider a lift
$\widetilde{v}$ to $\mathbb{H}^{3}$ and push slightly a supporting geodesic
plane to $\widetilde{M}(g)$ at $\widetilde{v}$ so its intersection $\alpha$
with $\partial\widetilde{M}$ is contained in the lift
$\widetilde{U}\ni\widetilde{v}$ of $U$. Then the neighborhood of
$\widetilde{v}$ bounded by $\alpha$ is the geodesic cone over $\alpha$ from
$\widetilde{v}$ and $\alpha$ is a closed planar convex curve. Hence, the
metric around $\widetilde{v}$ is locally isometric to a convex hyperbolic
cone. ∎

As discussed in Section 3.1.1, the support $\lambda_{\overline{g}}$ of the
bending lamination of $\partial C(\overline{g})$ consists of finitely many
disjoint simple closed geodesics and of minimal irrational laminations. Let
$\lambda$ be a minimal irrational component of $\lambda_{\overline{g}}$ and
$\psi\subset\lambda$ be a geodesic. If $\psi\subset\partial M(g)$, then since
$\partial M(g)\cap\partial C(\overline{g})$ is closed and the closure of
$\psi$ is $\lambda$, we see that $\lambda\subset\partial M(g)$. Another simple
observation is

######  Lemma 3.13.

Let $\lambda$ be a minimal irrational sub-lamination of
$\lambda_{\overline{g}}$ such that $\lambda\subset\partial M(g)$,
$\psi\subset\lambda$ be a boundary edge and $F\subset\partial C(\overline{g})$
be a face adjacent to $\psi$. Then $F\subset\partial M(g)$.

###### Proof.

Lift all to the universal cover, let $\widetilde{F}$ and $\widetilde{\psi}$ be
the respective lifts. By Lemma 3.4, $\widetilde{C}(\overline{g})$ has a unique
supporting plane $\Pi$ at $\widetilde{\psi}$, which contains $\widetilde{F}$.
We have $\widetilde{\psi}\subset\partial\widetilde{M}(g)$, and since any
supporting plane to $\widetilde{M}(g)$ at $\widetilde{\psi}$ is a supporting
plane to $\widetilde{C}(\overline{g})$, $\Pi$ is a supporting plane to
$\widetilde{M}(g)$ and
$(\Pi\cap\partial\widetilde{C}(\overline{g}))\subset(\Pi\cap\partial\widetilde{M}(g))$.
It follows that $\widetilde{F}\subset\partial\widetilde{M}(g)$. ∎

We use it first to prove

######  Lemma 3.14.

Let $g$ be a bent metric on $M$. No edge of $\widetilde{M}(g)$ has one end in
$\widetilde{V}(g)$ and another in $\Lambda(\overline{g})$.

###### Proof.

Suppose that $\widetilde{\chi}$ is such an edge ending at
$\widetilde{v}\in\widetilde{V}(g)$ and $q\in\Lambda(\overline{g})$. Let
$\chi$, $v$ be the projections of $\widetilde{\chi}$, $\widetilde{v}$ to
$\partial M(g)$. Note that the distance to $\widetilde{C}(\overline{g})$ goes
to zero along $\widetilde{\chi}$, but it never attains zero. Indeed, if it is
zero at $p\in\widetilde{\chi}$ and $p$ is the closest point to $\widetilde{v}$
with this property (in terms of the intrinsic distance of $\widetilde{\chi}$),
then $p$ is an edge-point of $\partial\widetilde{C}(\overline{g})$ and belongs
to the relative interior of an edge $\widetilde{\psi}$ of
$\partial\widetilde{C}(\overline{g})$ with a different tangent at $p$ than
$\widetilde{\chi}$. But then
$\widetilde{\psi}\subset\partial\widetilde{M}(g)$, so $p$ is regular in
$\partial\widetilde{M}(g)$ and $\widetilde{\chi}$ is not an edge.

Consider the closure $\lambda_{0}$ of $\chi$ in $\partial M(g)$. Because the
distance to $C(\overline{g})$ goes to zero along $\chi$, the set
$\lambda=\lambda_{0}\backslash\chi$ is at distance zero to $C(\overline{g})$,
thus it is a subset of $\partial M(g)\cap\partial C(\overline{g})$. It is
closed as the intersection of two closed subsets, $\lambda_{0}$ and $\partial
M(g)\cap\partial C(\overline{g})$. The points of $\lambda$ belong to the
boundary of the set $\partial M(g)\cap\partial C(\overline{g})$ in $\partial
M(g)$ and hence $\lambda$ consists of edge-points of $\partial
C(\overline{g})$. If $p\in\lambda$, then one can see that the whole edge
containing $p$ belongs to $\lambda$. Hence $\lambda$ is a sub-lamination of
$\lambda_{\overline{g}}$.

Cut $\partial M$ along $\lambda$, consider the connected component containing
$\chi$ and let $S$ be its metric completion. Due to Lemma 3.10, the cone-
points of $\partial M(g)$ are isolated from $\lambda$. Hence, the local
picture at the boundary points of $S$ is the same as the local picture for
metric completions of faces of geodesic laminations on hyperbolic surfaces.
Similarly to, e.g., [12, Chapter 1.4], one can see that $S$ is a convex
hyperbolic cone-surface of finite type with geodesic boundary, whose
components may be either closed or open geodesics corresponding to edges of
$\lambda$. Each boundary component has a convex neighborhood not containing
cone-points, so isometric to a convex subset of $\mathbb{H}^{2}$. It is not
hard to see that if $\chi$ gets close enough to a closed component, then it
either starts spiraling on it (i.e., in the universal cover of a convex
neighborhood developed to $\mathbb{H}^{2}$ a lift of $\chi$ is asymptotically
parallel to the respective lift of the boundary component), or $\chi$
intersects itself. If $\chi$ gets close enough to an open boundary component,
then either it is asymptotic to this component (i.e., goes to a spike from one
side of the component), or it intersects the boundary. The geodesic $\chi$ is
simple, disjoint from the boundary of $S$ and does not have accumulation
points in the interior of $\chi$, hence it either spirals onto a closed
geodesic component or goes to a spike.

In the first case $\lambda$ is a simple closed geodesic $\psi$. There is its
lift $\widetilde{\psi}\subset\partial\widetilde{M}(g)$ that is incident to
$q$. Indeed, there is a lift $\widetilde{\psi}$ of $\psi$ such that the
intrinsic distance in $\partial\widetilde{M}(g)$ from a point of
$\widetilde{\chi}$ to $\widetilde{\psi}$ tends to zero along
$\widetilde{\chi}$ towards $q$, hence also the extrinsic distance in
$\mathbb{H}^{3}$ tends to zero and such $\widetilde{\psi}$ must end in $q$.
Note that $\widetilde{\psi}$ is an isolated edge of
$\partial\widetilde{C}(\overline{g})$. Then $\widetilde{\psi}$ is the axis of
a pure loxodromic isometry $\gamma\in\rho_{\overline{g}}(\pi_{1}(M))$. The
$\gamma$-orbit of $\widetilde{v}$ is contained in the plane spanned by
$\widetilde{\chi}$ and $\widetilde{\psi}$. But then $\widetilde{\chi}$ belongs
to the relative interior of the convex hull of this orbit, thus
$\widetilde{\chi}$ is not an edge of $\partial\widetilde{M}(g)$.

In the second case there are edges $\psi_{1}$ and $\psi_{2}$ of $\lambda$
corresponding to the boundary components of $S$ ending in the same spike of
$S$ as $\chi$. As the whole $\lambda$ is contained in $\partial M(g)$,
$\psi_{1}$ and $\psi_{2}$ are boundary edges of the lamination $\lambda$ in
$\partial\widetilde{C}(\overline{g})$. There are their lifts
$\widetilde{\psi}_{1}$, $\widetilde{\psi}_{2}$ such that the intrinsic
distances in $\partial\widetilde{M}(g)$ from a point of $\widetilde{\chi}$ to
$\widetilde{\psi}_{1}$ and $\widetilde{\psi}_{2}$ tend to zero along
$\widetilde{\chi}$ towards $q$. Hence so do the extrinsic distances in
$\mathbb{H}^{3}$ and $\widetilde{\psi}_{1},\widetilde{\psi}_{2}$ end in $q$.
Lemma 3.7 says that they belong to the boundary of a face $\widetilde{F}$ of
$\partial\widetilde{C}(\overline{g})$. By Lemma 3.13, $\widetilde{F}$ belongs
to $\partial\widetilde{M}(g)$. The segment $\widetilde{\chi}$ must belong to
the support plane to $\widetilde{F}$ and lie between $\widetilde{\psi}_{1}$
and $\widetilde{\psi}_{2}$. Thus, it passes through interior points of
$\widetilde{F}$ and can not be an edge. ∎

Now we can show that any bent metric is polyhedral around vertices. This
follows from

######  Corollary 3.15.

Let $g$ be a bent metric on $M$ and $v\in V(g)$. Then there are finitely many
edges incident to $v$.

###### Proof.

Consider the universal cover $\widetilde{M}(g)\subset\mathbb{H}^{3}$. Let
$\widetilde{v}$ be a lift of $v$. Suppose that there are infinitely many edges
incident to $\widetilde{v}$. We claim that there is an edge $\widetilde{\chi}$
from $\widetilde{v}$ to $\Lambda(\overline{g})$. Indeed, this follows from the
fact that $\rho_{\overline{g}}(\pi_{1}(M))$ acts properly discontinuously and
the fact that the set of all edges and vertices is closed in $\partial M$.
Thus, we get a contradiction with Lemma 3.14. ∎

We see from Lemma 3.14 that every edge of $\widetilde{M}(g)$ either has both
ends at $\widetilde{V}(g)$ or both ends at $\Lambda(\overline{g})$, in which
case it belongs to $\widetilde{C}(\overline{g})\cap\partial\widetilde{M}(g)$.
In the latter case it is either a simple closed geodesic from $\partial
C(\overline{g})$, or belongs to an irrational lamination of $\partial
C(\overline{g})$. A bent metric is polyhedral if and only if there are no
edges of the third type.

It remains to understand how an irrational bending lamination in the boundary
of a bent metric can coexist with the presence of vertices.

######  Lemma 3.16.

Let $F$ be a face of a bent metric $g$ on $M$ such that it is adjacent to a
vertex $v\in V(g)$ and to an edge $\psi$ belonging to a minimal irrational
sub-lamination $\lambda$ of $\lambda_{\overline{g}}$. Then there exists a
simple closed geodesic $\chi\subset(\overline{F}\cap\partial C(g))$, where
$\overline{F}$ is the closure of $F$, separating $v$ from $\psi$.

In particular, if $g$ has vertices and the bending lamination
$\mu_{\overline{g}}$ has only irrational components, then $g$ is strictly
polyhedral.

###### Proof.

Let $\widetilde{F}\subset\widetilde{M}(g)$ be a lift of $F$ adjacent to a lift
$\widetilde{v}$ of $v$ and to a lift $\widetilde{\psi}$ of $\psi$. By
$\widetilde{\lambda}$ we denote the full preimage of $\lambda$ in
$\partial\widetilde{M}(g)$ and by $\Pi$ we denote the supporting plane to
$\widetilde{M}(g)$ containing $\widetilde{F}$. Note that since
$\widetilde{\psi}\subset\Pi$, the plane $\Pi$ is also supporting for
$\widetilde{C}(\overline{g})$. The geodesic $\widetilde{\psi}$ divides
$\partial\widetilde{C}(\overline{g})$ in two parts and the nearest-point
projection to $\widetilde{C}(\overline{g})$ maps $\widetilde{F}$ to one of
these parts (it is easy to see that $\widetilde{F}$ cannot be mapped on
$\widetilde{\psi}$). We claim that there is a face $\widetilde{F}^{\prime}$ of
$\partial\widetilde{C}(\overline{g})$ adjacent to $\widetilde{\psi}$ from this
side. Indeed, the only other option would be that there is a sequence of
geodesics from $\widetilde{\lambda}$ approaching $\widetilde{\psi}$ from this
side. As $\lambda$ is minimal, we have
$\widetilde{\lambda}\subset\partial\widetilde{M}(g)$, so all but finitely many
of these geodesics belong to $\widetilde{F}\subset\Pi$. As $\Pi$ is a
supporting plane to $\widetilde{C}(\overline{g})$, all these geodesics that
are sufficiently close to $\widetilde{\psi}$ can not be edges of
$\partial\widetilde{C}(\overline{g})$, which is a contradiction.

Since $\lambda$ is irrational, Lemma 3.13 implies that
$\widetilde{F}^{\prime}$ belongs to $\partial\widetilde{M}(g)$. Moreover, it
belongs to the same half-plane of $\Pi$ with respect to $\widetilde{\psi}$ as
$\widetilde{v}$. As the distance from $\widetilde{v}$ to
$\widetilde{C}(\overline{g})$ is positive, $\widetilde{v}$ is disjoint from
the closure of $\widetilde{F}^{\prime}$. Thus, there must be a geodesic line
$\widetilde{\chi}$ that is a boundary component of $\widetilde{F}^{\prime}$
and that separates $\widetilde{v}$ from $\widetilde{\psi}$. It remains to show
that the projection $\chi$ of $\widetilde{\chi}$ is not from a minimal
irrational sub-lamination $\lambda^{\prime}$ of $\lambda_{\overline{g}}$.
Indeed, otherwise $\lambda^{\prime}\subset\partial M(g)$ and there is a
sequence of geodesics of its full preimage $\widetilde{\lambda}^{\prime}$
approaching $\widetilde{\chi}$ from the other side than
$\widetilde{F}^{\prime}$. By the previous reasoning, all but finitely many of
these geodesics belong to $\Pi$, which is a contradiction. Hence, $\chi$ is a
simple closed geodesic that separates $v$ and $\psi$ in $\overline{F}$. ∎

######  Corollary 3.17.

The connected components of $\partial M(g)\cap\partial C(g)$ are simple closed
geodesics, which are edges of $\partial C(g)$, and compact hyperbolic surfaces
with geodesic boundary.

Recall now Definition 1.11 of strictly and controllably polyhedral metrics on
$M$. We can show

######  Corollary 3.18.

Let $g$ be a bent metric on $M$ such that the induced cone-metric $d$ on
$\partial M$ is balanced. Then $g$ is controllably polyhedral.

###### Proof.

Suppose the converse, then $\partial M(g)\cap\partial C(\overline{g})$
contains a compact hyperbolic surface with geodesic boundary. Pick a pair of
pants $P$ in this surface. Since
$\operatorname{cosp}(d)<\delta(\operatorname{sys}(d))$, Corollary 2.8 implies
that $V(d)\cap P$ must be non-empty, which is a contradiction. ∎

Together with Theorem B, Corollary 3.18 implies Proposition 1.16. Now we deal
with strictly polyhedral metrics. As we can see, this property has strong
implications on the boundary structure of $\partial M(g)$.

######  Corollary 3.19.

Let $g$ be a strictly polyhedral metric on $M$. Then each edge is a segment
between vertices and the closures of all faces are isometric to compact
hyperbolic polygons. Conversely, if the closures of all faces of a bent metric
are compact hyperbolic polygons, then the metric is strictly polyhedral.

###### Proof.

Let $g$ be a strictly polyhedral hyperbolic metric on $M$. The claim on the
edges follows from our description of the edges of a bent metric. The closure
of each face is isometric to a hyperbolic surface with convex boundary. As
every edge ends at vertices, the boundary of a face is compact. If a face has
more then one boundary component, then it contains a homotopically non-trivial
geodesic loop. Lifting this loop to the universal cover we see that it goes to
$\Lambda(\overline{g})$, hence it belongs to $\partial C(\overline{g})$, which
is a contradiction.

In the other direction, suppose the converse. If $\partial M(g)\cap\partial
C(\overline{g})$ is non-empty, we saw that there is a face of $\partial M(g)$
containing a simple closed geodesic either in its interior or in its boundary.
The closure of this face can not be isometric to a compact hyperbolic polygon,
which finishes the proof. ∎

We end this section by sketching a construction of non-polyhedral bent metrics
with vertices.

######  Example 3.20.

We assume that $N$ admits a convex cocompact hyperbolic metric $\overline{g}$
such that the bending lamination $\lambda_{\overline{g}}$ contains an
irrational component, and that there is a face $F$ of $\partial
C(\overline{g})$ with boundary consisting of simple closed geodesics. (The
works [17, 50] of Bonahon–Otal and Lecuire imply that such $\overline{g}$
exists on most manifolds under our consideration. In particular, it exists if
$M$ has an incompressible boundary component of genus at least three.)

Let $\psi\subset\partial\widetilde{C}(\overline{g})$ be a boundary edge of a
lift of $F$. By $\Pi_{\psi}$ we denote the supporting plane to
$\widetilde{C}(\overline{g})$ at $\psi$ that makes equal angles with the both
extremal supporting planes to $\widetilde{C}(\overline{g})$ at $\psi$. We call
a supporting plane $\Pi$ to $\widetilde{C}(\overline{g})$ _bad_ if it contains
a boundary edge $\psi$ of a lift $\widetilde{F}$ of $F$ and if the dihedral
angle between $\Pi$ and $\widetilde{F}$ is smaller than the dihedral angle
between $\Pi_{\psi}$ and $\widetilde{F}$. Otherwise, we call a supporting
plane _good_. By $\widetilde{A}$ we denote the intersection in
$\mathbb{H}^{3}$ of all closed supporting half-spaces to
$\widetilde{C}(\overline{g})$ bounded by good supporting planes. We note that
the set $\widetilde{A}$ is closed and $\rho_{\overline{g}}$-invariant, hence
it projects to a closed set $A\subset N(\overline{g})$. We also remark that

  * •

every face of $\partial\widetilde{C}(\overline{g})$ that is not a lift of $F$
belongs to $\partial\widetilde{A}$;

  * •

$\widetilde{A}\backslash\widetilde{C}(\overline{g})$ is non-empty.

Pick a point $\widetilde{v}$ in
$\widetilde{A}\backslash\widetilde{C}(\overline{g})$, consider its
$\rho_{\overline{g}}$-orbit $\widetilde{V}$ and take the closed convex hull of
$\widetilde{V}$ in $\mathbb{H}^{3}$. This determines a bent metric $g$ on $M$
with vertices. We have $\widetilde{M}(g)\subset\widetilde{A}$, hence
$M(g)\subset A$ and every face of $C(\overline{g})$ except $F$ belongs to
$\partial M(g)$. In particular, $\partial M(g)$ contains the irrational
component of $\lambda_{\overline{g}}$, thus $g$ is not polyhedral.

####  3.1.3 Every convex realization of a convex cone-metric is bent

In this subsection we prove Theorem B. We need some small preparations. In
Section 1.3 we mentioned that Alexandrov gave a complete intrinsic
characterization of the induced intrinsic metrics on the boundaries of general
convex sets in $\mathbb{E}^{3}$. He similarly did this in other model spaces.
For convex sets in $\mathbb{H}^{3}$ such metrics are called _CBB( $-1$)
metrics_. Again, we are not going to give a definition, but refer to [4, 22].
Particularly, every convex hyperbolic cone-metric is CBB($-1$). If $S$ is a
surface and $d$ is a CBB($-1$) metric, one can introduce two natural Borel
measures: _the area_ ${\rm area}(.)$ and _the intrinsic curvature_ ${\rm
curv}_{I}(.)$. The former is the standard Hausdorff measure, while the latter
is a non-smooth analogue of the Gaussian curvature measure. We define _the
extrinsic curvature measure_ ${\rm curv}(.):={\rm curv}_{I}(.)+{\rm area}(.)$.
If $d$ is a convex hyperbolic cone-metric and $U\subset S$ is a Borel set,
then ${\rm curv}(U)$ is equal to the sum of the curvatures of all cone-point
belonging to $U$. The curvature of a cone-point $v$ is equal to $2\pi$ minus
the total angle of $v$.

We use the duality between $\mathbb{H}^{3}$ and $d\mathbb{S}^{3}$. If
$C\subset\mathbb{H}^{3}$ is a closed convex set, one can define its _dual_
$C^{*}\subset d\mathbb{S}^{3}$ as the set of all planes in $\mathbb{H}^{3}$
that do not separate $C$ and are oriented outwards. If $C$ has a non-empty
interior and $U\subset\partial C$ is a Borel set, its dual
$U^{*}\subset\partial C^{*}$ is defined as the set of all support planes to
$G$ at points of $U$. An analogue of the Gauss theorem in this setting states
that ${\rm curv}(U)$ is equal to the area of $U^{*}$. See [72, Lemma 2.2.5]
for a proof. It is easy to deduce from this

######  Lemma 3.21 ([70], Lemma 3.2.2).

Let $C\subset\mathbb{H}^{3}$ be a compact convex set with non-empty interior
and $\Pi$ be a supporting plane such that the set $R:=\partial C\cap\Pi$ has
non-empty relative interior in $\partial C$. Define $R^{c}:=\partial
C\backslash R$. Then ${\rm curv}(R^{c})>0$.

Now we can prove Theorem B.

###### Proof of Theorem B.

Consider $\widetilde{M}(g)$ embedded in $\mathbb{H}^{3}$ in the Klein model.
By $\widetilde{V}$ we denote the full preimage of $V(d)$. By $B$ we denote
$\widetilde{M}(g)\cup\Lambda(\overline{g})$, which is a compact convex set in
$\mathbb{R}^{3}$ with non-empty interior. We recall few standard notions from
convex geometry. A point $p\in\partial B$ is called _extreme_ if it does not
belong to the relative interior of a segment with endpoints in $\partial B$. A
point $p\in\partial B$ is called _exposed_ if there exists a plane
$\Pi\subset\mathbb{R}^{3}$ such that $\Pi\cap B=\\{p\\}$. Obviously, exposed
points are extreme, but not necessarily vice versa. All points from
$\Lambda(\overline{g})$ are exposed. It follows directly from the definition
that $B$ is the convex hull of all its extreme points.

The Straszewicz theorem [85] (see also [80, Theorem 1.4.7]) says that the set
of extreme points is contained in the closure of the set of exposed points. We
are going to show that if $p\in\partial B$ is exposed, then
$p\in\widetilde{V}\cup\Lambda(\overline{g})$. Since the latter set is closed,
this also is true for every extreme point. In turn, this implies that every
$p\in\partial\widetilde{M}(g)$ such that $p\notin\widetilde{V}$ belongs to the
relative interior of a segment. Due to the equivariance of $\widetilde{M}(g)$,
this proves that $g$ is bent.

Suppose the converse that there is $p\in\partial\widetilde{M}(g)$,
$p\notin\widetilde{V}$ such that $p$ is exposed. Then $p$ has a neighborhood
$U$ in $\partial B$ distinct from $\widetilde{V}\cup\Lambda(\overline{g})$.
Let $\Pi$ be a plane such that $\Pi\cap B=\\{p\\}$. Push $\Pi$ slightly
towards $B$ such that the curve $\Pi\cap\partial B$ stays in $U$, by
$U^{\prime}$ we denote the open set bounded by this curve. By the choice of
$U$ we have ${\rm curv}(U)=0$, thus also ${\rm curv}(U^{\prime})=0$. On the
other hand, Lemma 3.21 implies ${\rm curv}(U^{\prime})>0$, which is a
contradiction. ∎

###  3.2 Triangulations

For our further investigations we need a geodesic triangulation of $M(g)$,
where $g$ is polyhedral, with vertices at some subset of $\partial M(g)$
containing $V(g)$ (the examples of non-strictly polyhedral metrics show that
just $V(g)$ might be not enough). Unfortunately, we can not prove the
existence of such a triangulation in full generality, but we can show two
weaker things, which are enough for our purposes: (1) there exists such a
decomposition of $M(g)$ into convex hyperbolic polyhedra; (2) in a finite
cover of $M(g)$ this decomposition can be triangulated. This situation is very
similar to the treatment of hyperbolic cusp-3-manifolds, where the existence
of an ideal geodesic triangulation would also be desirable for many
applications. In that case the analogue of (1) is the well-known
Epstein–Penner decomposition [32], and the analogue of (2) is shown in the
work [52] of Luo–Schleimer–Tillmann. A similar treatment was considered by
Schlenker in [78] for polyhedral hyperbolic 3-manifolds with ideal vertices.
We will follow the mentioned works and will show that their approaches are
also applicable in our situation. Note that the construction of the
Epstein–Penner decomposition is itself very similar to a well-known convex-
hull construction of the Delaunay decomposition of a point set in the
Euclidean space via lifting the set to a surface in one dimension higher. In
[30] DeBlois considers generalizations of this construction to discrete
subsets of $\mathbb{H}^{n}$ invariant with respect to torsion-free lattices of
finite covolume.

In the above-mentioned situations the first step is to prove that the visible
part of the convex hull in the Minkowski space has only space-like supporting
planes. From this and from the discretness of the vertex set it follows that
the convex hull provides a locally finite polyhedral decomposition. In our
setting we can have also light-like and time-like supporting planes, so these
conclusions become slightly more involved to make. Before we start, let us
prove an auxiliary result.

######  Lemma 3.22.

Let $g$ be a polyhedral metric on $M$, $s\in\Lambda(\overline{g})$ and $\Pi$
be a plane passing through $s$, but not intersecting $\widetilde{M}(g)$. Then
there exists a supporting plane $\Pi_{0}$ to $\widetilde{M}(g)$ passing
through $s$ and asymptotically parallel to $\Pi$. (By saying that a plane is
supporting to $\widetilde{M}(g)$ we particularly mean that it contains points
of $\partial\widetilde{M}(g)$.)

###### Proof.

Consider the dual convex set $\widetilde{M}(g)^{*}\subset d\mathbb{S}^{3}$.
Its boundary points can be divided in two types: (1) dual to supporting planes
to $\widetilde{M}(g)$ and (2) dual to planes $\Pi$ such that
$\Pi\cap\widetilde{M}(g)=\emptyset$, but
$\partial_{\infty}\Pi\cap\Lambda(\overline{g})\neq\emptyset$. We use the
notions of exposed and extreme points from the previous section. We make a
convention that $\partial\widetilde{M}(g)^{*}$ means
$\partial(\widetilde{M}(g)^{*})$. (Note that it is not the same as
$(\partial\widetilde{M}(g))^{*}$. The latter is exactly the set of points of
type (1).)

Let us show that points of type (2) cannot be exposed for
$\widetilde{M}(g)^{*}$. Suppose that $\Pi^{*}\in\partial\widetilde{M}(g)^{*}$
is such a point, dual to a plane $\Pi\subset\mathbb{H}^{3}$. Note that
$\partial_{\infty}\Pi\cap\Lambda(\overline{g})$ consists of a single point
$s$. Indeed, otherwise $\Pi$ contains a line between two points of
$\Lambda(\overline{g})$, hence $\Pi\cap\widetilde{M}(g)\neq\emptyset$ and
$\Pi^{*}$ is not of type (2). Assume that there is a plane $p^{*}\subset
d\mathbb{S}^{3}$ such that $p^{*}\cap\widetilde{M}(g)^{*}=\Pi^{*}$. Note that
$p^{*}$ cannot be time-like, thus the dual point $p$ is in
$\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$. If there is another plane
in $\mathbb{H}^{3}$ passing through $p$ and not intersecting
$\widetilde{M}(g)$, then $p^{*}$ contains another point of
$\widetilde{M}(g)^{*}$, which contradicts to our choice. If $p\neq s$, then
because $\widetilde{M}(g)\cup\Lambda(\overline{g})$ is closed in
$\mathbb{H}^{3}\cup\partial_{\infty}\mathbb{H}^{3}$ and
$(\Pi\cup\partial_{\infty}\Pi)\cap(\widetilde{M}(g)\cup\Lambda(\overline{g}))=s$,
we can rotate $\Pi$ slightly around an axis in $\Pi$ through $p$, but not
through $s$, so that the obtained plane does not intersect
$\widetilde{M}(g)\cup\Lambda(\overline{g})$. If $p=s$, then we can act on
$\Pi$ by a parabolic isometry preserving $s$ and sending $\Pi$ to an
asymptotically parallel plane in the direction opposite from
$\widetilde{M}(g)$. The resulting plane again does not intersect
$\widetilde{M}(g)$. Hence, $\Pi^{*}$ cannot be exposed.

One can see that when $g$ is polyhedral, a point
$\Pi^{*}\in\partial\widetilde{M}(g)^{*}$ of type (1) is exposed if and only if
the dual plane $\Pi$ contains a regular point of $\partial\widetilde{M}(g)$.
The set of such points $\Pi^{*}$ is discrete. The Straszewicz theorem applied
to $\widetilde{M}(g)^{*}\cup\Lambda(\overline{g})$ says that the extreme
points of $\widetilde{M}(g)^{*}\cup\Lambda(\overline{g})$ are in the closure
of the set of exposed points. Then all exposed points of
$\widetilde{M}(g)^{*}$ are extreme. In particular, no point of type (2) is
extreme.

Now consider the situation in the statement of the lemma. The point $\Pi^{*}$
is of the type (2). The segment from $s$ to $\Pi^{*}$ belongs to
$\partial\widetilde{M}(g)^{*}$. Let us extend it behind $\Pi^{*}$. It can not
be extended infinitely, thus it ends at a point
$\Pi_{0}^{*}\in\partial\widetilde{M}(g)^{*}$. It remains to show that
$\Pi_{0}^{*}$ is of type (1). Indeed, otherwise it is not extreme and belongs
to the relative interior of a segment $I$ in $\partial\widetilde{M}(g)^{*}$
such that each endpoint of $I$ either is in $\Lambda(\overline{g})$, or is an
extreme point of $\partial\widetilde{M}(g)^{*}$. If one of the endpoints is
$s_{0}\in\Lambda(\overline{g})$, then by construction $s_{0}\neq s$, hence
$\partial_{\infty}\Pi_{0}\cap\Lambda(\overline{g})$ contains $s$ and $s_{0}$,
so $\Pi^{*}_{0}$ is of type (1). If both endpoints are extreme, then they are
of type (1). We will show that so is $\Pi^{*}_{0}$.

Denote the supporting planes to $\widetilde{M}(g)$, dual to the endpoints of
$I$, by $\Pi_{1}$ and $\Pi_{2}$. They must be distinct. Note that because they
pass through $s$ and are supporting to $\widetilde{M}(g)$, from Lemma 3.14
they contain points of $\widetilde{C}(\overline{g})$, hence are supporting to
it. Suppose that one of them, say $\Pi_{1}$, supports a face $\widetilde{F}$
of $\widetilde{C}(\overline{g})$ such that there is no edge of this face
ending in $s$. Then $s$ is a limit point for points of
$\Lambda(\overline{g})\cap\partial_{\infty}\Pi_{1}$ from both sides. Thus,
every geodesic ray emanating from $s$ in $\Pi_{1}$ contains points of
$\widetilde{F}$. Then $\Pi_{1}$ is the only supporting plane to
$\widetilde{C}(\overline{g})$ through $s$, which is a contradiction with that
$\Pi_{2}$ is distinct from $\Pi_{1}$. Hence, there is an edge of
$\partial\widetilde{C}(\overline{g})$ in $\Pi_{1}$ adjacent to $s$. Similarly,
there is an edge adjacent to $s$ in $\Pi_{2}$. If these edges are distinct,
then by Lemma 3.8 they belong to an irrational component of
$\lambda_{\overline{g}}$, which is contained in $\partial\widetilde{M}(g)$.
This contradicts to $g$ being polyhedral. Thus, it is the same edge, i.e.,
$\Pi_{1}$ and $\Pi_{2}$ intersects and the intersection line between $\Pi_{1}$
and $\Pi_{2}$ is an edge of $\partial\widetilde{M}(g)$. Hence, $I$ consists of
points of type (1), which finishes the proof.

∎

######  Lemma 3.23.

Let $g$ be a polyhedral hyperbolic metric on $M$. Let $W\subset\partial M(g)$
be a finite set of points containing $V(g)$ and containing at least one point
in each edge that is a closed geodesic. Then there exists a decomposition of
$M(g)$ into finitely many convex hyperbolic polyhedra with vertices in $W$.

Here by a decomposition we mean that every point in $M(g)$ belongs to at least
one polyhedron, the interiors of the polyhedra are disjoint, and if the
intersection of two polyhedra is non-empty, then this intersection is a face
of both polyhedra. In this section by a face of a polyhedron we sometimes mean
a face of any dimension.

###### Proof.

Consider the universal cover $\widetilde{M}(g)\subset\mathbb{H}^{3}$ in the
hyperboloid model, let $\widetilde{W}$ be the full preimage of $W$. It is easy
to see that after we added to $W$ a point on each edge that is a closed
geodesic, $\widetilde{M}(g)$ is the convex hull of $\widetilde{W}$ in the
usual sense, without taking the closure. So each point of $\widetilde{M}(g)$
is a convex combination of finitely many points of $\widetilde{W}$.

Let $C^{\prime}$ be the convex hull of $\widetilde{W}$ in $\mathbb{R}^{3,1}$
and $C$ be the set of points $p$ of $\partial C^{\prime}$ visible from the
origin $o$, i.e., such that the segment $op$ in $\mathbb{R}^{3,1}$ does not
intersect $C^{\prime}$ except $p$. Take $p_{0}\in\widetilde{M}(g)$. As it is a
convex combination of finitely many points of $\widetilde{W}$, the ray
$op_{0}$ intersects $C^{\prime}$. By definition, the first intersection point
$p$ belongs to $C$. One can see that $C$ is homeomorphic to $\widetilde{M}(g)$
via the central projection from the origin, and is closed in
$\mathbb{R}^{3,1}$.

Consider a supporting plane $\Pi$ to $C$ that does not pass though the origin,
let $H:=\Pi\cap\mathbb{H}^{3}$. Depending if $\Pi$ is space-like, light-like
or time-like, $H$ is a sphere, a horosphere or a hypersphere respectively. We
prove that in the latter two cases

######  Claim 3.23.1.

$\partial_{\infty}H\cap\Lambda(\overline{g})=\emptyset.$

Note that as $\Pi\cap C\neq\emptyset$, we have
$\Pi\cap\widetilde{W}\neq\emptyset$. The set $H$ bounds an open convex set
$H_{+}$, which does not contain points of $\widetilde{W}$ since $\Pi$ is a
supporting plane for $C$.

In order to prove Claim 3.23.1 we need to introduce a special notion. A point
$s\in\partial_{\infty}\mathbb{H}^{3}$ is called a _conical limit point_ for a
Kleinian group $\Gamma$ if for every geodesic line $\psi$ through $s$ and
every $p\in\mathbb{H}^{3}$ there exists a sequence $\gamma_{i}\in\Gamma$ such
that $\gamma_{i}(p)$ converges to $s$ and stays at a bounded distance from
$\psi$. A theorem of Beardon–Maskit [7] says that if $\Gamma$ is convex
cocompact, then every limit point of $\Gamma$ is conical. See also a
discussion in [65, Chapter 2], particularly Theorem 2.7.2 there.

Suppose that $H$ is a horoball and that there is a point
$s\in\partial_{\infty}H\cap\Lambda(\overline{g})$. We have a point
$s\in\Lambda(\overline{g})$ with an open horoball $H_{+}$ centered at it,
which does not contain points of $\widetilde{W}$. In particular it does not
contain points from the $\rho_{\overline{g}}$-orbit of a point in
$\mathbb{H}^{3}$. This contradicts to the claim that $s$ is a conical limit
point for $\rho_{\overline{g}}(\pi_{1}(M))$.

Suppose that $H$ is a hypersphere and that there is a point
$s\in\partial_{\infty}H\cap\Lambda(\overline{g})$. The set $H$ is equidistant
from a plane $H_{0}\subset H_{+}$ with
$\partial_{\infty}H=\partial_{\infty}H_{0}$. As $H_{+}$ does not contain
points of $\widetilde{W}$, $H_{0}$ is disjoint from $\widetilde{M}(g)$. We
remark that $\partial_{\infty}H\cap\Lambda(\overline{g})=s$. Indeed, if there
were more than one point, then $H_{0}$ would contain a line from
$\widetilde{C}(\overline{g})\subset\widetilde{M}(g)$, which is a
contradiction.

By Lemma 3.22, there exists a a supporting plane $\Pi_{0}$ to
$\widetilde{M}(g)$ passing through $s$ and asymptotically parallel to $\Pi$.
Because of the conditions on $\Pi_{0}$, the set $\Pi_{0}\cap\widetilde{M}(g)$
is either an edge, or a face of $\partial\widetilde{M}(g)$. In the first case
denote the edge by $\widetilde{\psi}$. Since $g$ is polyhedral,
$\widetilde{\psi}$ is the axis of a purely hyperbolic isometry
$\gamma\in\rho_{\overline{g}}(\pi_{1}(M))$. Either $\gamma$ or $\gamma^{-1}$
sends $H$ to $H_{+}$. This contradicts with
$H_{+}\cap\widetilde{W}=\emptyset$.

Otherwise denote the face $\Pi_{0}\cap\widetilde{M}(g)$ by $\widetilde{F}$ and
remark that $H\cap\Pi_{0}$ is a horocycle based at $s$. Indeed, there exists a
unique one-parametric parabolic subgroup of ${\rm Iso}^{+}(\mathbb{H}^{3})$
fixing $s$ and fixing $\Pi$ and $\Pi_{0}$ as sets. Thus, it also fixes as sets
$H$ and $H\cap\Pi_{0}$. Hence, the orbit of every point from $H\cap\Pi_{0}$ is
the whole $H\cap\Pi_{0}$. On the other hand, this orbit must be a horocycle.

Now consider a subgroup $\Gamma$ of $\rho_{\overline{g}}(\pi_{1}(M))$
consisting of all elements fixing $\Pi_{0}$. This is a convex cocompact
Kleinian group, and $s$ must be a conical limit point for $\Gamma$. Thus, the
$\Gamma$-orbit of any point of $\widetilde{W}\cap\widetilde{F}$ must enter
$H_{+}\cap\widetilde{\Pi}_{0}$. This again contradicts with
$H_{+}\cap\widetilde{W}=\emptyset$ and finishes the proof of Claim 3.23.1.

Now we claim that $\Pi\cap\widetilde{W}$ is finite. If $\Pi$ is space-like,
then $H:=\Pi\cap\mathbb{H}^{3}$ is compact, so the claim follows from the
discreteness of $\widetilde{W}$ in $\mathbb{H}^{3}$. If $\Pi$ is time-like or
light-like and $\Pi\cap\widetilde{W}$ is infinite, then the discreteness of
$\widetilde{W}$ implies that there is
$s\in\partial_{\infty}H\cap\Lambda(\overline{g})$, which is a contradiction
with Claim 3.23.1.

Let $p$ be in the relative interior of $C$ and $\Pi$ be a support plane at
$p$. The point $p$ is a convex combination of at least two points of
$\widetilde{W}$, which are contained in $\Pi$. From the previous argument,
$\Pi\cap\widetilde{W}$ is finite, so $\Pi\cap C$ is the convex hull of
finitely many points of $\widetilde{W}$. We claim that there exists a
supporting plane through $p$ containing at least four points of
$\widetilde{W}$. Indeed, if it is less than four and it is not possible to
rotate $\Pi$ to increase $\Pi\cap\widetilde{W}$, then there exists a sequence
$\\{\widetilde{w}_{i}\\}\subset\widetilde{W}$ such that the distance from
$\widetilde{w}_{i}$ to $\Pi$ tends to zero. Due to the discreteness of
$\widetilde{W}$, up to taking a subsequence we assume that
$\\{\widetilde{w}_{i}\\}$ converges to $s\in\Lambda(\overline{g})$. It follows
that $\Pi$ is not space-like and $s\in\partial_{\infty}\Pi$, which is a
contradiction.

We see that $C$ decomposes into polyhedral 3-dimensional faces. We want to
show that this decomposition is locally finite. It is enough to show that for
each point $\widetilde{w}\in\widetilde{W}$ there are only finitely many
segments from $\widetilde{w}$ to other points of $\widetilde{W}$ belonging to
$C$. Indeed, otherwise from the discreteness of $\widetilde{W}$ and the
closeness of $C$ we obtain that there exists a light-like segment from
$\widetilde{w}$ belonging to $C$ pointing to $\Lambda(\overline{g})$. A
supporting plane $\Pi$ containing this segment is light-like and for
$H=\Pi\cap\mathbb{H}^{3}$ we have
$\partial_{\infty}H\cap\Lambda(\overline{g})\neq\emptyset$, which is a
contradiction with Claim 3.23.1.

By the central projection from the origin, this decomposition projects to a
locally finite decomposition of $\widetilde{M}(g)$ into convex hyperbolic
polyhedra with vertices in $\widetilde{W}$. As $C$ is invariant with respect
to $\rho_{\overline{g}}$, so is the decomposition, and it produces a
decomposition of $M(g)$ into convex polyhedra with vertices in $W$ as desired.

∎

######  Lemma 3.24.

Let $g$ be a polyhedral hyperbolic metric on $M$ and let $M(g)$ be decomposed
into finitely many convex hyperbolic polyhedra with vertices in a set $W$ such
that $V(g)\subseteq W\subset\partial M(g)$. Then there exists a finite cover
$M^{\prime}(g^{\prime})$ of $M(g)$, where $g^{\prime}$ is the lift of the
metric $g$, such that the lift of the decomposition to
$M^{\prime}(g^{\prime})$ can be subdivided to a triangulation without adding
new vertices, and the obtained triangulation is super-large.

###### Proof.

######  Claim 3.24.1.

There exists a finite cover $M^{\prime}(g^{\prime})$ of $M(g)$ such that the
polyhedral decomposition lifted to $M^{\prime}(g^{\prime})$ is super-large.

Here the notion of super-largeness from Definition 2.32 is applied to
polyhedral decompositions in a straightforward way. Indeed, let
$P_{1},\ldots,P_{r}$ be an enumeration of all polyhedra of the decomposition
of $M$. Choose a lift $\widetilde{P}_{i}$ for each $P_{i}$. Consider the set
$\Gamma$ of all $\gamma\in\pi_{1}(M)$ such that for some vertex
$\widetilde{w}$ of some $\widetilde{P}_{i}$, the vertex $\gamma\widetilde{w}$
is a vertex of the same $\widetilde{P}_{i}$. As the stabilizer of each
$\widetilde{w}$ is trivial, the set $\Gamma$ is finite. As $\pi_{1}(M)$ is
linear, it is residually finite [53, Theorem III.7.11], and there exists a
normal subgroup $S\triangleleft\pi_{1}(M)$ of finite index not containing any
element of $\Gamma$.

Consider the cover $M^{\prime}$ of $M$ corresponding to $S$. Suppose that it
does not satisfy our claim. Then there exists $\widetilde{P}^{\prime}$ of
$\widetilde{M}(g)$, a vertex $\widetilde{w}^{\prime}$ of
$\widetilde{P}^{\prime}$ and an element
$\gamma^{\prime}\in\pi_{1}(M^{\prime})$ such that
$\gamma^{\prime}\widetilde{w}^{\prime}$ is a vertex of
$\widetilde{P}^{\prime}$. Note that
$\widetilde{P}^{\prime}=\gamma\widetilde{P}_{i}$ for some
$\gamma\in\pi_{1}(M)$. Then $\widetilde{w}:=\gamma^{-1}\widetilde{w}^{\prime}$
is a vertex of $\widetilde{P}_{i}$. Hence,
$\gamma^{-1}\gamma^{\prime}\gamma\widetilde{w}$ is again a vertex of
$\widetilde{P}_{i}$. Thus, $\gamma^{-1}\gamma^{\prime}\gamma\in\Gamma$. On the
other hand, as $S$ is normal, $\gamma^{-1}\gamma^{\prime}\gamma\in S$, which
contradicts our construction of $S$. This finishes the proof of the claim.

Now we consider the cover $M^{\prime}$ of $M$ as from Claim 3.24.1. The lift
of the polyhedral decomposition to $M^{\prime}$ is super-large and it is well-
known that any such decomposition is triangulable. We give a quick proof of
this fact.

Let us construct a correspondence $f:\mathcal{C}\rightarrow W$ from the set of
polyhedra $\mathcal{C}$ to the set of vertices with the following properties.
(1) For every $P$ in $\mathcal{C}$, $f(P)$ is a vertex of $P$.
(2) If $P_{1}$ and $P_{2}$ are adjacent by a 2-face and both $f(P_{1})$,
$f(P_{2})$ belongs to this 2-face, then $f(P_{1})=f(P_{2})$.

To construct this correspondence, start from $P_{1}$ and let $f(P_{1})$ be any
its vertex $w$. Next, for each $P_{i}$ adjacent to $w$ set $f(P_{i})=w$. Now
at each next step take $P_{j}$ for which $f(P_{j})$ is still not defined. Take
any its vertex $w$ that is not in the image of $f$ and set $f(P_{i})=w$ for
each polyhedron $P_{i}$ adjacent to $w$. Note that for some of them $f(P_{i})$
could have been already defined, but we redefine it now. After finitely many
steps we obtain the map $f$ defined over all $\mathcal{C}$ satisfying the
required properties.

Now for each $P\in\mathcal{C}$ consider all 2-faces of $P$ adjacent to $f(P)$
and triangulate them by coning from the vertex $f(P)$. Clearly, there are no
contradictions due to the condition (2). Next, triangulate all remaining
2-faces arbitrarily. For each $P$ the obtained triangulation of $\partial P$
can be easily extended to a triangulation of $P$ by coning it from $f(P)$.
This provides the desired triangulation $\mathcal{T}$. ∎

###  3.3 Topology of the space of bent metrics

Let $V$ be a finite set of cardinality $n$. By $\mathcal{CH}(N,V)$ denote the
space of equivalence classes of pairs $(\overline{g},f)$ where $\overline{g}$
is a convex cocompact hyperbolic metric on $N$, ${f:V\hookrightarrow N}$ is an
injective map, and two pairs $(\overline{g}_{1},f_{1})$,
$(\overline{g}_{2},f_{2})$ are equivalent if there is an orientation-
preserving isometry $\iota:N(\overline{g}_{1})\rightarrow N(\overline{g}_{2})$
such that $f_{2}=\iota\circ f_{1}$ and $\iota$ is isotopic to identity by an
isotopy sending $f_{2}(v)$ to $f_{1}(v)$ for each $v\in V$. We will call the
points in $f(V)$ _marked_. We will write elements of $\mathcal{CH}(N,V)$ as
$[\overline{g},f]$. As in Section 2.4, we similarly define the space
$\mathcal{CH}_{h}(N,V)$ with isotopy replaced by homotopy. We are going to
endow $\mathcal{CH}(N,V)$ with a natural smooth structure such that the
forgetful map $\mathcal{CH}(N,V)\rightarrow\mathcal{CH}(N)$ is a smooth
bundle. This is very similar to the treatment of the space of metrics with
marked oriented planes performed by us in [72, Section 2.5].

Again, it is easier to define a smooth topology on $\mathcal{CH}_{h}(N,V)$. By
$(\mathbb{H}^{3})^{*V}$ we denote the space of injective maps
$\widetilde{f}:V\hookrightarrow\mathbb{H}^{3}$. Consider the direct product

$\mathcal{R}(\pi_{1}(M),G)\times(\mathbb{H}^{3})^{*V}.$

Let $G$ act on it from the left, by conjugation on the first factor and by
isometries of $\mathbb{H}^{3}$ on the second factor. Additionally,
$\pi_{1}(M)$ acts from the left on each fiber by
$\gamma(\rho,\widetilde{f})=(\rho,\rho(\gamma)\widetilde{f})$, where
$(\rho,\widetilde{f})\in\mathcal{R}(\pi_{1}(M),G)\times(\mathbb{H}^{3})^{*V}$.
One can check that these actions commute. By taking the quotient over both
actions we obtain a fibration over $\mathcal{X}(\pi_{1}(M),G)$. We denote the
total space of the fibration by $\mathcal{X}(\pi_{1}(M),G,V)$. With the help
of developing and holonomy maps the set $\mathcal{CH}_{h}(N,V)$ injects in
$\mathcal{X}(\pi_{1}(M),G,V)$ as an open smooth subset so that the forgetful
map $\mathcal{CH}_{h}(N,V)\rightarrow\mathcal{CH}_{h}(N)$ is a smooth bundle
with fibers homeomorphic to $N^{*V}$, the space of injective maps
$f:V\hookrightarrow N$. The group $MCG_{h}(M)$ acts on $\mathcal{CH}(N)$
freely and properly discontinuously determining a covering map
$\mathcal{CH}(N)\rightarrow\mathcal{CH}_{h}(N).$ We have the commutative
diagram

${\mathcal{CH}(N,V)}$${\mathcal{CH}_{h}(N,V)}$${\mathcal{CH}(N)}$${\mathcal{CH}_{h}(N)}$

from which we can locally pull back the topology on $\mathcal{CH}_{h}(N,V)$ to
the topology on $\mathcal{CH}(N,V)$ preserving the bundle structure. This
endows $\mathcal{CH}(N,V)$ with the topology of a smooth manifold of dimension
$3(n-k)$ where $k$ is the Euler characteristic of $\partial M$.