\section{Introduction}

\subsection{Metric geometry of convex polyhedra}

We start from a basic definition

\begin{dfn}
Let $S$ be a closed surface. We say that $d$ is a \emph{Euclidean cone-metric} (resp. \emph{spherical} or \emph{hyperbolic}) on $S$ if it is locally isometric to the Euclidean plane (resp.  the standard sphere or the hyperbolic plane) except finitely many points where it is locally isometric to a Euclidean cone (resp. spherical or hyperbolic) of total angle $\neq 2\pi$. We say that $d$ is \emph{convex} if all cone angles are $<2\pi$. We say that $d$ is \emph{concave} if all cone angles are $>2\pi$.
\end{dfn}

A theorem of Alexandrov~\cite[Chapter VI]{Ale} states that a convex Euclidean cone-metric on the 2-sphere is realized as the induced metric on the boundary of a unique up to isometry convex polyhedron in the Euclidean 3-space $\mathbb{E}^3$. The induced metric here means the induced path metric and the realization might degenerate to the double cover of a convex polygon. This is a polyhedral version of a fundamental result in differential geometry: every Riemannian metric of positive Gaussian curvature on the 2-sphere is realized as the induced metric on the boundary of a unique up to isometry smooth convex body in $\mathbb{E}^3$. This completely characterizes the induced metrics on the boundaries of smooth strictly convex bodies, i.e., those with a positive-definite shape operator at every boundary point. The existence part is known as the famous \emph{Weyl problem} resolved by Nirenberg~\cite{Nir} from one side, and by Alexandrov and Pogorelov~\cite{Ale, Pog} from another side. The uniqueness part is known as the global rigidity of closed smooth strictly convex surfaces in $\mathbb{E}^3$. We refer to Cohn-Vossen~\cite{CoV} and to Herglotz~\cite{Her} as to one of the main contributors.

Alexandrov noticed that his theorem can be transferred without changes to polyhedra in the standard 3-sphere $\S^3$ and in the hyperbolic 3-space $\H^3$. Convex bodies in these spaces have another interesting associated metric structure, which we call \emph{the dual metric}, and which is somewhat degenerate in the Euclidean 3-space $\mathbb{E}^3$. In the standard 3-sphere $\S^3$ there is a natural duality between points and oriented 2-planes. For a closed convex body $C \subset \S^3$ this allows to define \emph{the Gauss map} as a multivalued map on $\partial C$ sending $p \in \partial C$ to the set of points dual to supporting planes to $C$ at $p$ oriented outwards $C$. The image of $\partial C$ under this map is a surface bounding another closed convex set $C^* \subset \S^3$, which we call \emph{the dual} of $C$. We can also define $C^*$ as the set of points dual to all 2-planes not intersecting ${\rm int}(C)$ oriented outwards $C$.

This duality is polar on closed convex sets, i.e., $(C^*)^*=C$. It follows from the Alexandrov theorem for $\S^3$ that a convex polyhedron $C \subset \S^3$ is uniquely determined by the induced metric on the boundary of $C^*$. This does not work the same in the Euclidean 3-space: the natural metric structure on \emph{the co-Euclidean space}, which is the space of oriented 2-planes in $\mathbb{E}^3$, is degenerate, and the induced metric on the dual convex set is always isometric to the standard 2-sphere $\S^2$. Parallel translations allow to consider $\S^2$ as the target for the Gauss map on convex surfaces in $\mathbb{E}^3$. We refer to~\cite{FS} for a more detailed exposition.

The situation is more interesting in the hyperbolic 3-space $\H^3$. The set of oriented 2-planes in $\H^3$ is naturally identified with \emph{the de Sitter 3-space} $d\mathbb{S}^3$, which is a model Lorentzian 3-space of constant curvature 1. See Section~\ref{minkow} for more details on this duality. A non-trivial work was required to understand metric properties of dual convex bodies in this setting. For the case of compact convex polyhedra in $\H^3$ this resulted in the following theorem of Rivin--Hodgson~\cite{HR}:

\begin{dfn}
A metric on a surface $S$ is called \emph{large} if the lengths of all closed contractible geodesics are greater than $2\pi$.
\end{dfn}

\begin{thm}
\label{HR}
For every concave large spherical cone-metric $d^*$ on the 2-sphere $S$ there is a unique up to isometry compact convex polyhedron $C \subset \H^3$ such that $(S, d^*)$ is isometric to the boundary of its dual $C^* \subset d\mathbb{S}^3$.
\end{thm}

One can see that the conditions on $d^*$ in Theorem~\ref{HR} are also necessary (the only non-trivial condition is largeness, and its necessity is proven in~\cite{HR}). Theorem~\ref{HR} can be seen as a generalization of the Andreev theorem~\cite{And} on compact convex polyhedra in $\H^3$ with non-obtuse dihedral angles, another generalization of which was famously used by Thurston in his hyperbolization theorem for Haken 3-manifolds, see~\cite{Thu, Mor}. Theorem~\ref{HR} was also used by Rivin in~\cite{Riv2} to describe the dihedral angles of ideal polyhedra in $\H^3$, which in turn implied a resolution of the Steiner problem. The latter asked how to distinguish the combinatorial types of convex Euclidean 3-polyhedra that admit a realization with all vertices on the standard 2-sphere. In~\cite{Sch} Schlenker proved a version of Theorem~\ref{HR} for smooth strictly convex bodies in $\H^3$. We note that if $C \subset \H^3$ is a smooth strictly convex body, then the pull-back of the dual metric by the Gauss map is exactly \emph{the third fundamental form} of $\partial C$.

\subsection{Hyperbolic 3-manifolds with convex boundary}

In seventies Thurston revolutionized the field of 3-dimensional topology by showing the importance of geometric structures on 3-manifolds, posing his famous geometrization conjecture (whose proof was completed by Perelman) and demonstrating a special role of hyperbolic geometry in dimension 3. In this paper we will be interested in studying compact hyperbolic 3-manifolds with convex polyhedral boundary.

Let $M$ be an oriented smooth compact 3-manifold with non-empty boundary and $N$ be its interior.

\begin{dfn}
A subset $C$ of a complete hyperbolic manifold is called \emph{totally convex} if it is closed and contains every geodesic segment between any two points of $C$.
\end{dfn}

\begin{dfn}
A hyperbolic metric $\overline g$ on $N$ is called \emph{convex cocompact} if it is complete and $(N, \overline g)$ contains a compact totally convex subset.
\end{dfn}

We assume that $N$ admits a convex cocompact hyperbolic metric. Then $M$ is irreducible and atoroidal. We also assume that $M$ is distinct from the solid torus. Then all its boundary components have genii at least two. Thurston's hyperbolization theorem implies that the converse holds: the interior of a compact 3-manifold satifying these conditions admits a convex cocompact hyperbolic metric, see~\cite{Mar2, Mor}. Convex cocompact hyperbolic metrics are generic as they constitute the interior of the moduli space of complete hyperbolic metrics on $N$.

Let $M$ be endowed with a hyperbolic metric $g$ such that the boundary is locally convex. By $d$ we denote the induced metric on $\partial M$. A developing map ${\rm dev}: \tilde M \rightarrow \H^3$ embeds isometrically the universal cover $(\tilde M, g)$ to $\H^3$ as a (non-compact) convex set equivariant with respect to the holonomy map $\rho: \pi_1(M) \rightarrow {\rm Iso}^+(\H^3)$, where ${\rm Iso}^+(\H^3)$ is the group of orientation-preserving isometries of $\H^3$. We look at the dual convex set $({\rm dev}(\tilde M))^* \subset d\mathbb{S}^3$. The isometry groups of both ambient spaces can be identified together, so ${\rm Iso}^+(\H^3)={\rm Iso}^+(d\mathbb{S}^3)$ and $({\rm dev}(\tilde M))^*$ is, of course, also equivariant with respect to $\rho$. However, the action may be not free on $({\rm dev}(\tilde M))^*$. But under some generic conditions (positive distance from $\partial M$ to the convex core of $(M, g)$, which will be defined later) it is free and properly discontinuous on $\partial({\rm dev}(\tilde M))^*$. By taking the quotient, we obtain \emph{the induced dual metric} $d^*$ on $\partial M$. In the case of smooth strictly convex boundary, $d^*$ is exactly the third fundamental form of $\partial M$.

Gabai-Meyerhoff-Thurson~\cite{GMT} showed that any homeomorphism between closed hyperbolic 3-manifolds is isotopic to an isometry, refining in dimension 3 the well-known Mostow rigidity~\cite{Mos}. Due to the rigidity and the dual rigidity of compact convex bodies in $\H^3$, it is interesting to ask if $d$ and $d^*$ determine $g$ up to isotopy, and if all metrics satisfying natural necessary conditions are realizied for some choice of $g$. For the case of smooth strictly convex boundary this was done by Schlenker in~\cite{Sch4}:

\begin{thm}
\label{sch1}
Let $M$ be as above. For every smooth Riemannian metric $d$ on $\partial M$ of Gaussian curvature $>-1$ there exists a unique up to isotopy hyperbolic metric $g$ on $M$ such that $\partial M$ is strictly convex in $g$ and the induced metric on $\partial M$ is $d$.
\end{thm}

\begin{dfn}
We say that a metric $d^*$ on $\partial M$ is \emph{large for $M$} if the lengths of all closed geodesics in $(\partial M, d^*)$ that are contractible in $M$ are greater than $2\pi$.
\end{dfn}

\begin{thm}
\label{sch2}
Let $M$ be as above. For every smooth large for $M$ Riemannian metric $d^*$ on $\partial M$ of Gaussian curvature $<1$ there exists a unique up to isotopy hyperbolic metric $g$ on $M$ such that $\partial M$ is strictly convex in $g$ and the induced dual metric on $\partial M$ is $d^*$.
\end{thm}

It is natural to expect that polyhedral counterparts of Theorem~\ref{sch1} and~\ref{sch2} should hold. Curiously enough, the approach to infinitesimal rigidity (which is the key part of the argument) from~\cite{Sch4} is too much specific to the smooth situation and does not seem to admit a discrete analogue. In the same time, standard methods of proving the infinitesimal rigidity of polyhedra in space-forms use fundamentally the fact that the boundary is the 2-sphere, and become not applicable in more non-trivial topological situations. Hence, we needed to work out a different technique to bridge this gap. For a polyhedral version of Theorem~\ref{sch1} an additional complication arises from the necessity to consider not just hyperbolic 3-manifolds with polyhedral boundary, but a more general class of so-called \emph{bent} hyperbolic 3-manifolds. A good example is provided by \emph{the convex core} of a convex cocompact hyperbolic 3-manifold, which is the inclusion-minimal totally convex subset, see more details in Section~\ref{cocomp}. Its boundary is convex and intrinsically locally isometric to the hyperbolic plane, but it is bent along a geodesic lamination. This lamination may consist of uncountably many disjoint open geodesics winding around each other in a sophisticated manner, see~\cite{Thu, BO} for explanations. It is not known currently, how to establish the infinitesimal rigidity for such manifolds, which would be an important step towards resolving our problem. However, in our very recent work~\cite{Pro3} we prove the existence part of the problem and obtain the uniqueness under mild restrictions on the boundary.

The mentioned effects, however, do not considerably impact the dual problem, as the dual metric degenerates in the mentioned cases (though then it can be replaced by another metric object, and it is a very interesting task to deal with the rigidity in this setting, see the end of this section). This paper is devoted to a resolution of the dual problem, i.e., to a polyhedral counterpart of Theorem~\ref{sch2}.

\begin{dfn}
We say that $g$ is a \emph{strictly polyhedral} hyperbolic metric on $M$ if the interior is locally hyperbolic, the boundary is locally isometric to convex polyhedral sets in $\H^3$, and the distance from $\partial M$ to the convex core is positive.
\end{dfn}

Again, for more explanations on convex cores we refer to Section~\ref{cocomp}. Our result is

\begin{thm}
\label{mt}
Let $M$ be as above. For each concave large for $M$ spherical cone-metric $d^*$ on $\partial M$ there exists a unique up to isotopy strictly polyhedral hyperbolic metric $g$ on $M$ such that the induced dual metric on $\partial M$ is $d^*$.
\end{thm}

This is a generalization of Theorem~\ref{HR} to arbitrary compact hyperbolic 3-manifolds with boundary. We note that the condition of strict convexity in Theorems~\ref{sch1} and~\ref{sch2} implies that the boundary is on positive distance from the convex core. Thus, Theorem~\ref{mt} includes a polyhedral version of Theorem~\ref{sch2}. When the boundary of $M$ touches the convex core, the dual metric degenerates to a pseudo-metric, hence the condition of strict polyhedrality in Theorem~\ref{mt} is necessary to have a well-definite dual metric on $\partial M$. In Section~\ref{polyhsec} we explain that the conditions on $d^*$ are also necessary (again, this is non-trivial only for the condition of largeness for $M$).

The only previously known case of Theorem~\ref{mt} was established by Schlenker in~\cite{Sch5} for so-called \emph{Fuchsian hyperbolic 3-manifolds of the first kind} with convex polyhedral boundary. They are homeomorphic to $S \times [-1, 1]$, where $S$ is a closed surface of genus $>1$, and contain an embedded totally geodesic surface isotopic to $S \times \{0\}$. See also a paper of Fillastre~\cite{Fil}. We note that our Theorem~\ref{mt} answers positively to Question~2 in~\cite{Fil}. In the Fuchsian case of the first kind Fillastre also established in~\cite{Fil2} a polyhedral counterpart to Theorem~\ref{sch1}. Similar results for the Minkowski and anti-de Sitter 3-spaces were obtained in~\cite{Fil}.

A particularly interesting case of Theorem~\ref{mt} is given by so-called \emph{quasi-Fuchsian hyperbolic 3-manifolds} with boundary. Those are also homeomorphic to $S \times [-1, 1]$, but the condition on the embedded geodesic surface is dropped. The quotient of $d\mathbb{S}^3$ under a holonomy map of such hyperbolic manifold is known in Lorentzian geometry as a \emph{globally hyperbolic maximal Cauchy compact} (GHMC for short) (2+1)-spacetime of constant curvature 1. See, e.g., the fundamental paper of Mess~\cite{Mes} for more details. This case of Theorem~\ref{mt} can be interpreted as a convex polyhedral realization and rigidity result in such spacetimes.

Our proof of Theorem~\ref{mt} follows the standard continuity approach popularized by Alexandrov. We consider the space $\mathcal P(M, V)$ of isotopy classes of strictly polyhedral metrics on $M$ with marked faces, and the space $\mathcal D^*_M(\partial M, V)$ of isotopy classes of concave large for $M$ spherical cone-metrics on $\partial M$ with marked vertices. We endow these spaces with appropriate topologies and consider the realization map $\mathcal I_V: \mathcal P(M, V) \rightarrow \mathcal D^*_M(\partial M, V)$. We show that $\mathcal I$ is a $C^1$-diffeomorphism.

The proof is decomposed into three pieces: establishing differentiability properties of $\mathcal I$ and its infinitesimal rigidity, showing its properness, and studying connectivity properties of the spaces $\mathcal P(M, V)$ and $\mathcal D^*_M(\partial M, V)$. The last item was basically done before us: it is a straightforward generalization of the connectivity results from~\cite{HR}, as it was noted by Schlenker in~\cite{Sch5}. Thus, we mostly focus on the infinitesimal rigidity and the properness. Note also that the connectivity results on the spaces above are obtained only in some weak sense, and a special topological argument is required to finish the proof.

We deduce the infinitesimal rigidity via the doubling construction from the works on the infinitesimal rigidity of hyperbolic cone-3-manifolds due to Weiss and Montcouquiol-Weiss~\cite{Wei, Wei2, MW}. Note that in the presence of vertices of valence $>3$ the infinitesimal rigidity in our problem is not the same as the infinitesimal rigidity with respect to the dihedral angles, which follows from the aforementioned works rather directly, and we need to extract the former with some care. Hyperbolic cone-3-manifolds have interesting applications to various geometric problems, see, e.g.,~\cite{BLP, Bro, HK2, Pro2, Pro}. The proof of properness is partially inspired by Schlenker's works~\cite{Sch2, Sch3}, which deal with Alexandrov-type problems for polyhedra in the union of $\H^3$ and $d\mathbb{S}^3$.

An interesting issue with the continuity method is that it gives no construction of a hyperbolic metric $g$ inducing the dual metric $d^*$. This might be resolved by a variational approach. A variational proof of Alexandrov's theorem was given by Volkov, a student of Alexandrov, in his thesis, published only recently as~\cite{Vol}. A refinement of Volkov's approach was given in~\cite{BI} by Bobenko--Izmestiev. It was applied to similar problems for polyhedral hyperbolic 3-manifolds, e.g., in~\cite{FI, Pro}. A particularly relevant for us reference is~\cite{FI2} where the problem of induced dual metrics on the so-called \emph{hyperbolic 3-cusps} with convex polyhedral boundary was resolved by variational techniques. It would be very interesting to obtain a variational proof of Theorem~\ref{mt} or a proof of a polyhedral analogue of Theorem~\ref{sch1}, but at the moment there are difficulties with attempts to implement such intentions.

Up to now we discussed in parallel questions for polyhedral and for smooth boundaries. It is very natural to ask if there is a common generalization unifying these two settings. This was famously initiated by Alexandrov in his work on general metrics on surfaces with curvature bounded from below~\cite{Ale}. He particularly proved a realization theorem, obtaining a complete intrinsic classification of the induced path metrics on the boundaries of convex bodies in $\mathbb{E}^3$, $\H^3$ or $\S^3$. The rigidity question for general convex bodies was settled by Pogorelov~\cite{Pog} and is known as notoriously difficult. In recent years there was a stream of work showing general realization results on hyperbolic 3-manifolds, e.g.,~\cite{Slu, FIV}. The only known general rigidity result for manifolds topologically more complicated than the 3-ball (for Fuchsian manifolds of the first kind) was done by the author in~\cite{Pro2}. The corresponding results for the dual metrics are still not known. An important particular case, again, is given by the convex cores of convex cocompact hyperbolic 3-manifolds. Their dual metric is somewhat degenerate, but can be replaced with another object called \emph{the pleating lamination} measuring how their boundary is bent. Thurston conjectured that convex cores are uniquely determined by their pleating lamination as well as by the induced boundary metric. The realization part of the former question was shown in~\cite{BO} and~\cite{Lec} by Bonahon--Otal and Lecuire, while for the latter it is, e.g., a particular corollary of the work~\cite{Lab} by Labourie, where another proof of the realization part of Theorem~\ref{sch1} was given. The rigidity part remains to be an open problem. This is reminiscent to the famous Ahlfors--Bers theorem (see Theorem~\ref{ab}), which could be stated as that convex cocompact hyperbolic metrics on 3-manifolds are determined up to isotopy by the conformal structure at infinity.

\vskip+0.2cm

\textbf{Acknowledgments.} We thank Fran\c{c}ois Fillastre, Ivan Izmestiev and Jean-Marc Schlenker for helpful comments, in particular, leading to a simplification of the proof of Lemma~\ref{differ}. We are also grateful to our mentor, Michael Eichmair, for his support.

\section{Preliminaries}

\subsection{Hyperbolic and de Sitter spaces and the duality}
\label{minkow}

For some expositions on the duality between hyperbolic and de Sitter spaces we refer to \cite{HR, Sch2, FS}.

By $\langle .,. \rangle_M$ we denote the \emph{Minkowski scalar product} on $\mathbb{R}^4$, i.e.,
$$\langle x, y \rangle_M := -x_0y_0+x_1y_1+x_2y_2+x_3y_3.$$
The hyperbolic 3-space $\H^3$ can be identified then with the upper sheet of the two-sheeted hyperboloid
$$\H^3=\{x \in \mathbb{R}^4: \langle x,~x \rangle_M = -1, x_0>0\}$$
endowed with the induced metric. The de Sitter 3-space $d\mathbb{S}^3$ can be identified with the one-sheeted hyperboloid
$$d\mathbb{S}^3=\{x \in \mathbb{R}^4: \langle x, x \rangle_M =1\}$$
endowed with the induced metric. It is a Lorentzian 3-manifold of constant curvature 1 homeomorphic to $S^2 \times \mathbb{R}$. As the hyperbolic 3-space $\H^3$ is compactified by adding the boundary at infinity $\partial_{\infty} \H^3$, the de Sitter 3-space is compactified similarly by adding the boundary at infinity $\partial_{\infty} d\mathbb{S}^3$ consisting of two connected components, $\partial_{\infty}^+ d\mathbb{S}^3$ and $\partial_{\infty}^- d\mathbb{S}^3$, corresponding to the two parts of the light cone of the origin in $\mathbb{R}^4$. The component $\partial_{\infty}^+ d\mathbb{S}^3$ is naturally identified with $\partial_{\infty} \H^3$.

There is a well-known duality between oriented planes in $\H^3$ and points in $d\mathbb{S}^3$, and a duality between space-like oriented planes in $d\mathbb{S}^3$ and points in $\H^3$. Indeed, each plane in $\H^3$ is the intersection of $\H^3$ with a linear hyperplane in $(\mathbb{R}^4, \langle .,. \rangle_M)$ passing through the origin. Its unit normal in the positive direction belongs to $d\mathbb{S}^3$. Conversely, for any point in $d\mathbb{S}^3$ the orthogonal hyperplane in $(\mathbb{R}^4, \langle .,. \rangle_M)$ intersects $\H^3$ and determines an oriented geodesic plane there. Similarly, one produces a correspondence between space-like planes in $d\mathbb{S}^3$ and points in $\H^3$.

If $C \subset \H^3$ is a convex set, then dual points to all outward oriented planes that do not intersect the interior of $C$ form the dual convex set $C^* \subset d\mathbb{S}^3$. We say that $C^*$ is \emph{the Gauss dual} of $C$. By \emph{the Gauss map} we mean the multivalued map $\partial C \rightrightarrows \partial C^*$ sending a point $p \in \partial C$ to the set of the outward normals to all supporting planes to $C$ at $p$. It is easy to see that no supporting plane to $C^*$ is time-like, and that $\partial_{\infty}^+ d\mathbb{S}^3$ belongs to the convex side of $C^*$. Conversely, for any convex set $C \subset d\mathbb{S}^3$ satisfying these two conditions we similarly define its dual convex set $C^* \subset \H^3$. This duality is polar, i.e., $C^{**}=C$ when $C \subset \H^3$ is closed and convex or $C \subset d\mathbb{S}^3$ and, in addition, satisfies the conditions we stated.

It will be sometimes helpful for us to consider the \emph{spherical model} of $\H^3$ and $d\mathbb{S}^3$. Denote the \emph{Euclidean scalar product} on $\mathbb{R}^4$ by $\langle .,. \rangle_E$, i.e.,
$$\langle x, y \rangle_E :=x_0y_0+x_1y_1+x_2y_2+x_3y_3.$$
By $\S^3 \subset \mathbb{R}^4$ we denote the standard unit sphere
$$\S^3=\{x \in \mathbb{R}^4: \langle x, x \rangle_E =1\}$$
endowed with the metric induced from $\langle .,. \rangle_E$. The central projection from the origin of $\mathbb{R}^4$ maps injectively $\H^3$ and $d\mathbb{S}^3$ to subsets of $\S^3$ denoted by $\H_S^3$ and $d\mathbb{S}_S^3$. Notably it sends geodesics to geodesics, and, thus, convex sets to convex sets. If $C \subset \H_S^3$ or $C \subset d\mathbb{S}_S^3$ is convex, then the dual set $C^*$ can be obtained from the usual polarity on $\S^3$ composed with the reflection with respect to the hyperplane $x_0=0$. The boundary at infinity $\partial_{\infty} d\mathbb{S}^3$ projects to two spherical caps denoted by $\partial_{\infty}^+ d\mathbb{S}_S^3$ and $\partial_{\infty}^- d\mathbb{S}_S^3$.

\subsection{Cone-metrics}

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

\begin{dfn}
A \emph{triangulation} $\mathcal T$ of $(S, V)$ is a collection of simple disjoint paths with endpoints in $V$ that cut $S$ into triangles. Here a triangle is a 2-disk with three marked points at the boundary (possibly coinciding as points of $V$). Two triangulations with the same vertex set $V$ are \emph{equivalent} if they are isotopic on $S$.
\end{dfn}

We denote by $E(\mathcal T)$ the set of edges of $\mathcal T$.

\begin{rmk}
We highlight that being isotopic on $S$ is a softer equivalence relation than being isotopic on $S \backslash V$, which is also used sometimes. Here is an example. Consider a cube, $S$ be its surface, $V$ be the set of its vertices, $\mathcal T$ be any triangulation refining its face decomposition. Let $A$ and $B$ be opposite faces of the cube. Rotate $B$ by $360^\circ$ while keeping $A$ fixed. Let $\mathcal T'$ be the new triangulation obtained from $\mathcal T$. It is equivalent to $\mathcal T$ in the above sense, but it is not isotopic to it on $S \backslash V$.
\end{rmk}

Let $d$ be a cone-metric on $S$ and $V \subset S$ be a finite set containing the singular set $V(d)$ of $d$. In this situation we say that $d$ is a cone-metric on $(S, V)$.

\begin{dfn}
A \emph{geodesic triangulation} of $(S, V, d)$ is a triangulation of $(S, V)$ such that all edges are geodesics in $d$.
\end{dfn}

\begin{dfn}
Let $\mathcal T$ be a triangulation of $(S, V)$ and $d$ be a cone-metric on $(S, V)$. We say that $\mathcal T$ is \emph{realized} by $d$ if there is a geodesic triangulation of $(S, V, d)$ equivalent to $\mathcal T$. If $d$ is spherical, then we require additionally that the lengths of all edges are less than $\pi$.
\end{dfn}

Note that a geodesic path of length $<\pi$ in a spherical cone-metric is isolated, i.e., it has a neighborhood where it is the only geodesic path between its endpoints. All geodesic paths are isolated in a Euclidean or a hyperbolic cone-metric. Thus, if $\mathcal T$ is realized by $d$, then the realization is unique. We remark that not all triples $(S, V, d)$, when $d$ is a spherical cone-metric, realize a triangulation even in a weaker sense, without the edge-lengths condition. An example is the standard sphere $\S^2$ with $V$ being three points contained in an open hemisphere. However, for concave large spherical cone-metrics there are no such obstructions.

\begin{lm}
\label{triang}
For each concave large spherical cone-metric $d$ on $(S, V)$ there exists a triangulation $\mathcal T$ of $(S, V)$ realized by $d$.
\end{lm}

This was basically shown by Rivin in~\cite{Riv} although he does not formulate the condition on the edge-lengths, but it follows from his proof. For surfaces of genus $>1$, which is the case we need, this also follows directly from the Fuchsian version of Theorem~\ref{sch2} due to Schlenker~\cite{Sch5}. It says that each concave large spherical cone-metric on $S$ can be realized as the dual metric on the boundary of a Fuchsian manifold of the first kind. When we consider the respective dual embedding of the universal cover to $d\mathbb{S}^3$, we see that any equivariant geodesic triangulation refining the face decomposition projects to a triangulation realized by the metric.

By $\mathcal D^*(S, V)$ we denote the set of spherical cone-metrics on $(S, V)$ that realize at least one triangulation, considered up to isometry fixing $V$ and isotopic to identity. We highlight in this definition that although the isometry must fix $V$, this is not required for the isotopy. 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 $\mathcal T$. The set $\mathcal D^*(S, \mathcal T)$ can be considered as an open polyhedron in $\mathbb{R}^{E(\mathcal T)}$ defined by strict triangle inequalities and the conditions on edge lengths to be in $(0, \pi)$. 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 $k:=\chi(S)$. Sometimes we will abuse the notation and write $d \in \mathcal D^*(S, V)$, when $d$ is a metric representing a class belonging to $\mathcal D^*(S, V)$, or $d_t \subset \mathcal D^*(S, V)$ for a path of metrics projecting to a path in $\mathcal D^*(S, V)$. We will need the following simple lemma:

\begin{lm}
\label{fintriang}
Let $d \in \mathcal D^*(S, V)$. There exists a neighborhood $U \ni d$ in $\mathcal D^*(S, V)$ such that there are only finitely many triangulations realized by metrics from $U$.
\end{lm}

\begin{proof}
We show that for any $v, w \in V$ there are only finitely many isotopy classes on $S \backslash V$ of paths between $v$ and $w$ containing a geodesic path of length $\leq \pi$ in $d$. Indeed, suppose the converse.
The set of all geodesic paths from $v$ to $w$ of length $\leq \pi$ is uniformly Lipschitz. Thus, by the Arzel\'a--Ascoli theorem it is compact. Hence, if there are infinitely many such classes, we can take a geodesic representative of length $\leq \pi$ from each, and extract a converging subsequence. However, every geodesic path of length $< \pi$ between $v$ and $w$ is isolated, and every geodesic path of length $\pi$ is isolated from geodesic representatives of other classes. Thus, no path can appear in the limit.

This argument implies that there are finitely many triangulations up to isotopy on $S\backslash V$ that have a geodesic representative in $d$ with all edges $\leq \pi$. Hence, there are also finitely many such triangulations up to isotopy on $S$. Thus, there are finitely many triangulations $\mathcal T$ such that $d$ belongs to the closure of $\mathcal D^*(S, \mathcal T)$. This finishes the proof.
\end{proof}

By $\ol {\mc D}^*_c(S, V) \subset \mc D^*(S, V)$ we denote the subset of isotopy classes of concave spherical cone-metrics on $(S, V)$ and by $\mc D^*_c(S, V) \subset \ol {\mc D}^*_c(S, V)$ we denote the subset of isotopy classes of metrics $d$ additionally satisfying $V(d)=V$.

Now let $M$ be as in the introduction, $V \subset \partial M$ be a finite set of points with at least one point at each boundary component. By $\ol{\mc  D}^*_M(\partial M, V) \subset \mc D^*(\partial M, V)$ we denote the subset of isotopy classes of concave large for $M$ spherical cone-metrics on $(\partial M, V)$. By $\mathcal D^*_M(\partial M, V) \subset \ol{\mc  D}^*_M(\partial M, V)$ we denote the subset of isotopy classes of metrics $d$ additionally satisfying $V(d)=V$. We will show that $\mathcal D^*_M(\partial M, V)$ is an open subset of $\mathcal D^*(\partial M, V)$. Clearly, the concavity condition is open, so we need to check the condition of largeness for $M$. For a metric $d$ on $\partial M$ by ${\rm ml}_M(d)$ denote the infimum of lengths of all closed geodesics in $(\partial M, d)$ that are contractible in $M$.

For a map $f: (S_1, d_1) \rightarrow (S_2, d_2)$ between two compact metric spaces we define its \emph{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 $d$ \emph{in the Lipschitz sense} if the identity maps $(S, d_i) \rightarrow (S, d)$ have distortions $\varepsilon_i \rightarrow 0$. We remark that some authors allow the action by homeomorphisms in the definitions of the Lipschitz convergence, e.g., in~\cite[Section 6]{HR}. In a more general setting the Lipschitz convergence may be defined between non-homeomorphic metric spaces, see~\cite[Section 7.2]{BBI}.
Note that when classes of metrics converge in $\mathcal D^*(S, V)$, they and the limit class can be represented by metrics $d_i$ and $d$ converging in the Lipschitz sense.

By $\S^1_l$ we denote the metric circle of length $l$. A \emph{closed global geodesic} in a metric space $(S, d)$ is the image of an isometric map $\S^1_l \rightarrow (S, d)$ (here it is crucial that it preserves distances in $\S^1_l$ globally). A \emph{closed $\varepsilon$-quasi-geodesic} is the image of $S^1_l$ under an $\varepsilon$-quasi-isometry, i.e., a map $S^1_l \rightarrow (S, d)$ with distortion at most $\varepsilon$. The following consequence of the Arzel\'a--Ascoli theorem was shown in~\cite[Theorem 6.1]{HR}:

\begin{lm}
\label{geodconv}
Let $\{d_i\}$ be a sequence of metrics on $S$ converging in the Lipschitz sense to a metric $d$. Let $\{\gamma_i \subset S\}$ be closed $\varepsilon_i$-quasi-geodesics in $d_i$ with $\varepsilon_i \rightarrow 0$ and the lengths $l_i$ of $\gamma_i$ converging to $l>0$. Then up to extracting a subsequence, $\gamma_i$ converge to a closed global geodesic $\gamma \subset S$ in $d$ of length $l$.
\end{lm}

From this we can deduce

\begin{crl}
Let $\{d_i\}$ be a sequence of metrics on $\partial M$ converging in the Lipschitz sence to a metric $d$. Then
$${\rm ml}_M(d) \leq \liminf_{i \rightarrow \infty} {\rm ml}_M(d_i).$$
\end{crl}

\begin{proof}
When a sequence of closed curves on $\partial M$ converges, their free homotopy classes stabilize. Hence, the limit of a sequence of curves in $\partial M$ that are contractible in $M$ is also contractible in $M$. Next, for each closed geodesic in $(\partial M, d_i)$ of length $l$ that is contractible in $M$ there exists a closed global geodesic of length at most $l$ that is contractible in $M$. Indeed, this is because there is a correspondence between closed geodesics in $(\partial M, d_i)$ contractible in $M$ and closed geodesics in $(\partial \tilde M, d_i)$, where $\tilde M$ is the universal cover of $M$. Thus, by Lemma~\ref{geodconv} the infimum ${\rm ml}_M(d_i)$ is realized by a closed global geodesic $\gamma_i$ contractible in $M$. Applying Lemma~\ref{geodconv} again, we see that a subsequence of $\gamma_i$ converges to a closed global geodesic in $(\partial M, d)$ contractible in $M$ of length $\liminf_{i \rightarrow \infty} {\rm ml}_M(d_i)$.
\end{proof}

\begin{crl}
\label{semicont}
The function ${\rm ml}_M$ is lower-semicontinuous in the Lipschitz topology of metrics on $\partial M$, i.e., for each $l \in \mathbb{R}$ the set of metrics $d$ on $\partial M$ satisfying ${\rm ml}_M(d)>l$ is open. In particular, $\mathcal D^*_M(\partial M, V)$ is an open subset of $\mathcal D^*(\partial M, V)$.
\end{crl}

Let $d \in \mc D^*(\partial M, \mc T)$ and $\lambda \in \mathbb{R}$ be a number. By $(\lambda *_{\mc T} d)$ we denote a metric obtained by changing the length $l_e$ of each edge $e \in E(\mc T)$ to $e^\lambda l_e$ if the lengths remain staying $<\pi$ under this operation. We will use some corollaries of the following easy lemma

\begin{lm}
Let $T$ be a spherical triangle with side lengths $a$, $b$ and $c$ less than $\pi$. Consider a spherical triangle $T'$ with the lengths $e^\lambda a$, $e^\lambda b$, $e^\lambda c$ for $\lambda > 0$ such that the lengths of the new triangle are $<\pi$. Then the angles of $T'$ are at least the respective angles of $T$.
\end{lm}

The proof is identical to the proof of a similar lemma for hyperbolic triangles (\cite[Lemma 2.3.9]{Pro2}) with all inequalities reversed.

\begin{crl}
\label{multipl}
Let $d \in \ol{\mc D}^*_c(\partial M, \mc T)$ and $\lambda>0$. Then $(\lambda *_{\mc T} d) \in \mc D^*_c(\partial M, \mc T)$.
\end{crl}

The proof is straightforward.

\begin{crl}
\label{locconnect}
Let $d \in \ol{\mc D}^*_M(\partial M, V)$. There exists an open neighborhood $U$ of $d$ in $\ol{\mc D}^*_M(\partial M, V)$ such that $U \cap {\mc D}^*_M(\partial M, V)$ is connected.
\end{crl}

\begin{proof}
If $d \in {\mc D}^*_M(\partial M, V)$, then the claim is trivial. Suppose that $d \notin {\mc D}^*_M(\partial M, V)$. Note that $\ol{\mc D}^*_c(\partial M, V)$ is a semi-analytic subset of the manifold $\mc D^*(\partial M, V)$, hence it is locally connected. Choose a connected open neighborhood $U$ of $d$ in $\ol{\mc D}^*_c(\partial M, V)$. Since ${\rm ml}_M(d)>2\pi$ and ${\rm ml}_M$ is upper-semicontinuous, we can assume that $U \subset \ol{\mc D}^*_M(\partial M, V)$. Pick any two metrics $d_0, d_1 \in U \cap {\mc D}^*_M(\partial M, V)$ and connect them by a path $d_t$, $t \in [0,1]$, in $U$. Choose a function $\lambda: [0,1] \rightarrow [0, \varepsilon]$ for some small $\varepsilon>0$ such that $\lambda(0)=\lambda(1)$ and $\lambda(t)>0$ for $t \in (0,1)$. Then, provided that $\varepsilon$ is sufficiently small, the path $(\lambda(t) *_{\mc T} d_t)$ connects $d_0$ and $d_1$ in $U \cap {\mc D}^*_M(\partial M, V)$.
\end{proof}

\subsection{Convex cocompact hyperbolic manifolds}
\label{cocomp}

Let $g$ be a hyperbolic metric on $M$ such that $\partial M$ is locally convex. By $G$ we denote the group ${\rm Iso^+(\H^3)}$ of orientation-preserving isometries of $\H^3$, identified with $PSL(2, \mathbb{C})$.

A developing map embeds the universal cover $(\tilde M, g)$ as a convex set in $\H^3$ (see~\cite[Section 1.4]{CEG}) and the respective holonomy map is a discrete and faithful representation $\rho: \pi_1(M) \rightarrow G$. The quotient $\overline M$ of $\H^3$ by $\rho$ is a hyperbolic 3-manifold, in which $M$ naturally embeds inducing an isomorphism of the fundamental groups. Denote this embedding by $\iota: M \hookrightarrow \overline M$.

It is not hard to see that $\overline M$ is homeomorphic to $N$. Indeed, the complement $\overline M \backslash \iota(M)$ can be decomposed into open intervals, each having one end on $\iota(\partial M)$. This is the projection of the decomposition of the complement of $\H^3$ to $\tilde M$ (embedded by a developing map) into outer normal rays. An outer normal ray to a convex set $C \subset \H^3$ at a point $p \in \partial C$ is a geodesic ray originating at $p$, directing outwards of $C$ and making an angle at least $\pi/2$ with every chord of $C$ passing through $p$. The set of all outer normal rays emanating from $p$ is either a single ray, or a cone over a closed segment, or a cone over a closed convex planar set. It follows that a connected component of $\overline M \backslash \iota(M)$ is homeomorphic to $S \times \mathbb{R}$, where $S$ is the connected component of $\partial M$, such that $\iota(S)$ bounds the respective end. We choose a small regular neighborhood of $S$ in $M$ and map onto it homeomorphically the connected component of $\overline M \backslash \iota(M)$ so that the induced map $S \rightarrow S$ is isotopic to identity. After doing so for each boundary component, this extends to a homeomorphism $\overline M \rightarrow N$. This construction allows us to define up to isotopy a convex cocompact hyperbolic metric on $N$, which we denote by $\overline g$.

We denote the pair $(N, \overline g)$ by $N(\overline g)$, so $N(\overline g)$ is a hyperbolic 3-manifold, while $N$ is the underlying topological 3-manifold. By $\partial_{\infty} N(\overline g)$ we denote its boundary at infinity, i.e., $\partial M$ equipped with the respective conformal structure. We denote by $M(g)$ the image of $(M, g)$ in $N(\overline g)$ under the embedding induced by $\iota$. Generally we will denote  by $\overline g$ the convex cocompact hyperbolic metric on $N$ obtained from a metric $g$ as above. However, sometimes we will abuse the notation, and will denote by $\overline g$ a convex cocompact hyperbolic metric on $N$ in the absence of a metric $g$.

\begin{dfn}
The \emph{convex hull} ${\rm conv}(V)$ of a subset $V$ in a complete hyperbolic manifold is the inclusion-minimal totally convex set containing $V$.
\end{dfn}

Abusing the notation, we denote the extension of $\overline g$ to the universal cover $\tilde N$ still by $\overline g$, and the pair $(\tilde N, \overline g)$ by $\tilde N(\overline g)$. We will frequently identify $\tilde N(\overline g)$ with $\H^3$ by an arbitrary developing map (recall that a developing map is defined up to a global isometry of $\H^3$).
By $\tilde M(g)$ we denote the respective embedding of the universal cover of $M(g)$ as a convex set.
Let $\rho_{\overline g}: \pi_1(M) \rightarrow G$ be the corresponding holonomy representation. By $\Lambda(\overline g)$ we denote its limit set and by $\tilde C(\overline g)$ we denote the convex hull of $\Lambda(\overline g)$. If $\Lambda(\overline g)$ is contained in a circle, we say that $\overline g$ and $\rho_{\overline g}$ are \emph{Fuchsian}. They are \emph{Fuchsian of the first kind} if $\Lambda(\overline g)$ is a circle, and \emph{of the second kind} otherwise. The set $\tilde C(\overline g)$ is $\rho_{\overline g}$-invariant and projects to the \emph{convex core} $C(\overline g) \subset N$. The latter is totally convex, and is homeomorphic to $M$, except the Fuchsian case. Since the $\rho_{\overline g}$-orbit of any point of $\H^3$ accumulates at $\Lambda(\overline g)$, $C(\overline g)$ is contained in any totally convex subset of $N(\overline g)$. It is also well-known that

\begin{lm}
The induced path metric on $\partial C(\overline g)$ is hyperbolic.
\end{lm}

See, e.g.,~\cite{EM} or~\cite{Smi}. Thus, a connected component of $\partial C(\overline g)$ is an embedded hyperbolic surface, which, however, may be not totally geodesic as it may be bent along a geodesic lamination. Here by $\partial C(\overline g)$ we mean the boundary in $N(\overline g)$, so in the Fuchsian case, when $C(\overline g)$ is 2-dimensional, we mean $\partial C(\overline g)=C(\overline g)$. In the Fuchsian case of the first kind $C(\overline g)$ is an embedded closed totally geodesic surface, and $M$ is an interval bundle over this surface. In the Fuchsian case of second kind $C(\overline g)$ is a totally geodesic surface with geodesic relative boundary, and the boundary of $M$ is compressible.

If $\overline g$ is a convex cocompact hyperbolic metric on $N$, the convex hull of any bounded point set $V$ is homeomorphic to $M$ except when $\ol g$ is Fuchsian and $V$ belongs to the projection of the plane in $\H^3$ spanned by the circle containing $\Lambda(\overline g)$.

Let $S_j$ be a connected component of $\partial M$ and $\nu_j: \pi_1(S_j) \rightarrow \pi_1(M)$ be the map induced by the inclusion of $S_j$ to $M$. By $D_j$ we denote the covering space of $S_j$ corresponding to $\ker(\nu_j)$. It is homeomorphic to a boundary component of $\tilde M$ (possibly to infinitely many of them). It is either homeomorphic to the open 2-disk if $S_j$ is incompressible, or is infinitely connected. The embedding of $\tilde M(g)$  to $\H^3$ gives an embedding $\phi_j: D_j \rightarrow \H^3$ equivariant with respect to a convex cocompact, discrete and faithful representation of ${\rm im}(\nu_j)$ in $G$. Here we say that a discrete representation in $G$ is convex cocompact if the respective quotient is a convex cocompact hyperbolic manifold.

Now we mention few results about the deformation space of convex cocompact metrics on $N$. First we need to pay some attention to the difference between equivalences of metrics up to homotopy and up to isotopy. By $\textrm{MCG}(M)$ we denote the \emph{mapping class group} of $M$, i.e., the group of isotopy classes of orientation-preserving self-homeomorphisms of $M$. By $\textrm{MCG}_h(M) \subset \textrm{MCG}(M)$ we denote the subgroup of isotopy classes of orientation-preserving self-homeomorphisms that are homotopic to the identity. There is a natural homomorphism $\textrm{MCG}(M) \rightarrow {\rm Out}(\pi_1(M))$, which sends a homeomorphism class to the induced outer automorphism of $\pi_1(M)$. One can see that $\textrm{MCG}_h(M)$ is exactly the kernel of this homomorphism.

The work of Waldhausen~\cite{Wal} establishes that when the boundary of $M$ is incompressible, the group $\textrm{MCG}_h(G)$ is trivial. However, this is no longer the case when the boundary is compressible. Let $\gamma \subset \partial M$ be an oriented simple closed curve that is contractible in $M$, and $D \subset M$ be an embedded 2-disk contracting $\gamma$, which exists thanks to the Dehn lemma. Do a Dehn twist around $\gamma$. Let $D \times [-1, 1] \subset M$ be a small neighborhood of $D$ such that $D \times \{0\}=D$ and $\partial D \times [-1, 1]$ is an annulus around $\gamma$ supporting the Dehn twist. The Dehn twist naturally extends to $D \times [-1, 1]$. Extending it by identity outside of $D \times [-1, 1]$, we obtain a self-homeomorphism of $M$ called a \emph{twist} of $M$. One can see that it is homotopic to identity, but not isotopic. McCullough and Miller showed in~\cite{MM2} that $\textrm{MCG}_h(M)$ is generated by twists. Let $S_j$ be a connected component of $\partial M$. By $\textrm{MCG}_{h, j}(M)$ we denote the subgroup of $\textrm{MCG}_h(M)$ generated by twists along curves belonging to $S_j$. As compressing disks of disjoint simple closed curves in $\partial M$ can be made disjoint, $\textrm{MCG}_h(M)$ is the direct product of all $\textrm{MCG}_{h, j}(M)$.

By $\mathcal R(\pi_1(M), G)$ we denote the space of representations of $\pi_1(M)$ in $G$. The group $G$ acts on $\mathcal R(\pi_1(M), G)$ by conjugation, and we denote the resulting quotient by $\mathcal X(\pi_1 (M), G)$. Since $G=PSL(2, \mathbb{C})$ is an algebraic group over $\mathbb{C}$, the space $\mathcal R(\pi_1(M), G)$ can be viewed as a complex algebraic variety, and we consider it with the induced topology. We endow the space $\mathcal X(\pi_1 (M), G)$ with the quotient topology. Note that $\mathcal X(\pi_1 (M), G)$ is not even a Hausdorff space, and should not be confused with \emph{the character variety}, which is a more accurate quotient that can be constructed by means of algebraic geometry. However, we are interested only in the quotient of an invariant open smooth subset of $\mathcal R(\pi_1(M), G)$, corresponding to the holonomies of convex cocompact metrics, over which the action behaves well. We will be anyway calling elements of $\mathcal X(\pi_1 (M), G)$ \emph{characters}.

By $\mathcal{HM}(N)$ we denote the space of complete hyperbolic metrics on $N$ up to isotopy. The group $MCG_h(N)$ acts on $\mathcal {HM}(N)$ and the quotient, which we denote by $\mathcal {HM}_h(N)$, is the space of complete hyperbolic metrics on $M$ up to homotopy. By $\mathcal{CH}(N) \subset \mathcal{HM}(N)$ and $\mathcal{CH}_h(N) \subset \mathcal{HM}_h(N)$ we denote the subspaces of convex cocompact hyperbolic metrics up to isotopy and homotopy respectively. The character of a holonomy sends injectively $\mathcal{HM}_h(N)$ to $\mathcal X(\pi_1(M), G)$. Indeed, if two hyperbolic metrics $\overline g_1$, $\overline g_2$ have the same character, then we have an orientation-preserving isometry between $N(\overline g_1)$ and $N(\overline g_2)$ inducing the identity map on the fundamental groups. Since $M$ is aspherical, $\overline g_1$ and $\overline g_2$ are homotopic by the Whitehead theorem. We endow $\mathcal{HM}_h(N)$ with the topology pull-backed from $\mathcal X(\pi_1(M), G)$. Note that we can endow $\mathcal{HM}_h(N)$ with a more natural topology, which comes from $C^{\infty}$-convergence on compact subsets for developing maps. Then the so-called Ehresmann-Thurston theorem says that these topologies are the same. See~\cite{Gol} for details. Note that we will be frequently writing $\overline g \in \mathcal{CH}(N)$ meaning that $\overline g$ is a convex cocompact hyperbolic metric on $N$.

Let $\overline g \in \mathcal{CH}(N)$ and $\rho_{\overline g}$ be its holonomy. It follows from Marden's isomorphism theorem~\cite[Section 8]{Mar} that a convex cocompact representation $\rho \in \mathcal R(\pi_1(M), G)$ is a holonomy representation of a convex cocompact metric on $N$ if and only if it is related to $\rho_{\overline g}$ by a quasi-conformal deformation. Thus, the set of holonomy representations of metrics from $\mathcal{CH}_h(N)$ can be identified with the set $\mathcal{QC}(\rho_{\overline g})$ of quasi-conformal deformations of $\rho_{\overline g}$. In particular, it is connected. Marden's stability theorem~\cite[Section 10]{Mar} implies also that it is an open subset of the smooth part of $\mathcal R(\pi_1(M), G)$. The set $\mathcal{QC}(\rho_{\overline g})$ stays invariant under the action of $G$ by conjugation, and $G$ acts freely and smoothly on it. Moreover,~\cite[Theorem 10.8]{Mar} implies that this action is proper, so the quotient is a complex manifold. Other approaches to the properness can be seen in~\cite[Proposition 1.1]{JM} or in~\cite[Lemma 6.24]{Wei}.
We identify $\mathcal{CH}_h(N)$ with this quotient. We refer to the expositions in~\cite[Chapter 5]{Mar2} and~\cite[Section 7]{CM}.

By $\mathcal{TS}(\partial M)$ we denote the Teichm\"uller space of $\partial M$. It is the direct product of the Teichm\"uller spaces of all connected components of $\partial M$. The group $\textrm{MCG}_h(M)$ acts on $\mathcal {TS}(\partial M)$ and the action splits on actions of $\textrm{MCG}_{h, j}(M)$ on the respective components of $\mathcal{TS}(\partial M)$. As the action of the mapping class group of $\partial M$ on $\mathcal {TS}(\partial M)$ is propely discontinuous (see, e.g.,~\cite[Theorem 12.2]{FM}), so is the action of $\textrm{MCG}_h(M)$. In~\cite[Section 3.1]{Mar3} Marden shows that this action is free. By $\mathcal{TS}_h(\partial M, M)$ we denote the quotient of $\mathcal {TS}(\partial M)$ over this action.

In~\cite[Section 3.1]{Mar3} Marden states a version of the Ahlfors--Bers theorem (based also on the works of Kra and Maskit), which says that if $\overline g$ is a convex cocompact hyperbolic metric on $N$, then the action of quasi-conformal deformations of $\rho_{\overline g}$ on $\partial_{\infty} \H^3$ defines a biholomorphic map from the quotient of $\mathcal{QC}(\rho_{\overline g})$ by conjugation of $G$ to $\mathcal{TS}_h(\partial M, M)$. We refer to~\cite{Mar3} and references therein. Additional expositions can be found in~\cite[Section 5.1]{Mar2} or \cite[Section 7]{CM}. Due to the discussion above, we can state this result as

\begin{thm}
\label{abh}
The map $\mathcal {AB}_h: \mathcal{CH}_h(N) \rightarrow \mathcal{TS}_h(\partial M, M)$ sending a class represented by a metric $\overline g$ to the class of its conformal structure at infinity $\partial_{\infty}N(\overline g)$ is a biholomorphic map.
\end{thm}

Finally, we return to $\mathcal{CH}(N)$. By $\mathcal {AB}: \mathcal{CH}(N) \rightarrow \mathcal{TS}(\partial M)$ we denote a similar map defined as above. Clearly, it is equivariant with respect to the action of $\textrm{MCG}_h(M)$. Together with Theorem~\ref{abh} it implies

\begin{thm}
\label{ab}
The map $\mathcal{AB}$ is bijective.
\end{thm}

We endow $\mathcal{CH}(N)$ with the induced topology. It becomes a connected complex manifold of real dimension $-3k$, where $k:=\chi(\partial M)$, and the natural projection $\mathcal{CH}(N)\rightarrow \mathcal{CH}_h(N)$ becomes a covering map.

Let $\overline g$ be a convex cocompact hyperbolic metric on $N$ and $s \in \partial_{\infty}\H^3 \backslash \Lambda(\overline g)$. Consider a small horosphere at $s$ such that it does not intersect $\tilde C(\overline g)$. Let $\mu(s) \in \partial \tilde C(\overline g)$ be the closest point from $\tilde C(\overline g)$ to this horosphere. Clearly, $\mu(s)$ is independent on the choice of a horosphere, and determines a map from $\partial_{\infty} N(\overline g)$ to $\partial C(\overline g)$ still denoted by $\mu$.
When $\overline g$ is Fuchsian of the first kind, we modify $\mu$ so that its target is two copies of $C(\overline g)$, and $\mu$ remembers from which side of $C(\overline g)$ the point of $\partial_{\infty}N(\overline g)$ was located. When $\overline g$ is Fuchsian of the second kind, we consider the double of $C(\overline g)$ along its relative boundary as the target of $\mu$ (keeping track on the side where $s$ was located except when $\mu(s)$ is in the relative boundary of $C(\overline g)$).

The uniformization theorem endows $\partial_{\infty}\H^3 \backslash \Lambda(\overline g)$ with a conformal $\rho_{\overline g}$-invariant hyperbolic metric, projecting to a conformal hyperbolic metric on $\partial_{\infty} N(\overline g)$. The following statement compares this metric structure with the indiced metric on $\partial C(\overline g)$:

\begin{thm}
\label{su}
There exist functions $a, b: (0, +\infty)\rightarrow (0, +\infty)$ such that if for $r_0>0$ the injectivity radius of $\partial C(\overline g)$ is at least $r_0$, then $\mu$ is $a(r_0)$-Lipschitz and has a $b(r_0)$-Lipschitz homotopy inverse. If $\partial M$ is incompressible, then $a,b$ can be chosen absolute constants.
\end{thm}

It was first sketched by Sullivan in~\cite{Sul} for the case of incompressible boundary, a more detailed proof can be found in~\cite{EM}. The case of general boundary was obtained in~\cite{BC}.

Finally, we need a well-known compactness criterion for subsets of the Teichm\"uller space (complementing the \emph{Mumford compactness theorem}).

\begin{thm}
\label{flp}
Let $S$ be a closed oriented surface. There exists a finite set of isotopy classes of simple closed curves $\{\gamma_1, \ldots, \gamma_n\}$ such that for any positive number $C$ the set of hyperbolic metrics, having geodesic representatives of all $\gamma_i$ with lengths at most $C$, is compact.
\end{thm}

It follows straightforwardly from Lemma 7.10 and Proposition 7.11 in~\cite{FLP}.

\subsection{Polyhedral hyperbolic manifolds}
\label{polyhsec}

\begin{dfn}
We say that a hyperbolic metric $g$ on $M$ is \emph{polyhedral} if each point $p \in \partial M(g)$ has a neighborhood isometric to a convex polyhedral set in $\H^3$. It is \emph{strictly polyhedral} if the distance between the boundary and the convex core is positive.
\end{dfn}

For $p \in \partial M$, the \emph{spherical link} of $p$ in $M(g)$ is the part of the unit sphere in $T_p N(\overline g)$ that is cut out by the directions to $M(g)$. If the spherical link of $p$ is a half-sphere, then $p$ is called \emph{regular} in $M(g)$. If it is a spherical lune, then $p$ is called a \emph{ridge point} in $M(g)$. Otherwise, it is called a \emph{vertex} of $M(g)$, and its spherical link is a convex spherical polygon. We denote the set of vertices by $V(g)$. The closures of the connected components of the set of regular points in $M(g)$ are called \emph{faces} of $M(g)$ and constitute \emph{the face decomposition} of $M(g)$. By $F(g)$ we denote the set of faces of $M(g)$. The \emph{edges} of $M(g)$ are closures of the connected components of the set of ridge points of $M(g)$.

\begin{lm}
\label{strpolyh}
Let $g$ be a strictly polyhedral metric on $M$. Then each edge is a segment between vertices and all faces are isometric to compact hyperbolic polygons.
\end{lm}

\begin{proof}
If an edge can be extended infinitely at least in one direction or is a closed geodesic, then its lift to the universal cover ends in the limit set $\Lambda(\overline g)$. Hence, the distance to $\tilde C(\overline g)$ would tend to zero along this lift, which is a contradiction as $\partial M$ is compact and the distance to the convex core of $g$ is positive.

Each face is isometric to a hyperbolic surface with convex piecewise-geodesic boundary. As no edge can be extended infinitely, the boundary of a face is compact. If a face contains a homotopically non-trivial loop, then, as the boundary of the face is convex, it contains a simple closed geodesic. Lifting this geodesic to the universal cover, we see that it ends in the limit set $\Lambda(\overline g)$. Again, this is a contradiction as the distance to $\tilde C(\overline g)$ would tend to zero along this lift.
\end{proof}

Let $g$ be a strictly polyhedral metric on $M$. Consider $\tilde M(g) \subset \H^3$ and its Gauss dual $\tilde M(g)^* \subset d\mathbb{S}^3$. As $g$ is strictly polyhedral, no supporting plane passes through a point of $\Lambda(\overline g)$. Indeed, this would imply the existence on $\partial \tilde M(g)$ of a curve, infinite in one direction, such that the distance to $\tilde C(\overline g)$ would tend to zero along it. Hence, each boundary component of $\partial \tilde M(g)^*$ is space-like, so the induced metric is a spherical cone-metric. Another consequence of strict polyhedrality is that $\rho_{\overline g}$ acts on $\partial \tilde M(g)^*$ freely. Indeed, a point in $d\mathbb{S}^3$ is fixed by $\rho_{\overline g}(\gamma)$ only if it is dual to a plane in $\H^3$ containing the axis of the isometry $\rho_{\overline g}(\gamma)$. But all such axes are contained in $\tilde C(\overline g)$. Therefore, by taking the quotient of $\partial \tilde M(g)^*$, we obtain a spherical cone-metric $d^*$ on $\partial M$. We say that $d^*$ is the \emph{dual metric} induced by $g$ on $\partial M$.

There is another way to construct $d^*$. For each vertex $v$ of $M(g)$ we consider its spherical link, which is a convex spherical polygon, and its spherical polar dual is another convex spherical polygon. By gluing together these dual polygons with respect to the face decomposition of $M(g)$, we obtain $d^*$. It is easy to see from this description that $d^*$ is concave.

The main result of the work~\cite{CD} implies that if $C \subset \H^3$ is a convex polyhedral set, then any two points in $\partial C^*$ at distance less than $\pi$ can be connected by a unique minimizing intrinsic geodesic segment in $\partial C^*$. Moreover, a refined version~\cite[Theorem 4.1.1]{CD} also states that if $\gamma \subset \partial C^*$ is a closed intrinsic geodesic of length $2\pi$, then it encloses a cusp point of $C^*$, i.e., a point of tangency with $\partial_{\infty} d\mathbb{S}^3$. This particularly implies

\begin{lm}
Let $g$ be a strictly polyhedral metric on $M$. Then the dual metric $d^*$ on $\partial M$ is large for $M$.
\end{lm}

\subsection{Topology on the space of strictly polyhedral metrics}

Let $V$ be a finite set, $g$ be a strictly polyhedral metric on $\partial M$ with $n:=|V|$ faces and $f: V \rightarrow F(g)$ be a bijection. We say that the pair $(g, f)$ is \emph{a strictly polyhedral metric on $M$ with faces marked by $V$}. By $\mathcal P(M, V)$ we denote the space of strictly polyhedral metrics $g$ on $M$ with faces marked by $V$ up to isometry preserving the marking and isotopic to identity. We are going to endow $\mathcal P(M, V)$ with a natural topology. We will frequently abuse the notation and write $g \in \mathcal P(M, V)$ meaning that some $f$ is also chosen and with the class of $g$ it is in $\mathcal P(M, V)$.

For $\overline g \in \mathcal{CH}(N)$, consider an oriented plane in $\tilde N(\overline g)=\H^3$. It gives rise to an immersed oriented plane in $N(\overline g)$. By $\mathcal{CH}(N, V)$ we denote the space of convex cocompact hyperbolic metrics on $N$ equipped with a set of immersed oriented planes $Z$ marked by $V$, considered up to isometry sending oriented marked planes to oriented marked planes, preserving the marking and isotopic to identity. So elements of $\mathcal {CH}(N, V)$ are represented by triples $(\overline g, Z, z)$, where $\overline g$ is a convex cocompact hyperbolic metric on $N$, $Z$ is a set of immersed planes and $z: V \rightarrow Z$ is a bijection. We will write $z_v$ instead of $z(v)$. We are going to endow the space $\mc{CH}(N, V)$ with a natural smooth structure (moreover, an analytic structure) so that the forgetful map $\mc{CH}(N, V)$ is a smooth bundle.

First, we similarly define a space $\mc{CH}_h(N, V)$ with isotopy replaced by homotopy. As with $\mc{CH}_h(N)$, it is easier to endow $\mc{CH}_h(N, V)$ with a topology. By $(d\mathbb{S}^3)^{*V}$ we denote the space of injective maps $\tilde f: V \hookrightarrow d\mathbb{S}^3$. We can interpret $(d\mathbb{S}^3)^{*V}$ as the configuration space of oriented planes in $\H^3$ marked by $V$. We consider the direct product
$$\mc R(\pi_1(M), G)\times (d\mathbb{S}^3)^{*V}$$
with a left action of $G$ on it, by conjugation on the first factor and by isometries on the second factor. There is also a left action of $\pi_1(M)$ given by $\gamma(\rho, \tilde f)=(\rho, \rho(\gamma)\tilde f)$, where $(\rho, \tilde f) \in \mc R(\pi_1(M), G)\times (d\mathbb{S}^3)^{*V}$ and $\gamma \in \pi_1(M)$. These two actions commute, and by taking the quotient over both we obtain a fibration over $\mc X(\pi_1(M), G)$. The standard approach via developing maps allows us to inject $\mc{CH}_h(N, V)$ in the total space of this fibration so that the forgetful map $\mc{CH}_h(N, V) \rightarrow \mc{CH}_h(N, V)$ is a smooth bundle. For $\ol g \in \mc{CH}_h(N, V)$ the fiber can be naturally identified with the space of oriented planes in $N(\ol g)$ marked by $V$. Recall that the action of ${\rm MCG}_h(M)$ on $\mc{CH}(N)$ is free, properly discontinuous and determines a covering map $\mc{CH}(N) \rightarrow \mc{CH}_h(N)$. Consider the commutative diagram
\begin{center}
\begin{tikzcd}
\mathcal{CH}(N, V) \arrow[r] \arrow[d]
& \mathcal{CH}_h(N, V) \arrow[d] \\
\mathcal{CH}(N) \arrow[r]
& \mathcal{CH}_h(N)
\end{tikzcd}
\end{center}
Using this diagram, we pull-back the topology on $\mathcal{CH}_h(N, V)$ to a topology on $\mc{CH}(N, V)$ making it a smooth manifold of dimension $3(n-k)$ where $k$ is the Euler characteristic of $\partial M$.

For $g \in \mathcal P(M, V)$, every face gives rise to an immersed plane, which we consider oriented outwards $M(g)$. We denote the set of such planes by $Z(g)$. This allows us to consider $\mathcal P(M, V)$ as a subset of $\mathcal{CH}(N, V)$. Clearly, it is open. We endow $\mathcal P(M, V)$ with the induced topology.

We note that when $n$ is at least the number of boundary components of $\partial M$, the space $\mathcal P(M, V)$ is non-empty, and, moreover, for every $\ol g \in \mc{CH}(N)$ there exists $g \in \mathcal P(M, V)$ such that $\ol g$ is the underlying metric on $N$. Indeed, for each component $S_i$ of $\partial M$ pick a corresponding component $\Omega_i$ of $\partial_{\infty} \H^3 \backslash \Lambda(\ol g)$. There exists a compact fundamental domain in $\Omega_i$ for the restriction of the action of $\rho_{\ol g}$ to $\Omega_i$. Choose a plane $\Pi_i$ such that the disk bounded by $\partial_\infty \Pi_i$ contains this domain and that is at positive distance from $\tilde C(\ol g)$. Consider the union of $\rho_{\ol g}$-orbits of all chosen planes. For each plane in this union there is a (closed) half-space bounded by this plane containing $\tilde C(\ol g)$. Take the intersection of all these half-spaces. It is evident that the $\rho_{\ol g}$-quotient of this set is homeomorphic to $M$ and is a strictly polyhedral manifold with exactly one face in each boundary component.

To every triple $(\ol g, Z, z)$ representing a class in $\mc{CH}(N, V)$ we can correspond the intersection $K$ of all closed negative immersed half-spaces in $N(\ol g)$ bounded by oriented immersed planes of $Z$. By $\ol{\mc P}(M, V)$ we denote the subset of $\mc{CH}(N, V)$ of classes of triples such that (1) $K$ is homeomorphic to $M$; (2) $\partial K$ is on positive distance from $C(\ol g)$; (3) each plane from $Z$ intersects $\partial K$. Note that the complement of $K$ in $N$ has a form $\partial K \times (0,1)$. By $g$ we denote  the metric on $M$ induced by a homeomorphism between $M$ and $K$ chosen so that for each component of $\partial M$ identified with $\partial K$ with the help of the mentioned trivialization, the induced map on the fundamental groups is identity. The metric $g$ is determined up to isotopy. We will represent the elements of $\ol{\mc P}(M, V)$ by pairs $(g, z)$. For every such pair there is a subset $V(g) \subseteq V$ where $v \in V(g)$ if and only if the intersection of the plane $z_v$ with $\partial M(g)$ has non-empty relative interior, i.e., corresponds to a face. A pair $(g, z)$ determines a class in $\mc P(M, V(g))$.

\begin{lm}
\label{connect}
Let $g'_0$, $g'_1 \in \mathcal P(M, V)$. Then there exists $W \supseteq V$ and a path $g_t \subset \overline{\mathcal P}(M, W)$ for $t \in [0, 1]$ such that $g_0$ and $g_1$ are lifts of $g'_0$ and $g'_1$ respectively.
\end{lm}

\begin{proof}
Let $\xi_i = \mathcal{AB}(\overline g_i) \in \mathcal{TS}(\partial M)$ (see Theorem~\ref{ab} for the definition of the map $\mathcal{AB})$. As $\mathcal{TS}(\partial M)$ is connected, we can connect $\xi_0$, $\xi_1$ by an analytic path $\xi_t$, $t \in [0, 1]$. From Theorem~\ref{ab} we get $\overline g_{t}:=\mathcal{AB}^{-1}(\xi_t)$, an analytic path connecting $\overline g_0$ and $\overline g_1$ via convex cocompact hyperbolic metrics on $N$.

The metrics $g_0$ and $g_1$ give rise to sets of immersed oriented planes $Z(g_0)$, $Z(g_1)$ marked by $V$ in $N(\overline g_0)$, $N(\overline g_1)$ respectively. We want to extend $\overline g_t$ from an analytic path in $\mathcal{CH}(N)$ to an analytic path in $\mathcal{CH}(N, V)$ taking $Z(g_0)$ to $Z(g_1)$ preserving the marking so that all oriented planes stay away from $C(\overline g_t)$. Let $z_{v, 0}$ be the oriented plane from $Z(g_0)$ marked by $v \in V$. Consider its lift $\tilde z_{v, 0} \subset \tilde N(\overline g_0) = \H^3$ and look at the line passing through the nearest points of $\tilde z_{v, 0}$ and $\tilde C(\overline g_0)$. By $\tilde \omega_{v, 0}$ we denote the intersection point of this line with $\partial_{\infty} \H^3$ outward to $\tilde C(\overline g_0)$. By $\omega_{v, 0}$ we denote the projection of $\tilde \omega_{v, 0}$ to $\partial_{\infty} N(\overline g_0)$. Similarly, define the point $\omega_{v, 1} \in \partial_{\infty} N(\overline g_1)$. We can connect $\omega_{v, 0}$, $\omega_{v,1}$ by an analytic path $\omega_{v, t}$, where $\omega_{v, t} \in \partial_{\infty} N(\overline g_t)$, $t \in [0,1]$ (this is a lift of a path in the space of conformal structures to a path in the space of conformal structures with one marked point). Let $\tilde \omega_{v, t}$ be a lift of $\omega_{v, t}$ to $\partial_{\infty} \H^3$. Consider the segment connecting $\tilde \omega_{v, t}$ with its closest point of $\tilde C(\overline g_t)$ (i.e., the closest point from $\tilde C(\overline g_t)$ to any small enough horosphere at $\tilde\omega_{v,t}$). If through any point of this segment we draw a perpendicular plane, then it projects to an immersed plane in $N(\overline g_t)$ at positive distance from $C(\overline g_t)$. The planes $\tilde z_{v,0}$, $\tilde z_{v, 1}$ are of this form. We can use this to connect $z_{v, 0}$ and $z_{v, 1}$ by an analytic path of immersed planes $z_{v, t} \subset N(\overline g_t)$ staying at positive distance from $C(\overline g_t)$. Doing this for every $v \in V$, we obtain a set of immersed oriented planes denoted by $Z_t$, where $Z_0=Z(g_0)$ and $Z_1=Z(g_1)$.

The intersection of the immersed negative half-spaces of planes in $Z_t$ is a totally convex set $K_t \subset N(\overline g_t)$. For $t=0,1$ we see that $K_t$ coincides with $M(g_t)$. Moreover, near $t=0,1$ the set $K_t$ is homeomorphic to $M$ and determines metrics $g_t \in \mathcal P(M, V)$. What can go wrong with $K_t$ for other $t$? First, the boundary $\partial K_t$ never touches $C(\overline g_t)$ by construction. Next, if one face gets away from $\partial K_t$ or two faces merge, then $K_t$ still determines a class in $\ol{\mc P}(M, V)$. The only remaining issue is if a vertex of $K_t$ escapes outside $N(\ol g_t)$, so $K_t$ becomes non-compact.

Look on what is going on in the Klein model. Even if a vertex escapes outside $\H^3$, the supporting planes to the universal cover $\tilde K_t$ still intersect outside of $\H^3$ determining a hyper-ideal polyhedral convex set equivariant with respect to $\rho_{\overline g_t}$. Let the vertex $u_t$ of $\tilde K_t$ escape outside $\H^3$ at some moment $t \in [0, 1]$. Consider the end of $\tilde K_t$ corresponding to $u_t$: the connected component of $\partial \tilde K_t \backslash \H^3$ containing $u_t$. It intersects $\partial_{\infty} \H^3$ at a compact convex set. Let $\tilde z_t$ be a plane in $\H^3$ separating this end from all other such ends of $\tilde K_t$. Take the orbit of $\tilde z_t$ under the respective representation $\rho_{\overline g_t}$. This determines an immersed oriented plane $z_t$ in $N(\overline g_t)$. We add a new element $w$ to $V$, creating a set $W'$, marking this new plane by the new element. Consider the maximal segment $[t_0, t_1] \subset [0, 1]$ when this end exists. We can move the plane $\tilde z_t$ so that it closes the corresponding end over $[t_0, t_1]$. Moreover, we can move the plane $\tilde z_t$ further to define it for all $t \in [0,1]$ so that it always intersects $\partial \tilde K_t$, but does not intersect the interior of $\tilde K_t$ for $t=0$ and $t=1$. We get a modified path in $\mathcal{CH}(N, W')$ and a modified totally convex subset $K'_t \subset N(\overline g_t)$, which is no longer non-compact at this end over $[t_0, t_1]$: we cut it out by introducing a new face. As there is an upper bound on the number of vertices of $K_t$ and the initial path in $\mathcal{CH}(N, V)$ is analytic, we need to do this operation only finitely many times to get rid of all non-compact ends of $K_t$ for all $t \in [0,1]$. This results in a path in $\mathcal {CH}(N, W)$ for some finite set $W \supset V$ and a totally convex compact set $K^{\#}_t \subset N(\overline g_t)$ with polyhedral boundary staying at positive distance from $C(\overline g)$. Thus, it can be considered as a path of metrics $g_t \in \overline{\mathcal P}(M, W)$ connecting metrics $g_0$ and $g_1$ that are lifts of metrics $g'_0$ and $g'_1$.
\end{proof}

Let $g \in \mathcal P(M, V)$ and $e$ be an edge of $M(g)$. Then $e$ remains to be an edge in some neighborhood of $g$ in $\mathcal P(M, V)$. From our definition of the topology on $\mathcal P(M, V)$, when we vary $g$ smoothly, the planes containing faces of $\tilde M(g)$ vary smoothly. This implies

\begin{lm}
\label{diffweak}
The dihedral angle of $e$ is a smooth function in a neighborhood of $g$ in $\mathcal P(M, V)$.
\end{lm}

For $g \in \mathcal P(M, V)$, the dual metric $d^*$ is defined up to isotopy. We also identify $V$ with the subset of $\partial M$ of the projections of the respective vertices of $\tilde M(g)^*$. Thus, $d^*$ determines an element of $\mathcal D^*_M(\partial M, V)$. For $(g, z)$ representing an element of $\ol{\mc P}(M, V)$ the dual points to the lifts of the planes $z_v$ belong to the boundary of the dual set $\tilde M(g)^*$. Hence, in this case we can consider the class of the dual metric in $\ol{\mc D}^*_M(\partial M, V)$. We denote the obtained map by $$\mathcal I_V: \ol{\mathcal P}(M, V) \rightarrow \ol{\mathcal D}^*_M(\partial M, V).$$
It is not hard to see that it is continuous.

\begin{lm}
\label{differ}
The map $\mathcal I_V$ is $C^1$ over $\mc P(M, V)$.
\end{lm}

\begin{proof}
Let $g \in \mathcal P(M, V)$ and $d^*$ be the respective dual metric. The boundary of $\tilde M(g)^*$ is decomposed into faces, which are compact convex spherical polygons, dual to the face decomposition of $\tilde M(g)$ and equivariant with respect to $\rho_{\overline g}$. So it projects to a decomposition of $(\partial M, V)$ into faces. Let $\mathcal T$ be any triangulation of $(\partial M, V)$ refining this decomposition. Thus, $d$ belongs to the chart $\mathcal D^*(\partial M, \mathcal T)$. We need to show that the edge-lengths of $\mathcal T$ are $C^1$-functions of $g$. Due to Lemma~\ref{diffweak}, it remains only to check what happens when there are non-triangular faces in the face decomposition of $\tilde M(g)^*$, or, equivalently, when there are non-simple vertices of $\tilde M(g)$.

Pick a basis for $\pi_1(M)$, let $m$ be its size. Consider three vertices of $\tilde M(g)^*$ and act on them by each basis element of $\pi_1(M)$. Additionally, for each $v \in V$ take a vertex $\tilde v$ of $\tilde M(g)^*$ corresponding to $v$. Let $\tilde V_0 \subset d\mathbb{S}^3$ be the set of these representatives together with all the images of the vertices above, so it is a subset of vertices of $\tilde M(g)^*$ of size $n+3m$. Since each element of ${\rm Iso}^+ d\mathbb{S}^3={\rm Iso}^+ \H^3$ is determined by its action on three non-collinear points, one can see that the position of the set $\tilde V_0$ in $d\mathbb{S}^3$ completely determines the representation $\rho_{\overline g}$ and the set $\tilde M(g)^*$. Thus, it determines $g$. Moreover, if we vary the position of $\tilde V_0$ small enough, we can reconstruct a new discrete and faithful representation $\rho'$ of $\pi_1(M)$, and all points of $\tilde V_0$ will still be in convex position with respect to their orbits under $\rho'$. This produces a metric $g' \in \mathcal P(M, V)$ and all sufficiently close metrics to $g$ arise like this. This shows that there is a neighborhood $U$ of $g$ in $\mathcal P(M, V)$, a neighborhood $N \subset (d\mathbb{S}^3)^{\tilde V_0}$ of the positions of $\tilde V_0$ and a smooth submersion $N \rightarrow U$. It is sufficient to show that the edge-lengths of $\mathcal T$ are $C^1$-functions in $N$.

Let $Q \subset d\mathbb{S}^3$ be a convex geodesic space-like polygon with more than three vertices, $V_Q$ be the set of its vertices, $e$ be its interior edge with endpoints $v_1$ and $v_2$. By $s$ we denote the de Sitter distance between $v_1$ and $v_2$.
When we slightly vary the positions of $V_Q$ and take the convex hull, we may obtain a 3-dimensional polytope, whose boundary is naturally divided into two parts, both of which can be considered as variations of the intrinsic metric of $Q$. Let us restrict the attention to one of the parts, we call it \emph{upper}. If the deformation is small enough, this polytope has only space-like faces, and the angles of the vertices in the intrinsic metric of the upper part stay less than $\pi$. Let $N_Q \subset (d\mathbb{S}^3)^{V_Q}$ be a small enough neighborhood of the initial position of $V_Q$, for which this holds. Then we can identify a deformation of $e$ in the upper part, which is a broken line if $e$ is not an edge of the obtained polytope. Consider the length $l$ of $e$ as a function over $N_Q$. We need to show that it is a $C^1$-function.

Now let $\mathcal T_1, \ldots, \mathcal T_r$ be all possible triangulations of $Q$. Fix some $\mathcal T_i$ and consider deformations of $Q$ triangulated with $\mathcal T_i$ under small variations of $V_Q$. Here the obtained polyhedral surface might become non-convex. We assume that $N_Q$ is small enough that for all $\mathcal T_i$ all faces of the deformed surface are space-like and all angles of the vertices in the intrinsic metric are less than $\pi$. Let $l_i$ be the length of $e$ under such deformations. Clearly, it is a smooth function in $N_Q$ as all the edge-lengths of $\mathcal T_i$ are smooth functions. For each $v \in V_Q$ we choose coordinates such that two of them, called \emph{horizontal}, are the coordinates of the projection of $v$ to the plane containing $Q$ in the initial moment, and the third, called \emph{vertical}, is the oriented length of a perpendicular from $v$ to this plane. The latter is a purely imaginary number, but we consider it as a real.

We claim that all vertical partial derivatives of $l_i$ at the initial moment are zero. Indeed, let $ABC$ be a de Sitter triangle with $BC$ time-like, $AB$ and $AC$ space-like and $AB$ orthogonal to $BC$. By $a$, $b$, $c$ denote the respective side-lengths (again, we treat the length of $BC$ as a real number). Then we have
$$\cos(b)=\cos(c)\cosh(a).$$
Thus, for fixed $c \neq 0$, $\partial b/\partial a=0$ at $a=0$. Thus, the vertical partial derivatives of all edge-lengths of $\mathcal T_i$ are zero. In turn, this shows that the vertical partial derivatives of $l_i$ are zero too. Now we remark that all horizontal partial derivatives of $l_i$ in the initial moment coincide with the horizontal partial derivatives of $s$, the de Sitter distance between the endpoints of $e$.

We return to the convex hull of $V_Q$. The domain $N_Q$ admits a piecewise-analytic decomposition into cells $E_1, \ldots, E_r$ corresponding to the face triangulations of the upper part of ${\rm conv}(V_Q)$. The function $l$ coincides with $l_i$ over $E_i$. The argument above shows that in the initial moment the differential of each $l_i$ coincides with the differential of $s$. Thus, $l$ is differentiable in the initial moment. Moreover, it follows that at any point of $N_Q$ belonging to several $E_i$, the differentials of the respective $l_i$ coincide. This implies that $l$ is differentiable over $N_Q$. Since all partial derivatives of $l_i$ are continuous over $E_i$ and coincide with the partial derivatives of $E_j$ at the common points of $E_i$ and $E_j$, we see that $l$ is $C^1$. This finishes the proof.
\end{proof}

We can refine now Theorem~\ref{mt} to a more appropriate form:

\begin{thm}
\label{mtr}
The restriction of the map $\mathcal I_V$ to $\mc P(M, V)$ is a $C^1$-diffeomorphism onto $\mc D^*_M(\partial M, V)$.
\end{thm}

In the next section we will show that $d\mathcal I_V$ is non-degenerate over $\mc P(M, V)$. Next, in Section~\ref{psec} we will show that $\mathcal I_V$ is proper. We finish the proof of Theorem~\ref{mtr} in Section~\ref{fsec}.

\section{Infinitesimal rigidity}
\label{irsec}

The goal of this section is to prove that $d\mathcal I_V$ is injective.

Take two copies of $M$ and identify the corresponding boundary points reversing the orientation. We obtain a closed 3-manifold $M^D$ called \emph{the double of $M$}. A strictly polyhedral hyperbolic metric $g$ on $M$ produces a metric $g^D$ on $M^D$, which is a hyperbolic cone-metric with the singular locus coming from the edges of $M(g)$ and with all cone-angles smaller than $2\pi$. We refer to~\cite{HK}, \cite{Wei} or \cite{MM3} for introductions to hyperbolic cone-3-manifolds. To uniformize the notation, we will write $M^D(g^D)$ for the hyperbolic manifold $(M^D, g^D)$. We denote by $\Sigma(g^D)$ the singular locus of $g^D$ and by $L(g^D):=M^D(g^D) \backslash \Sigma(g^D)$ denote the smooth part of $M^D(g^D)$ equipped with the induced metric. We can see that

\begin{lm}
If $g_0$, $g_1$ belong to the same connected component of $\mathcal P(M, V)$, then $L(g^D_0)$, $L(g^D_1)$ are isotopic as subsets of $M^D$.
\end{lm}

\begin{proof}
Connect $g_0, g_1$ by a smooth path $g_t$. The faces of $\partial M(g_t)$ change smoothly except times when some vertices can merge or split. One can see that these changes do not affect the isotopy type of $L(g^D_t)$ in $M^D$.
\end{proof}

We need now some preliminaries on the deformation spaces of hyperbolic cone-metrics similar to our treatment of convex cocompact hyperbolic metrics in Section~\ref{cocomp}. Fix a connected component $U$ of $\mathcal P(M, V)$ and let $L$ be a smooth open manifold diffeomorphic to $L(g^D)$ for all $g \in U$. Note that $L$ is diffeomorphic to the interior of a compact manifold with boundary, obtained by deleting from $M^D(g^D)$ a small regular neighborhood of the singular set. By $\mathcal H(L)$ we denote the space of hyperbolic metrics on $L$ up to equivalence, where the equivalence is generated by isotopies and thickenings. Here the metrics are not necessarily complete. If a hyperbolic metric $g_1$ on $L$ extends to a hyperbolic metric $g_2$ on $L \cup (\partial L \times [0, \varepsilon))$ for $\varepsilon>0$, then $g_2$, considered as a metric on $L$, is called \emph{a thickening} of $g_1$. We refer to~\cite[Section 1.6]{CEG} or to~\cite[ Section 5.3]{CHK} for more details on this equivalence. By $\mathcal C(M^D) \subset \mathcal H(L)$ we denote the subspace of metrics such that the metric completion is a hyperbolic cone-manifold homeomorphic to $M^D$.

Recall that by $G$ we denote ${\rm Iso}^+(\H^3)=PSL(2, \mathbb{C})$. It will be convenient here to consider its universal cover $\tilde G=SL(2, \mathbb{C})$. As usual, to every hyperbolic metric from $\mathcal H(L)$ can be assigned a holonomy map $\pi_1(L) \rightarrow G$ defined up to conjugation. It can be lifted to a holonomy map $\pi_1(L) \rightarrow \tilde G$ by a result of Culler~\cite{Cul}. Again, we consider the space of representations $\mathcal R(\pi_1(L), \tilde G)$ and its quotient $\mathcal X(\pi_1(L), \tilde G)$ by conjugation of $\tilde G$. The characters of the holonomies determine a map $\mathcal H(L) \rightarrow \mathcal X(\pi_1(L), \tilde G)$. The Ehresmann-Thurston theorem~\cite[Theorem 3.1]{Gol} shows that this map is injective and each point of the image has a neighborhood, on which the map is surjective. Actually, it is a local homeomorphism for some natural topology on $\mathcal H(L)$, but, as previously, we prefer to just endow $\mathcal H(L)$ with the pull-back topology (as it also provides a convenient differentiable structure). The works of Montcouquiol and Weiss~\cite{Mon, Wei2} show that around the characters of cone-manifolds, $\mathcal X(\pi_1(L), \tilde G)$ locally has the topology of a smooth manifold of real dimension $6(n-k)$ (recall that $k$ is the Euler characteristic of $\partial M$), therefore so is $\mathcal H(L)$.

The map $g \rightarrow g^D$ induces a map $D: \mathcal P(M, V) \rightarrow \mathcal C(M^D) \subset \mathcal H(L)$. We can show

\begin{lm}
The map $D$ is a smooth immersion.
\end{lm}

\begin{proof}
Let $g \in \mathcal P(M, V)$, $p \in {\rm int}(M)$ be a point and $(e_1, e_2, e_3) \in T_p M$ be a basis in its tangent space. Fix a natural embedding ${\rm int}(M) \hookrightarrow L$, and let $\tilde p \in \tilde L$ be a lift of $p$, $( \tilde e_1, \tilde e_2, \tilde e_3 ) \in T_{\tilde p} \tilde L$ be the images of $( e_1, e_2, e_3 )$. Consider $\gamma \in \pi_1(L)$. From the description of topology of $\mathcal P(M, V)$ it follows that when $g$ varies smoothly, $\gamma(\tilde p)$ and $\gamma_*(\tilde e_i)$ vary smoothly in $\tilde L$ for all $i=1,2,3$. Fix a developing map ${\rm dev}: \tilde L(g^D) \rightarrow \H^3$. Consider a variation of $g$ and a respective variation of ${\rm dev}$, we can always do it so that  the metric on $T_p M(g)$ does not change, and ${\rm dev}$ is constant on $\tilde p$ and $(\tilde e_1, \tilde e_2, \tilde e_3)$. Then ${\rm dev}(\gamma (\tilde p))$ and all ${\rm dev}_*(\gamma_* (\tilde e_i))$ vary smoothly. Every isometry of $\H^3$ is completely determined by its action on a point of $\H^3$ and on a basis in the tangent space of this point. Hence, $\rho(\gamma)$ varies smoothly, where $\rho$ is the respective holonomy representation of $\pi_1(L)$ in $G$. This shows that $D$ is a smooth map.

Now suppose that there is $x \in T_g \mathcal P(M, V)$ such that $dD(x)=0$. The image of the natural embedding ${\rm int}(M) \hookrightarrow L$ is a retract in $L$, hence it induces an embedding $\pi_1(M) \hookrightarrow \pi_1(L)$. Thus, $x$ induces zero variation on the holonomy representation $\rho_{\overline g}$ of $\pi_1 (M)$, and consequently, induces zero variation on $\overline g$.

It follows that $x$ is vertical with respect to the fibration $\mathcal {CH}(N, V) \rightarrow \mathcal {CH}(N)$. Thus, $x$ can be represented by a Killing vector field $X_v$ on each immersed plane $z_v$ of $Z(g)$ in $N(\overline g)$. Let $\tilde z_v \subset \H^3$ be a lift of $z_v$ to $\tilde N(\overline g)=\H^3$. The vector field $X_v$ lifts to a vector field $\tilde X_{\tilde z_v}$, which is the restriction of a global Killing field of $\H^3$. By $\tilde f_v$ denote the face of $\tilde M(g)$ belonging to the plane $\tilde z_v$. In the dual picture of $d\mathbb{S}^3$ the face $\tilde f_v$ corresponds to a vertex of $\tilde M(g)^*$, which we denote by $\tilde v$, and $\tilde X_{\tilde z_v}$ corresponds to a vector in $T_{\tilde v}d\mathbb{S}^3$, which we denote by $\tilde X^*_{\tilde v}$. Consider the closed star of $\tilde v$ in $\partial \tilde M(g)^*$. As $dD(x)=0$, $x$ induces zero variation on all lengths and dihedral angles of $M(g)$, including all ``hidden'' edges of zero length at vertices of valence greater than three. Thus, the collection of vectors $\tilde X^*_{\tilde w}$, where $\tilde w$ is either $\tilde v$ or a vertex of $\partial \tilde M(g)^*$ adjacent to $\tilde v$, induce zero deformations on the de Sitter lengths and the dihedral angles of the closed star of $\tilde v$. Hence, they all are restrictions of the same global Killing field, and so are the respective $\tilde X_{\tilde z_w}$. It follows that all $\tilde X_{\tilde z}$, where $\tilde z$ is any supporting plane containing a face of $\tilde M(g)$, are restrictions of the same Killing field on $\H^3$. On the other hand, they should be equivariant with respect to $\rho_{\overline g}$. The only such option is that all $\tilde X_{\tilde z}$ are zero, and so are $X_v$, implying that $x=0$.
\end{proof}

Now we quickly recall the basics of the infinitesimal theory of the character space $\mathcal X(\pi_1(L), \tilde G)$ via the bundle of infinitesimal isometries on $L$. This goes back to the works of Weil~\cite{Weil} and Matsushima--Murakami~\cite{MM} notably expounded in the book~\cite{Rag} of Raghunathan. The applications of this theory to cone-manifolds are due to Hodgson--Kerckhoff, Montcouquiol and Weiss~\cite{HK, Wei, Wei2, MW}. We describe the theory quite briefly, and mostly refer to the above-mentioned fundamental works.

Let $g^D \in \mathcal C(M^D)$. Here we use the notation $g^D$ to highlight that $g^D \in \mathcal C(M^D)$, but we do not actually assume now that $g^D$ comes as a double of $g \in \mathcal P(M, V)$. Fix a developing map from $\tilde L(g^D)$ to $\H^3$, let $\rho \in \mathcal R(\pi_1(L), \tilde G)$ be a lift of its holonomy and $\chi \in \mathcal X(\pi_1(L), \tilde G)$ be its character. As mentioned above, $\mathcal X(\pi_1(L), \tilde G)$ is a smooth manifold around $\chi$. The work of Weil~\cite{Weil} establishes the identification
$$T_{\chi}\mathcal X(\pi_1(L), \tilde G)\cong H^1(\pi_1(L), Ad\circ \rho),$$
where $H^1(\pi_1(L), Ad\circ \rho)$ is the first cohomology group of $\pi_1(L)$ with respect to the representation $Ad \circ \rho$ on the Lie algebra $\mathfrak g=\mathfrak{sl}(2, \mathbb{C})$ of $\tilde G$, and $Ad$ is the adjoint representation of $\tilde G$ on $\mathfrak g$. See the details in~\cite[Chapter VI]{Rag},~\cite[Section 4]{HK},~\cite[Section 6.2]{Wei}.

Let $\tilde E \rightarrow \tilde L$ be the trivial bundle, where $\tilde E$ is identified with $\tilde L \times \mathfrak g$ and $\tilde L$ is endowed with the metric $g^D$. The trivialization furnishes the bundle with a flat connection. The group $\pi_1(M)$ acts naturally on $\tilde E$ via the representation $Ad \circ \rho$. By taking the quotient, we obtain a flat bundle $E \rightarrow L$ of the infinitesimal isometries of $L$. The flat connection induces an exterior derivative $d$ on the complex $\Omega^*(L, E)$ of $E$-valued differential forms on $L$. Consider the respective cohomology complex $H^*(L, E)$. An analogue of the de Rham theorem in this setting establishes that
$$H^1(\pi_1(L), Ad\circ \rho) \cong H^1(L, E).$$
The details can be found in~\cite[Chapter VII]{Rag},~\cite[Section 1]{HK},~\cite[Section 6]{Wei}. By $H^1_{L^2}(L, E)$ we denote the first $L^2$-cohomologies, i.e., the cohomologies of the  subcomplex of $L^2$-bounded differential forms with $L^2$-bounded differential.

Let $e_i$ be an edge of $\Sigma_g$. By $\gamma_i \in \pi_1(L)$ we denote the class of the meridian around $e_i$. By ${\rm tr}_{\gamma_i}$ we denote the trace of $\rho(\gamma_i)$. As trace is invariant with respect to conjugation, it is defined for $\chi$. It was shown by Weiss that

\begin{lm}[\cite{Wei2}, Corollary 4.17]
\label{L_2}
Let $y \in H^1(L, E)$. If for all $i$ we have $y\in {\rm ker}(d{\rm tr}_{\gamma_i})_{\chi}$, then $y \in H^1_{L^2}(L, E)$.
\end{lm}

Now let $u_j$ be a vertex of $\Sigma_g$ and $\Xi_j$ be the smooth part of its spherical link. The restriction of $\rho$ to $\pi_1(\Xi_j)$ preserves a point $p_j \in \H^3$, so $\rho|_{\pi_1 (\Xi_j)}$ is conjugate to a representation to $SU(2)$. We have $\mathfrak{sl}(2, \mathbb{C})=\mathfrak{su}(2)\oplus i\mathfrak{su}(2)$ and the adjoint representation of $SU(2)$ acts diagonally on it. Thus, it preserves the decomposition and the bundle $E_j=E|_{\Xi_j}$ is decomposed as $E_j\cong E_j^1\oplus E_j^2$. This decomposition corresponds to the infinitesimal rotations and the infinitesimal translations at $p_j$ and is not the same as the usual decomposition of $E$ in rotations and translations around each point. Moreover, this decomposition is flat and $E_j$ can be endowed with a flat metric inherited from $E_{p_j}=\mathfrak{so}(T_{p_j}\H^3)\oplus T_{p_j}\H^3$. Weiss proved the following vanishing theorem for $L^2$-cohomology:

\begin{thm}[\cite{Wei2}, Theorem 3.15]
\label{vanishing}
Let $y \in H^1_{L^2}(L, E)$ with the property that for every vertex $u_j$ either
$$y|_{H^1_{L^2}(\Xi_j, E^1_j)}=0,~~~~~~{\rm or}~~~~~y|_{H^1_{L^2}(\Xi_j, E^2_j)}=0.$$
Then $y=0$.
\end{thm}

We also note that

\begin{lm}[\cite{Wei2}, Lemma 3.9]
\label{degree}
$$\dim_{\mathbb{C}} H^1_{L^2}(\Xi_j, E)=2(\deg(u_j)-3).$$
\end{lm}

We can establish the infinitesimal rigidity from these results.

\begin{lm}
\label{infrig}
If $x \in T_g\mathcal P(M, V)$ is such that $(d\mathcal I_V)_g(x)=0$, then $x=0$.
\end{lm}

\begin{proof}
Denote $d(\chi\circ D)_g(x) \in H^1(L, E)$ by $y$. Recall that for $\gamma \in \pi_1(L)$, the element $\rho(\gamma)$ is elliptic if and only if its trace ${\rm tr}_{\gamma} \in (-2, 2)$. In particular, then $\Im({\rm tr}_{\gamma})=0$. Moreover, in this case ${\rm tr}_{\gamma}=2\cos(2\theta)$, where $2\theta$ is the rotation angle of $\rho(\gamma)$.

As $y$ is tangent to the $D$-image of $\mathcal P(M, V)$, in particular, it is tangent to $\mathcal C(M^D)$. Then for all meridians $\gamma_i$ we get $\Im(d{\rm tr}_{\gamma_i})_{\chi}(y)=0$. Next, as $(d\mathcal I_V)_g(x)=0$, then for each dihedral angle $\theta_i$ we have $(d\theta_i)_g(x)=0$. Thus, $\Re(d{\rm tr}_{\gamma_i})_{\chi}(y)=0$, so $(d{\rm tr}_{\gamma_i})_{\chi}(y)=0$ and by Lemma~\ref{L_2} we get $y \in H^1_{L^2}(L, E)$.

Now consider a vertex $u_j$ of $\Sigma_g$. We want to show that $y|_{H^1_{L^2}(\Xi_j, E^1_j)}=0$. If $\deg(u_j)=3$, then by Lemma~\ref{degree} we have $H^1_{L^2}(\Xi_j, E)=0$ and the claim follows. Consider the case of $\deg(u_j)>3$. Look at $u_j$ as at a vertex of $M(g)$. Its Gauss image is a spherical polygon $G_j$. The vector $x$ is tangent to a smooth path $g_t$, for which some diagonals $\mu_1, \ldots, \mu_k$ of $G_j$ become dual to edges of $M(g_t)$. Extend this set of diagonals to a triangulation of $G_j$. Now we have the set $\mu_1, \ldots, \mu_{\deg(u_j)-3}$ of diagonals. These diagonals correspond to simple closed curves $\delta_1, \ldots, \delta_{\deg(u_j)-3}$ on $\Xi_j$ producing a pair of pants decomposition $\mathcal P_x$ of $\Xi_j$. Now we need

\begin{dfn}[\cite{MW}, Definition 2.3]
A pair of pants decomposition $\mathcal P$ of a surface $\Xi$ is called \emph{$\rho$-admissible} for an irreducible representation $\rho: \pi_1(\Xi) \rightarrow SL_2(\mathbb{C})$ if the restriction of $\rho$ to each pair of pants of $\mathcal P$ is irreducible, i.e., it does not leave invariant any plane in $\H^3$.
\end{dfn}

It is easy to see that in our setting the decomposition $\mathcal P_x$ is admissible for the restriction of $\rho$ to $\pi_1(\Xi_j)$. Indeed, $\rho(\delta_i)$ is a rotation around $p_j$ by angle $2\theta_i<2\pi$ where $\theta_i$ is the dihedral angle between the respective faces of $M(g)$. The planar angles between rays obtained in the intersections of these faces are smaller than $\pi$. After conjugating $\rho|_{\pi_1(\Xi_j)}$ to $SU(2)$ we see that its restriction to any pair of paints is the holonomy of a spherical cone-metric obtained by gluing two copies of a convex spherical triangle. Since the cone-angles are smaller than $2\pi$ and the side lengths are smaller than $\pi$, the holonomy can not leave invariant a plane in $\H^3$.

Now we need the following

\begin{lm}[\cite{MW}, Lemma 2.5]
\label{linind}
Let $\Xi$ be a $d$-punctured sphere, $\{\gamma_1, \ldots, \gamma_d\}$ be the peripheral curves, $\rho: \pi_1(\Xi) \rightarrow SL_2(\mathbb{C})$ be an irreducible representation and $\mathcal P=\{\delta_1, \ldots, \delta_{d-3}\}$ be a $\rho$-admissible representation. Then
$$\{d{\rm tr}_{\gamma_1}, \ldots, d{\rm tr}_{\gamma_d}, d{\rm tr}_{\delta_1},\ldots, d{\rm tr}_{\delta_{d-3}}\}$$
are $\mathbb{C}$-linearly independent at $\chi=[\rho]$.
\end{lm}

Let $\gamma_1, \ldots, \gamma_{\deg(u_j)}$ be the meridians around edges adjacent to $u_j$. As the restriction of $\rho$ to $\pi_1(\Xi_j)$ preserves a point $p_j$, all $\rho(\gamma_k)$, $\rho(\delta_k)$ are elliptic elements fixing $p_j$. Thus,
$$H^1_{L^2}(\Xi_j, E^2_j) \subseteq \left(\bigcap_{k=1}^{\deg(u_j)}\ker(d{\rm tr}_{\gamma_k})_{\chi}\right)\cap\left(\bigcap_{k=1}^{\deg(u_j)-3}\ker(d{\rm tr}_{\delta_k})_{\chi}\right).$$
By comparing the dimensions, from Lemma~\ref{linind} and Lemma~\ref{degree} we get
$$H^1_{L^2}(\Xi_j, E^2_j) = \left(\bigcap_{k=1}^{\deg(u_j)}\ker(d{\rm tr}_{\gamma_k})_{\chi}\right)\cap\left(\bigcap_{k=1}^{\deg(u_j)-3}\ker(d{\rm tr}_{\delta_k})_{\chi}\right).$$

Now as $y$ is tangent to $\mathcal C(M^D)$, we get $\Im(d{\rm tr}_{\delta_k})_{\chi}(y)=0$ for all $k$. But as $(d\mathcal I_V)_g(x)=0$, by construction of $\mathcal P_x$ we see that also $\Re(d{\rm tr}_{\delta_k})_{\chi}(y)=0$. Thus,
$$y \in \left(\bigcap_{k=1}^{\deg(u_j)}\ker(d{\rm tr}_{\gamma_k})_{\chi}\right)\cap\left(\bigcap_{k=1}^{\deg(u_j)-3}\ker(d{\rm tr}_{\delta_k})_{\chi}\right).$$
Thus, $y|_{H^1(\Xi_j, E)} \in H^1_{L^2}(\Xi_j, E^2_j)$, or, in other words, $y|_{H^1_{L^2}(\Xi_j, E^1_j)}=0$ as desired. This happens for every vertex $u_j$ of $\Sigma_g$. Now we can apply Theorem~\ref{vanishing} and get $y=0$. Thus, also $x=0$ and the proof is over.

\end{proof}

\section{Properness}
\label{psec}

The goal of this section is to prove

\begin{lm}
\label{proper}
Let $\{g_i\}$ be a sequence in $\mathcal P(M, V)$ such that the sequence of the induced dual metrics $\{d^*_i\}$ converges to a metric $d^* \in \ol{\mathcal D}^*_M(\partial M, V)$. Then, up to extracting a subsequence, $\{g_i\}$ converges to $g \in \ol{\mathcal P}(M, V)$ with $V(g)=V(d^*)$.
\end{lm}

Let $S_j$ be a connected component of $\partial M$ and $V_j \subset V$ be the subset of $V$ marking faces of $S_j$. We continue to denote the restriction of $d^*$ to $S_j$ just by $d^*$, so $d^*$ can be considered as an element of $\mc D^*(S_j, V_j)$. Recall that by $\nu_j: \pi_1(S_j) \rightarrow \pi_1(M)$ we denote the push-forward homomorphism induced by the inclusion of $S_j$ to $M$, and by $D_j$ we denote the covering space of $S_j$ corresponding to $\ker(\nu_j)$. Denote the lift of $V_j$ to $D_j$ by $\tilde V_j$. We continue to denote the lift of $d^*$ to $D_j$ by $d^*$. Note that all closed geodesics in $(D_j, d^*)$ have lengths greater than $2\pi$. Let $\mathcal T$ be a triangulation of $(S_j, V_j)$ realized by $d^*$ such that its lift to $D_j$ is a simplicial complex. We continue to denote this lift by $\mathcal T$.

\begin{dfn}
\label{regemb}
An isometric embedding $\phi: (D_j, d^*) \rightarrow d\mathbb{S}^3$ is called \emph{$\mathcal T$-regular} if

1) it is polyhedral with respect to $\mathcal T$,

2) it is future-convex, i.e., it is convex and the future cone of each point on the edges of $\phi(D_j)$ belongs to the convex side of $\phi(D_j)$.
\end{dfn}

We note that since $\phi$ is an isometric embedding, then all faces are necessarily space-like. Consider the spherical model $d\mathbb{S}_S^3 \subset \S^3$ and assume that $\phi$ maps there. It follows from Definition~\ref{regemb} that $\partial \phi(D_j) \subset \partial_{\infty}^+ d\mathbb{S}^3$. The convexity condition also implies that $\phi(D_j) \subset \partial {\rm conv} (\phi(D_j)).$ It is evident that
\begin{equation}
\label{part}
\big(\partial{\rm conv}(\phi(D_j)) \backslash \phi(D_j)\big) \subset \big(\partial_{\infty}^+ d\mathbb{S}_S^3 \cup \H_S^3\big).
\end{equation}

In the setting of Lemma~\ref{proper} we have a sequence of isometric embeddings $\phi_i: (D_j, d^*_i) \rightarrow d\mathbb{S}^3$. Due to Lemma~\ref{fintriang}, there are only finitely many triangulations realized by metrics $d^*_i$. By passing to a subsequence, we may assume that there is a triangulation $\mathcal T$ realized by all $d^*_i$ such that its lift to $D_j$ is a face triangulation of all $\phi_i(D_j)$. Note that the lift of $\mathcal T$ to $D_j$ is a simplicial complex and all $\phi_i$ are $\mathcal T$-regular.

\begin{lm}
\label{dualconv}
After possibly composing each $\phi_i$ with an isometry $\eta_i$ of $d\mathbb{S}^3$ and passing to a subsequence, $\phi_i$ converge to a $\mathcal T$-regular embedding $\phi: (D_j, d^*) \rightarrow d\mathbb{S}^3$.
\end{lm}

\begin{proof}
We rely on the spherical model $d\mathbb{S}_S^3$ of $d\mathbb{S}^3$ in $\S^3$ assuming that all $\phi_i$ map to $d\mathbb{S}_S^3$. After passing to a subsequence, we can assume that for all $v \in \tilde V_j$ the sequence $\phi_i(v)$ converges to a point in $$\overlined\mathbb{S}_S^3=d\mathbb{S}_S^3 \cup \partial_{\infty}^+ d\mathbb{S}_S^3 \cup \partial_{\infty}^- d\mathbb{S}_S^3.$$ We denote this point $\phi(v)$, so $\phi$ is a map from $\tilde V_j$ to $\overlined\mathbb{S}_S^3$.

Note that due to the future-convexity of $\phi_i$, the intersection $\phi(\tilde V_j) \cap \partial_{\infty}^- d\mathbb{S}_S^3$ consists of maximum one point. If this intersection is a point $s$, then we can compose each $\phi_i$ with an isometry $\eta_i$ of $d\mathbb{S}^3$ that has $s$ as its fixed repulsive point so that this intersection becomes empty. Abusing the notation, we continue to denote the corrected maps $\eta_i \circ \phi_i$ just by $\phi_i$. By a similar procedure we can also assume that $\phi(\tilde V_j)\cap d\mathbb{S}_S^3$ consists of at least one point, continuing to keep $\phi(\tilde V_j) \cap \partial_{\infty}^- d\mathbb{S}_S^3$ empty.

Now we extend $\phi$ from $\tilde V_j$ to $D_j$ by continuity as the pointwise limit of $\phi_i$. We continue to denote the resulting map by $\phi$. Thus, $\phi$ becomes a continuous $\mathcal T$-polyhedral map from $D_j$ to $d\mathbb{S}_S^3 \cup \partial_{\infty}^+ d\mathbb{S}_S^3$.

Note that if $v,w \in \tilde V_j$ are adjacent in $\mathcal T$ and $\phi(v), \phi(w) \in \partial_{\infty}^+d\mathbb{S}_S^3$, then $\phi(v)=\phi(w)$, as otherwise for some $\phi_i$ the distance between $\phi_i(v)$ and $\phi_i(w)$ is not space-like.

For $p \in (D_j \backslash \tilde V_j)$ belonging to the interior of a triangle of $\mathcal T$, we define the open star ${\rm ost}(p)$ of $p$ to be the interior of this triangle. If $p$ belongs to the relative interior of an edge of $p$, then by ${\rm ost}(p)$ we denote this relative interior together with the interiors of both triangles adjacent to this edge. When $p \in \tilde V_j$, by ${\rm ost}(p)$ we mean the relative interiors of all edges and triangles adjacent to $p$ together with $p$ itself. The open star ${\rm ost}(U)$ of a set of points $U \subset D_j$ is defined as the union of ${\rm ost}(p)$ for all $p\in U$.

Define $X:=D_j\backslash {\rm ost}(\phi^{-1}(\partial_{\infty}^+d\mathbb{S}_S^3))$. By construction, $X$ is non-empty. By continuity, the restriction of $\phi$ to $X$ is a path isometry. As by continuity $\phi(D_j) \subset {\rm conv}(\phi(D_j))$, the restriction to $X$ is injective.

Let $s \in \phi(D_j)\cap  \partial_{\infty}^+ d\mathbb{S}_S^3$, $\Pi_s$ be the tangent plane to $ \partial_{\infty}^+ d\mathbb{S}_S^3$ at $s$, and $Y_s:=  \phi(D_j) \cap \Pi_s$. Then $Y_s$ is a compact convex set. Indeed, $\Pi_s \cap {\rm conv} (\phi(D_j))$ is compact and convex, and $\Pi_s \cap \partial_{\infty}^+ d\mathbb{S}_S^3=s$. By continuity, as (\ref{part}) holds for all $\phi_i$, it holds also for $\phi$. Hence, (\ref{part}) implies that $$\Pi_s \cap {\rm conv}(\phi(D_j))=\Pi_s \cap \phi(D_j)=Y_s.$$ Moreover, $Y_s$ is the $\phi$-image of the closure of ${\rm ost}(\phi^{-1}(s))$. In particular, $Y_s\neq s$.

Suppose that $Y_s$ is 1-dimensional and $s \in \partial Y_s$. Then the boundary of ${\rm ost}(\phi^{-1}(s))$ consists just of one point $v \in \tilde V_j$. Indeed, the boundary should be mapped isometrically to $Y_s$, but it belongs to a light-like direction, thus all distances should be zero. Then ${\rm ost}(\phi^{-1}(s))$ is exactly the complement to $v$ in $D_j$. Hence, $\phi$ maps the complement of ${\rm ost}(v)$ to a single point $s \in \partial_{\infty}^+ d\mathbb{S}_S^3$, $v$ to $d\mathbb{S}_S^3$ and ${\rm ost}(v)$ to the segment between $s$ and $\phi(v)$. This contradicts with $\phi$ being the limit of future-convex maps. Indeed, for any future-convex map $\phi: D_j \rightarrow d\mathbb{S}^3$ the future cone of every point $p \in \phi(D_j)$ belongs to ${\rm conv}(\phi(D_j))$. Clearly, this remains true when $\phi$ is the pointwise limit of future-convex maps. However, it is not true in our case for the future cone of $\phi(v)$. If $Y_s$ is 1-dimensional and $s$ is in the relative interior of $Y_s$, then the argument is the same, but $\partial {\rm ost}(\phi^{-1}(s))$ are two points.

Thus, $Y_s$ is 2-dimensional. Suppose that $s \in \partial Y_s$. So $\partial {\rm ost}(\phi^{-1}(s))$ is mapped to $\partial Y_s$ excluding $s$ and, possibly, excluding light-like segments from $s$ (either closed or half-open). This means that $\partial {\rm ost}(\phi^{-1}(s))$ is homeomorphic to a segment, either open or half-open or closed. Since the length of $\partial Y_s$ is finite, $\partial {\rm ost}(\phi^{-1}(s))$ can not contain infinitely many points of $\tilde V_j$. Thus, it is a finite simple path. Then it is mapped by $\phi$ to a polygonal simple path in $d\mathbb{S}_S^3$, the complement of its open star in $D_j$ is mapped to $s$ and the rest of the open star is mapped to their linear span. This also contradicts with $\phi$ being the limit of future-convex maps by the same reasons as above.

Hence, $s$ belongs to the relative interior of $Y_s$ and $\partial {\rm ost}(\phi^{-1}(s))$ is a finite simple closed polygonal curve in $D_j$. As $\partial Y_s$ is a convex curve in $d\mathbb{S}_S^3$ around a point at infinity in a degenerate plane $\Pi_s$, it has length $2\pi$. One can also see that its angles on the side to ${\rm ost}(\phi^{-1}(s))$ are $\pi$.  This is shown in~\cite[Lemma 6.6]{Sch}. We quickly sketch the argument. Take a vertex $q$ of $Y_s$ and consider a small enough convex neighborhood $C$ of $q$ in $Y_s$. Let $v=\phi^{-1}(q)$ and $U=\phi^{-1}(Y_s)$. By $C_i$ we denote convex neighborhoods of $\phi_i(v)$ in $\phi_i(U)$ converging to $C$. The sets $\phi_i^{-1}(C_i)$ are convex neighborhoods of $v$ in $U$ with metrics $d^*_i$. Up to taking a subsequence, they converge to a convex set $A$ in $U$ with the metric $d^*$ containing $v$. But two parts of the boundary of $C$, the one belonging to $\partial Y_s$ and its complement, have the same length. Hence, so is with the boundary of $A$, thus $A$ is a geodesic segment on the side of $U$ (having angle $\pi$ at $v$). As all cone angles of $d^*$ are greater than $2\pi$, $\partial {\rm ost}(\phi^{-1}(s))$ is a simple closed geodesic of length $2\pi$ in $(D_j, d^*)$. This is a contradiction.

Thus, $\phi(D_j) \subset d\mathbb{S}_S^3$ and $\phi$ is a path isometry over all $(D_j, d^*)$. In particular, this means that all faces of $\phi(D_j)$ are space-like. The future-convexity of $\phi$ follows by continuity. Hence, $\phi$ is a $\mathcal T$-regular isometric embedding of $(D_j, d^*)$.
\end{proof}

Each embedding $\phi_i$ is equivariant with respect to a convex cocompact, discrete and faithful representation $\rho_{j, i}: {\rm im}(\nu_j) \rightarrow G={\rm Iso^+}(d\mathbb{S}^3)={\rm Iso}^+(\H^3)$. As $\phi_i$ converge to $\phi$, for each $\gamma \in \pi_1(S_j)$, $\rho_{j,i}(\gamma)$ converge to an element of $G$. A theorem of J{\o}rgensen~\cite{Jor} says that the set of discrete and faithful representations to $G$ is closed in the space of representations. Thus, $\rho_{j,i}$ converge to a discrete and faithful representation $\rho_j: {\rm im}(\nu_j) \rightarrow G$ and $\phi$ is equivariant with respect to $\rho_j$.

The dual to the embedded surface $\phi(D_j)$ is a convex polyhedral embedding of $D_j$ to $\H^3$ equivariant with respect to $\rho_j$. As ${\rm im}(\mu_j)$ contains no torsion, the image of $\rho_j$ contain no elliptic elements, so the action is free. By taking the quotient, we obtain a hyperbolic convex cone-metric $d$ on $S_j$. By continuity, we get that the metrics $d_i$ on $S_j$ induced by $g_i$ at the boundary of $M$ converge uniformly to a convex hyperbolic cone-metric $d$ on $S_j$.

We do these procedures for each connected component $S_j$ of $\partial M$. We obtain a convex hyperbolic cone-metric on $\partial M$, which we continue to denote by $d$.

\begin{lm}
Let $\{g_i\}$ be a sequence in $\mathcal P(M, V)$ such that the sequence of the induced metrics $\{d_i\}$ on $\partial M$ converges uniformly to a convex hyperbolic cone-metric $d$ on $\partial M$. Then up to extracting a subsequence, the sequence $\{\overline g_i\}$ converges to a convex cocompact hyperbolic metric $\overline g \in \mathcal{CH}(N)$.
\end{lm}

\begin{proof}
Because $d_i$ converge uniformly to $d$, the length of a smallest curve in each free homotopy class of closed curves on $\partial M$ is uniformly bounded from above in $\{d_i\}$. Due to the hyperbolic Busemann--Feller lemma~\cite{Mil}, this implies that they are also uniformly bounded from above in $\partial C(\overline g_i)$. Theorem~\ref{flp} implies that the injectivity radii of $\partial C(\overline g_i)$ are uniformly bounded from below. Due to the Theorem~\ref{su}, the length of a smallest curve in each free homotopy class is also uniformly bounded in the hyperbolic representatives of conformal structures in $\partial_{\infty}N(\overline g_i)$. Due to Theorem~\ref{ab} and Theorem~\ref{flp}, this means that $\overline g_i$ stays in a compact subset of $\mathcal{CH}(N)$, and thus, up to taking a subsequence, converges to $\overline g \in \mathcal{CH}(N)$.
\end{proof}

Now we can finish the proof of Lemma~\ref{proper}.

\begin{proof}
Let $v \in V$ and $z_{v,i}$ be the immersed plane in $N(\overline g_i)$ containing the face of $M(g_i)$ marked by $v$. By $h_{v, i}$ we denote the distance from $z_{v, i}$ to $C(\overline g_i)$. We see that $h_{v, i}$ is bounded from above. Indeed, otherwise the diameters of the induced metrics on $\partial M(g_i)$ grow to infinity, which contradicts to the fact that they converge to a cone-metric.

Thus, up to taking a subsequence, all $z_{v,i}$ converge to some immersed planes $z_v$ in $N(\overline g) \backslash {\rm int}(C(\overline g))$. Hence, polyhedral surfaces $\partial M(g_i)$ converge to polyhedral surfaces in $N(\overline g)$ bounding together a totally convex set. It is compact, as otherwise, again, the diameters of $d_i$ would grow to infinity. Thus, it is homeomorphic to $M$, and we denote it by $M(g)$.

We need to show that $z_{v, i}$ are bounded from below. To see this, let us pass to the universal cover. The surface of $\partial \tilde C(\overline g)$ consists of totally geodesic pieces bounded by lines, which extend to the boundary at infinity. Hence, if a face of $\partial \tilde M(g)$ falls to $\partial \tilde C(\overline g)$, then this face contains a line in $\H^3$. Let $\tilde v \subset \partial M(g)^*$ be the vertex dual to this face. Then there are faces of $\tilde M(g)^*$ adjacent to $\tilde v$ that are touching $\partial_{\infty}^+ d\mathbb{S}^3$. Arguments from the proof of Lemma~\ref{dualconv} show that this can not happen. It is evident that $V(g)=V(d^*)$.

\end{proof}

\section{Final steps}
\label{fsec}

Here we finish the proof of Theorem~\ref{mtr}. We need the following weak connectivity results for spaces $\mathcal D^*_M(\partial M, V)$:

\begin{lm}
\label{connect1}
For every $d^*_0, d^*_1 \in \mathcal D^*_M(\partial M, V)$ there exist $W \supseteq V$, a triangulation $\mc T$ of $(\partial M, W)$ and a path $\alpha: [0,1] \rightarrow \mc D(\partial M, \mc T)$ such that $\alpha(0)$ and $\alpha(1)$ are lifts of $d^*_0$ and $d^*_1$ respectively, and that for all $t\in (0,1)$ we have $\alpha(t) \in \mc D^*_M(\partial M, W)$.
\end{lm}

\begin{lm}
\label{connect2}
For every continuous map $\beta: S^1 \rightarrow \overline{\mathcal D}^*_M(\partial M, V)$ there exist $W \supseteq V$, a triangulation $\mc T$ of $(\partial M, W)$ and a continuous map $\alpha: \overline D^2 \rightarrow \mc D(\partial M, \mc T)$ of the closed 2-disk $\overline D^2$ with the boundary identified to $S^1$ such that $\alpha|_{S^1}$ is a lift of $\beta$, and that $\alpha(D^2) \subset \mathcal D^*_M(\partial M, W)$, where $D^2$ is the interior of $\overline D^2$.
\end{lm}

By $\mathcal D^*_{cl}(S, V)$ denote the space of isotopy classes of concave large spherical cone-metrics $d$ on $(S, V)$ with $V(d)=V$. Lemma~\ref{connect1} was first proven in~\cite[Section~9]{HR} for the space $\mathcal D^*_{cl}(S, V)$ when $S$ is the 2-sphere.  Then Lemmas~\ref{connect1} and~\ref{connect2} for $\mathcal D^*_{cl}(S, V)$ and arbitrary closed $S$ appeared in their present form in a manuscript~\cite{Sch5} of Schlenker. Brief proof sketches were given there. Basically, Schlenker noticed that a proof of Lemma~\ref{connect1} can be given exactly in the same way as it was done for spheres in~\cite{HR}. For Lemma~\ref{connect2} Schlenker remarks that a proof is an easy generalization of the proof of Lemma~\ref{connect1}. We claim that these results hold straightforwardly also for the spaces $\mathcal D^*_M(\partial M, V)$. Indeed, we only need to modify the proofs to gain control of the smallest geodesic that is contractible not just in $\partial M$, but in $M$. For a metric $d$ on $S$ by ${\rm ml}(d)$ we denote the infimum of lengths of closed geodesics in $(S, d)$ that are contractible in $S$. The control on ${\rm ml}(d)$ in~\cite{HR} follows from two facts. The first is that if $\lambda>0$ is a number and $d$ is a Riemannian metric on $S$, then ${\rm ml}(\lambda d)=\lambda {\rm ml}(d)$, which clearly holds also for the function ${\rm ml}_M$. The second fact is that ${\rm ml}$ is upper-semicontinuous in the Lipschitz topology. For ${\rm ml}_M$ this was shown in Corollary~\ref{semicont}.

We now provide proof sketches of Lemmas~\ref{connect1} and~\ref{connect2}. All the details can be reconstructed with the help of~\cite{HR}.

\begin{proof}
We continuously smoothen the initial metrics ($d^*_0$ and $d^*_1$ for Lemma~\ref{connect1} or $\beta(S^1)$ for Lemma~\ref{connect2}) in the class of Riemannian metrics with conical singularities and obtain in the end a pair or a loop of Riemannian metrics. As the set of Riemannian metrics on $S$ is connected and simply connected, we can connect our pair of smooth metrics by a path or we can contract the loop. Now we multiply each metric of our path or our disk by a scalar (equal 1 near the boundary of the path or of the disk) to achieve that the sectional curvature is bounded from above by 1 and that the length of the shortest geodesic loop in $\partial M$ that is contractible in $M$ is greater than some constant greater than $2\pi$. This is easy to perform due to the compactness of the path or the disk. For the involved cone-metrics we also take care that the cone-angles stay greater than $2\pi$ during our procedure.

We approximate the obtained metrics by spherical cone-metrics. To this purpose one constructs a very small triangulation $\mc T$ of $\partial M$ with a vertex set $W$, which can be geodesically realized in all involved metrics. Next, for each metric we replace each triangle by a spherical triangle with the same lengths. If the triangulation was sufficiently fine, then the distances in the obtained cone-metrics got distorted by at most $\varepsilon$ comparing with the approximated metrics for an arbitrarily small globally defined $\varepsilon>0$. From comparison geometry one shows that the resulting cone-metrics are concave. Provided that $\varepsilon$ is sufficiently small, it also follows from the upper semi-continuity of the function ${\rm ml}_M$ that the length of the shortest geodesic in $\partial M$ that is contractible in $M$ remains greater than $2\pi$.
\end{proof}

Now we can collect together the proof of the main result.

\begin{proof}[Proof of Theorem~\ref{mtr}]
Consider the restriction of the map $\mc I_V$ to $\mathcal P(M, V)$. From Lemma~\ref{differ}, it is $C^1$. Due to Lemma~\ref{infrig}, the differential $d\mathcal I_V$ is non-degenerate. By the inverse function theorem, $d\mathcal I_V$ is locally injective. Since manifolds $\mathcal P(M, V)$ and $\mathcal D^*_M(\partial M, V)$ have the same dimensions, by Brower's invariance of the domain theorem, the restriction of $\mathcal I_V$ is a local homeomorphism. Due to Lemma~\ref{proper}, it is proper. Thus, it is a covering map. We highlight that we can not really show the connectedness of $\mathcal P(M, V)$ and $\mathcal D^*_M(\partial M, V)$, so we take into account that it is a covering map between possibly disconnected spaces.

Due to Lemma~\ref{connect1}, for any two metrics $d^*_0, d^*_1 \in \mathcal D^*_M(\partial M, V)$ there exists $W \supseteq V$ and a path $\alpha: [0, 1] \rightarrow \ol{\mc D}^*_M(\partial M, W)$ such that $\alpha(0)$ and $\alpha(1)$ are lifts of $d^*_0$ and $d^*_1$, and that the relative interior of the path belongs to $\mathcal D^*_M(\partial M, W)$. We take $d^*_1$ arbitrary and take $d^*_0$ in the image of $\mathcal I_V$, hence $\alpha(0)$ in the image of $\mc I_W$. We denote its preimage by $g_0$. We can modify the path so that for all small $t$ the metrics $\alpha(t)$ are in the image of $\mathcal I_W$. Indeed, consider any map $f_W: W \rightarrow F(g_0)$, where $F(g_0)$ is the set of faces of $g_0$, extending the marking map $f: V \rightarrow F(g_0)$ of $g_0$. We can slightly perturb the respective faces by splitting them in parts to obtain a metric $g'_0 \in \mathcal P(M, W)$. By Corollary~\ref{locconnect}, there exists a neighborhood $U$ of $\alpha(0)$ such that $U \cap \mc D^*_M(\partial M, W)$ is connected. Hence, we can choose $g'_0$ so that $\mc I_W(g'_0)$ can be connected with the metric $\alpha(t)$ for some small $t$ by a path in $\mc D^*_M(\partial M, W)$. Taking the concatenation of this path with the rest of the initial path, we obtain the one satisfying our demand. Then the infinitesimal rigidity and the properness of $\mathcal I_W$ imply that all $\alpha(t)$ are in the image of $\mathcal I_W$. Hence, also $d^*_1$ is in the image of $\mc I_V$, which means that $\mathcal I_V$ is surjective.

It remains to show that each metric in $\mathcal D^*_M(\partial M, V)$ has only one preimage. Suppose the converse, that for $g_0, g_1 \in \mathcal P(M, V)$ we have $\mathcal I_V(g_0)=\mathcal I_V(g_1)$. Due to Lemma~\ref{connect}, there exists $W' \supseteq V$ and a path $\eta: [0,1] \rightarrow \mc P(M, W')$ such that $\eta(0)$ and $\eta(1)$ are lifts of $g_0$ and $g_1$ respectively. Consider $S^1$ as $[0,1]$ with identified endpoints. Then $\mc I_{W'}\circ \eta$ is a loop, which we denote by $\beta: S^1 \rightarrow \ol{\mc D}^*_M(\partial M, W')$. Due to Lemma~\ref{connect2}, there exist $W \supseteq W'$, a triangulation $\mc T$ of $(\partial M, W')$ and a map $\alpha: \ol D^2 \rightarrow \ol{\mc D}^*_M(\partial M, W)$ such that $\alpha|_{S^1}$ is a lift of $\beta$, and that $\alpha(D^2) \subset \mathcal D^*_M(\partial M, W)$.

There exists a lift $\zeta: [0,1] \rightarrow \ol{\mc P}(\partial M, W)$ of the map $\eta$ such that $\alpha|_{S^1}=\mc I_W \circ \zeta$. We claim that there exists a homotopy $\zeta_s$ of $\zeta_0=\zeta$ with $s \in [0, \varepsilon]$ for some small $\varepsilon>0$ such that for all $s>0$ we have $\zeta_s \subset \mc P(M, W)$. Indeed, for $s>0$ and every $t \in [0,1]$ we move every immersed oriented plane in $N(\ol{\zeta}(t))$ marked by $v \in V$ at distance $s$ from its previous position along the segment connecting the closest point on this plane to the closest point on $C(\ol{\zeta}(t))$ towards $C(\ol{\zeta}(t))$. One can see that for sufficiently small $s$ the intersection of the immersed negative half-spaces of these planes defines a subset of $N(\ol\zeta(t))$ homeomorphic to $M$, and that every marked plane intersects the boundary of this subset by a set with non-empty relative interior. This determines an element $\zeta_s(t) \in \mc P(M, W)$.

Define $\theta_s(t):=\mc I_W \circ \eta_s(t)$. Since $\mc D^*_c(\partial M, \mc T)$ is open, we may assume that $\varepsilon$ is sufficiently small so that $\theta_s(t) \in \mc D^*(\partial M, \mc T)$ for all $t \in [0,1]$ and $s \in [0, \varepsilon]$.  Consider the arcs
$$\theta'_s:=\{\theta_r(0), r\in [0, s]\} \cup \{\theta_r(1), r \in [0, s]\}.$$
For each $s$ the union $\theta_s \cup \theta'_s$ is a loop. We parametrize them as $\iota_s: S^1 \rightarrow \ol{\mc D}^*_M(\partial M, W)$ so that the arc $\theta'_s$ corresponds to the restriction of $\iota_s$ on the interval $[1/2-s, 1/2+s]$. In particular $\iota_0(1/2)=\theta_0(0)=\theta_0(1)$.

Recall that for a number $\lambda$ and a spherical cone-metric $d^*$ on $(\partial M, W)$ realizing a triangulation $\mc T$ by $(\lambda *_{\mc T} d^*)$ we denote the metric obtained from $d^*$ by multiplication of all the side-lengths of $\mc T$ by $e^\lambda$. Consider a 1-parameter family of functions $\lambda_s: S^1 \rightarrow [0, \varepsilon]$, where $s \in [0,\varepsilon]$, such that $\lambda_0$ is identically zero, and for all $s>0$ we have $\lambda_s(t)>0$ for $t \in (1/2-s, 1/2+s)$, but $\lambda_s$ is zero outside $(1/2-s, 1/2+s)$. Define loops $\delta_s:=(\lambda_s *_{\mc T} \iota_s)$. Provided that $\varepsilon$ is sufficiently small, for $s>0$ from Corollary~\ref{multipl} we get that $\delta_s \subset \mc D^*_c(\partial M, \mc T)$. We also assume that $\delta_s \subset \mc D^*_M(\partial M, W)$ for $s>0$. We note that the loop $\delta_0$ is a reparametrization of $\alpha|_{S^1}$.

Because $\delta_0$ is contractible in $\ol{\mc D}^*_M(\partial M, W) \cap \mc D^*(\partial M, \mc T)$, so are the loops $\delta_s$. We note that it means that this means that for $s>0$ the loops $\delta_s$ are contractible in ${\mc D}^*_M(\partial M, W) \cap \mc D^*(\partial M, \mc T)$. Indeed, let $\alpha_s: \ol D^2 \rightarrow (\ol{\mc D}^*_M(\partial M, W) \cap \mc D^*(\partial M, \mc T))$ be a homotopy contracting $\delta_s$ in $\ol{\mc D}^*_M(\partial M, W) \cap \mc D^*(\partial M, \mc T)$. Choose a function $\lambda_s: \ol D^2 \rightarrow [0, \varepsilon]$ such that $\lambda_s|_{S^1}=0$ and $\lambda_s|_{D^2}>0$. Consider now a homotopy $\alpha'_s:=(\alpha_s *_{\mc T} \lambda_s)$. By Corollary~\ref{multipl}, provided that $\varepsilon$ is sufficiently small, we get
$$\alpha'_s \subset ({\mc D}^*_M(\partial M, W) \cap \mc D^*(\partial M, \mc T)).$$
Because the restriction of $\mc I_W$ to $\mc P(M, W)$ is a covering map, the preimage $\mc I_W^{-1}(\delta_s)$ consists of loops for all $s>0$. This means that for $s>0$ there are paths $\kappa_s: [0, 1] \rightarrow \mc P(M, W)$ with $\kappa_s(0)=\eta_s(0)$, $\kappa_s(1)=\eta_s(1)$ such that
$$\mc I_W \circ \kappa_s(t) = \delta_s(1/2-s+2ts).$$
In particular, the arcs $\mc I_W \circ \kappa_s$ converge to $\delta_0(1/2)$.

Consider a compact neighborhood $U$ of $\delta_0(1/2)$ such that all arcs obtained from the restrictions of $\delta_s$ to $[1/2-s, 1/2+s]$ belong to $U$. Because $\mc I_W$ is proper, the set $X:=\mc I_W^{-1}(U)$ is compact. Hence, there exists a Hausdorff limit $\kappa \subset X$ of the arcs $\kappa_s$ (here we perceive $\kappa_s$ and $\kappa$ not as maps, but as point-sets). By continuity, $\mc I_W(\kappa)=\delta_0(1/2)$. Because $\mc I_V$ is a local homeomorphism, the points $\eta(0)$, $\eta(1)$ are isolated in $\mc I_W^{-1}(\delta_0(1/2))$. Hence, $\kappa = (\eta(0) \cup \eta(1))$. This can not happen as $\kappa$ is the Hausdorff limit of the connected arcs $\kappa_s$ and $\ol{\mc P}(M, V)$ is a Hausdorff space. This contradiction finishes the proof.

\end{proof}

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