Assume that $V\subset\partial M$. By $\mathcal{P}_{b}(M,V)$ we denote the
space of bent hyperbolic metrics $g$ on $M$ with $V(g)=V$ considered up to
isometries preserving $V$ pointwise and isotopic to identity. We will
frequently abuse the notation and write $g\in\mathcal{P}_{b}(M,V)$ meaning
that $g$ is a bent hyperbolic metric on $M$ representing a class in
$\mathcal{P}_{b}(M,V)$. A class in $\mathcal{P}_{b}(M,V)$ determines a class
in $\mathcal{CH}(N,V)$ and this defines an embedding of $\mathcal{P}_{b}(M,V)$
to $\mathcal{CH}(N,V)$, so we will consider $\mathcal{P}_{b}(M,V)$ as a subset
of $\mathcal{CH}(N,V)$ and endow it with the induced topology.

######  Lemma 3.25.

$\mathcal{P}_{b}(M,V)$ is open in $\mathcal{CH}(N,V)$.

###### Proof.

A pair
$(\rho,\widetilde{f})\in\mathcal{R}(\pi_{1}(M),G)\times(\mathbb{H}^{3})^{*V}$
projects to an element of $\mathcal{P}_{b}(M,V)$ if $\rho$ is a holonomy of a
convex cocompact hyperbolic metric on $N$ and if $\Lambda(\rho)$ and
$\widetilde{V}$, where $\widetilde{V}$ is the $\rho$-orbit of
$\widetilde{f}(V)$, are in a convex position. Take $g\in\mathcal{P}_{b}(M,V)$
and lift it to
$(\rho,\widetilde{f})\in\mathcal{R}(\pi_{1}(M),G)\times(\mathbb{H}^{3})^{*V}$.
Lemma 2.36 says that when convex cocompact representations converge in
$\mathcal{R}(\pi_{1}(M),G)$, their limit sets converge in the Hausdorff sense.
Hence, if $g$ does not have a neighborhood in $\mathcal{CH}(N,V)$ belonging to
$\mathcal{P}_{b}(M,V)$, then for some $\widetilde{v}\in\widetilde{V}$ there
exists a sequence of 4-tuples $(p^{1}_{i},p^{2}_{i},p^{3}_{i},p^{4}_{i})$,
where $p^{j}_{i}$ is either in $\widetilde{V}$ or in $\Lambda(\rho)$, such
that the distance from $\widetilde{v}$ to the convex span of the $i$-th
4-tuple tends to zero. Up to passing to subsequences, we can assume that all
four sequences converge to points $p^{1},p^{2},p^{3},p^{4}\in\Lambda(\rho)$.
Then $\widetilde{v}$ belongs to the convex span of $p^{1},p^{2},p^{3},p^{4}$,
which is a contradiction with that $(\rho,\widetilde{f})$ projects to
$\mathcal{P}_{b}(M,V)$. ∎

By $\overline{\mathcal{P}}_{b}(M,V)$ we denote the space of bent hyperbolic
metrics $g$ on $M$ with $V(g)\subseteq V$ considered up to isometries
preserving $V$ pointwise and isotopic to identity. Similarly to
$\mathcal{P}_{b}(M,V)$, the set $\overline{\mathcal{P}}_{b}(M,V)$ embeds in
$\mathcal{CH}(N,V)$ and is endowed with the induced topology. It is easy to
see that $\overline{\mathcal{P}}_{b}(M,V)$ is exactly the closure of
$\mathcal{P}_{b}(M,V)$ in $\mathcal{CH}(N,V)$.

######  Lemma 3.26.

The space $\overline{\mathcal{P}}_{b}(M,V)$ is connected.

###### Proof.

By $\mathcal{CH}_{f}(N,V)$ denote the set of
$[\overline{g},f]\in\mathcal{CH}(N,V)$ such that $\overline{g}$ is Fuchsian
and $f(V)\subset C(\overline{g})$, by $\mathcal{CH}_{nf}$ we denote the
complement of $\mathcal{CH}_{f}(N,V)$ in $\mathcal{CH}(N,V)$. It is easy to
see that $\mathcal{CH}_{nf}(N,V)$ is connected. Indeed, $\mathcal{CH}(N,V)$ is
connected and $\mathcal{CH}_{f}(N,V)$ has codimension greater than one.
Consider $g_{0},~{}g_{1}\in\overline{\mathcal{P}}_{b}(M,V)$, represent them as
$[\overline{g}_{0},f_{0}],~{}[\overline{g}_{1},f_{1}]\in\mathcal{CH}(N,V)$.
Connect them by a path
$\\{[\overline{g}_{t},f_{t}],t\in[0,1]\\}\subset\mathcal{CH}_{nf}(N,V)$. By
$M(g_{t})$ denote the closed convex hull of $f_{t}(V)$, it is clear that since
$[\overline{g}_{t},f_{t}]\notin\mathcal{CH}_{f}(N,V)$, the set $M(g_{t})$ is
homeomorphic to $M$. Fix a family of homeomorphisms $h_{t}:M(g_{t})\rightarrow
M$ such that the resulting map

$h:\\{(p,t):p\in M(g_{t}),~{}t\in[0,1]\\}\rightarrow M$

is continuous with respect to the topology induced from $N\times[0,1]$. For
$v\in V$ define

$O_{v}:=\\{t\in[0,1]:f_{t}(v)\in{\rm int}(M(g_{t}))\\}.$

Clearly, $0,1\notin O_{v}$ and $O_{v}$ is open in $[0,1]$. Hence, it consists
of at most countably many open intervals. Let $I$ be such an interval with
boundary points $t_{1},~{}t_{2}\notin O_{v}$, so $f_{t_{1}}(v)\in\partial
M(g_{t_{1}})$ and $f_{t_{2}}(v)\in\partial M(g_{t_{2}})$. Connect
$h_{t_{1}}\circ f_{t_{1}}(v)$ and $h_{t_{2}}\circ f_{t_{2}}(v)$ by a connected
path $\iota:[t_{1},t_{2}]\rightarrow\partial M$ such that for each
$t\in[t_{1},t_{2}]$ and each $v^{\prime}\in V\backslash\\{v\\}$ we have
$\iota(t)\neq h_{t}\circ f_{t}(v)$ (it is easy to achieve this). Modify the
path $\\{[\overline{g}_{t},f_{t}],t\in[0,1]\\}$ by setting
$f_{t}(v)=h^{-1}_{t}\circ\iota(t)$ over $[t_{1},t_{2}]$. Do this for each
interval $I\subset O_{v}$ and for each $v\in V$. This clearly does not modify
$M(g_{t})$ as a subset of $N(\overline{g}_{t})$. Since for all $t$ now
$f_{t}(V)\subset\partial M(g_{t})$, the obtained path
$\\{[g_{t},f_{t}],t\in[0,1]\\}$ indeed belongs to
$\overline{\mathcal{P}}_{b}(M,V)$. ∎

From the description of the topology on $\overline{\mathcal{P}}_{b}(M,V)$ it
follows that if a sequence $\\{g_{i}\\}$ converges to $g$, then one can choose
the embeddings of $\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to
$\mathbb{H}^{3}$ such that the respective holonomy maps
$\\{\rho_{\overline{g}_{i}}\\}$ converge to $\rho_{\overline{g}}$.

######  Lemma 3.27.

Let a sequence $\\{g_{i}\\}$ converge to $g$ in
$\overline{\mathcal{P}}_{b}(M,V)$. Choose the embeddings of
$\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to $\mathbb{H}^{3}$ as above.
Then $\\{\widetilde{M}(g_{i})\\}$ converges to $\widetilde{M}(g)$ in the
Hausdorff sense.

###### Proof.

Let $\widetilde{V}_{i}$ and $\widetilde{V}$ be the lifts of $V$ in
$\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to $\mathbb{H}^{3}$. From the
description of the topology on $\overline{\mathcal{P}}_{b}(M,V)$ the discrete
sets $\widetilde{V}(g_{i})$ converge to $\widetilde{V}$ in the Hausdorff
sense. Consider the Klein model of $\mathbb{H}^{3}$. Recall that the Hausdorff
convergence of closed subsets of a metric space depends only on the topology
of the space. By Lemma 2.36, $\Lambda(\overline{g}_{i})$ converge to
$\Lambda(\overline{g})$ in the Hausdorff sense. As
$\widetilde{M}(g_{i})\cup\Lambda(\overline{g}_{i})={\rm
conv}(\Lambda(\overline{g}_{i})\cup\widetilde{V}_{i})$ and
$\widetilde{M}(g)\cup\Lambda(\overline{g})={\rm
conv}(\Lambda(\overline{g})\cup\widetilde{V})$, we get that the closed subsets
$\widetilde{M}(g_{i})\cup\Lambda(\overline{g}_{i})$ of the unit ball converge
in the Hausdorff sense to $\widetilde{M}(g)\cup\Lambda(\overline{g})$. It is
easy to check that then the same do their intersections with the interior of
the unit ball, i.e., $\widetilde{M}(g_{i})$ converge in the Hausdorff sense to
$\widetilde{M}(g)$. ∎

It is easy to conclude from this

######  Corollary 3.28.

Let a sequence $\\{g_{i}\\}$ converge to $g$ in
$\overline{\mathcal{P}}_{b}(M,V)$. Choose the embeddings of
$\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to $\mathbb{H}^{3}$ as above.
Then $\\{\partial\widetilde{M}(g_{i})\\}$ converges to
$\partial\widetilde{M}(g)$ in the Hausdorff sense.

We will make use from the following standard fact

######  Lemma 3.29 ([2], Theorem 4).

Let $S_{i}\subset\mathbb{H}^{3}$ be a sequence of complete convex surfaces
converging to a complete convex surface $S\subset\mathbb{H}^{3}$ in the
Hausdorff sense, and $p_{i},q_{i}\in S_{i}$ be two sequences of points
converging to $p,q\in S$. Then the intrinsic distances between $p_{i}$ and
$q_{i}$ on $S_{i}$ converge to the intrinsic distance between $p$ and $q$ on
$S$.

For every $g\in\overline{\mathcal{P}}_{b}(M,V)$ the induced intrinsic metric
on $\partial M(g)$ is a convex hyperbolic cone-metric $d$ with $V(d)\subseteq
V$ defined up to isometries preserving $V$ pointwise and isotopic to identity.
Hence, this defines the realization map

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

(note that in Section 1.2 in order to simplify the exposition we reduced the
domain and the codomain of $\mathcal{I}_{V}$). We can deduce

######  Lemma 3.30.

The map ${\mathcal{I}}_{V}$ is continuous.

###### Proof.

Let $\\{g_{i}\\}\subset\overline{\mathcal{P}}_{b}(M,V)$ be a sequence
converging to $g\in\overline{\mathcal{P}}_{b}(M,V)$. We note that here by
$g_{i}$ and $g$ we mean some concrete metrics on $M$ such that their classes
converge in $\overline{\mathcal{P}}_{b}(M,V)$. Choose the embeddings of
$\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to $\mathbb{H}^{3}$ such that
$\rho_{\overline{g}_{i}}$ converge to $\rho_{\overline{g}}$. Corollary 3.28
implies that $\partial\widetilde{M}(g_{i})$ converges to
$\partial\widetilde{M}(g)$ in the Hausdorff sense. Let $d_{i}$ and $d$ be the
induced intrinsic metrics on $\partial M(g_{i})$ and $\partial M(g)$.

We claim that there exists a finite set $W\subset\partial M$, $W\supseteq V$,
and a geodesic triangulation $\mathcal{T}$ of $(\partial M,W,d)$ such that
each edge is the unique shortest arc between its endpoints. Indeed, first we
claim that there exists a finite set $W^{\prime}\subset\partial M$,
$W^{\prime}\supseteq V$ and a geodesic triangulation $\mathcal{T}^{\prime}$ of
$(\partial M,W^{\prime},d)$ such that each edge is a shortest arc between its
endpoints, but possibly not unique. To this purpose it is enough to start with
any geodesic triangulation of $(\partial M,V,d)$ and do the barycentric
subdivision sufficiently many times. Next, it is easy to see that every proper
sub-arc of a shortest arc is the unique shortest arc between the endpoints.
Thus, if we divide every triangle of $\mathcal{T}^{\prime}$ into six triangles
by drawing all the medians, we will obtain the desired set $W$ and the
triangulation $\mathcal{T}$.

By $\widetilde{W}_{i}$ and $\widetilde{W}$ we denote the full preimages of $W$
in $\partial M(g_{i})$ and $\partial M(g)$ respectively. We make an isotopy of
$g_{i}$ (and respectively of $d_{i}$) fixing $V$ such that for each $w\in W$
the orbit of $w$ under $\rho_{\overline{g}_{i}}$ converge to the orbit under
$\rho_{\overline{g}}$.

We prove that after this the classes of $\\{d_{i}\\}$ converge to the class
$d$ in $\overline{\mathcal{D}}_{c}(\partial M,W)$, which implies also that
their classes converge in $\overline{\mathcal{D}}_{c}(\partial M,V)$ and
proves the lemma. To this purpose we show that for all sufficiently large $i$
the triangulation $\mathcal{T}$ is realized in $d_{i}$ and that the lengths of
the edges converge to those in $d$. Let $e$ be an edge of $\mathcal{T}$ of
length $l_{e}$ in $d$, $\widetilde{e}$ be a lift of $e$ with endpoints
$\widetilde{v},\widetilde{w}\in\widetilde{W}$, let
$\widetilde{v}_{i},\widetilde{w}_{i}\in\widetilde{W}_{i}$ be the respective
points of $\widetilde{W}_{i}$ converging to $\widetilde{v}$, $\widetilde{w}$.
Lemma 3.29 shows that the distance between $\widetilde{v}_{i}$ and
$\widetilde{w}_{i}$ converges to the distance between $\widetilde{v}$ and
$\widetilde{w}$. Connect $\widetilde{v}_{i}$ and $\widetilde{w}_{i}$ by an
intrinsic shortest arc $\widetilde{e}_{i}$ in $\partial\widetilde{M}(g_{i})$,
so the lengths of $\widetilde{e}_{i}$ converge to the length of
$\widetilde{e}$. Parametrize the curves $\widetilde{e}_{i}$ and
$\widetilde{e}$ by the arc length. We claim that $\widetilde{e}_{i}$ converge
to $\widetilde{e}$ as parametrized curves. By [4, Chapter II.1, Theorems 4 and
5] from any subsequence of $\\{\widetilde{e}_{i}\\}$ one can extract a further
subsequence converging to a curve
$\widetilde{e}^{\prime}\subset\partial\widetilde{M}(g)$ of length at most
$l_{e}$. Because $e$ is the unique shortest arc between its endpoints, the
same is true for $\widetilde{e}$, and we get that every arc
$\widetilde{e}^{\prime}$ obtained this way is actually $\widetilde{e}$. In
particular, this argument implies that a subsequence of
$\\{\widetilde{e}_{i}\\}$ converges to $\widetilde{e}$. If, however,
$\\{\widetilde{e}_{i}\\}$ does not converge to $\widetilde{e}$ itself, by the
Hausdorff maximal principle we can choose a maximal subsequence
$\\{\widetilde{e}_{i_{j}}\\}$ of $\\{\widetilde{e}_{i}\\}$ converging to
$\widetilde{e}$. Then the complement to $\\{\widetilde{e}_{i_{j}}\\}$ in
$\\{\widetilde{e}_{i}\\}$ should be infinite, but this means that we can again
extract from the complement a subsequence converging to $\widetilde{e}$, which
contradicts to the maximality of $\\{\widetilde{e}_{i_{j}}\\}$.

Let $e_{i}$ be the projection of $\widetilde{e}_{i}$ to $\partial M$. It
follows from the construction that for all sufficiently large $i$ the arc
$e_{i}$ belongs to the same isotopy class of arcs in $\partial M\backslash W$
as $e$. Doing the described procedure for all edges of $\mathcal{T}$ we obtain
a collection of intrinsic geodesic arcs in $d_{i}$ constituting a
triangulation equivalent to $\mathcal{T}$ for all sufficiently large $i$. The
lengths of the respective arcs converge to the lengths of the arcs of
$\mathcal{T}$. This finishes the proof. ∎

Denote the space of classes of controllably polyhedral metrics by
$\mathcal{P}_{cp}(M,V)$ and the space of classes of strictly polyhedral
metrics by $\mathcal{P}_{sp}(M,V)$. We have

$\mathcal{P}_{sp}(M,V)\subset\mathcal{P}_{cp}(M,V)\subset\mathcal{P}_{p}(M,V)\subset\mathcal{P}_{b}(M,V).$

We also denote by $\overline{\mathcal{P}}_{p}(M,V)$,
$\overline{\mathcal{P}}_{cp}(M,V)$, $\overline{\mathcal{P}}_{sp}(M,V)$ the
respective subsets of metrics of $\overline{\mathcal{P}}_{b}(M,V)$.

######  Definition 3.31.

A hyperbolic metric $g$ on $M$ in
$\mathcal{P}_{cp}(M,V)\backslash\mathcal{P}_{sp}(M,V)$ is called _peculiar_.

######  Lemma 3.32.

$\overline{\mathcal{P}}_{sp}(M,V)$ is open in
$\overline{\mathcal{P}}_{b}(M,V)$.

###### Proof.

One needs to check that the function $g(\partial M(g),\partial
C(\overline{g}))=d_{\mathbb{H}^{3}}(\partial\widetilde{M}(g),\partial\widetilde{C}(\overline{g}))$
is continuous at zero over $\overline{\mathcal{P}}_{b}(M,V)$. Consider
$g\in\mathcal{\overline{}}{\mathcal{P}}_{b}(M,V)$ such that

$g(\partial M(g),\partial C(\overline{g}))=0.$

Hence, there is a common point
$p\in\partial\widetilde{M}(g)\cap\partial\widetilde{C}(\overline{g})$. Let
$\\{g_{i}\\}$ be a sequence converging to $g$ in
$\overline{\mathcal{P}}_{b}(M,V)$. Due to Corollary 3.28, we can choose the
embeddings of $\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to
$\mathbb{H}^{3}$ such that $\\{\partial\widetilde{M}(g_{i})\\}$ converges to
$\partial\widetilde{M}(g)$ in the Hausdorff sense. Corollary 3.28 also implies
$\partial\widetilde{C}(\overline{g}_{i})$ converges to
$\partial\widetilde{C}(\overline{g})$ in the Hausdorff sense. Hence, there
exists a sequence of points $\\{p_{i}\in\partial\widetilde{M}(g_{i})\\}$ and
$\\{p^{\prime}_{i}\in\partial\widetilde{C}(\overline{g}_{i})\\}$ converging to
$p$. Thus,

$g_{i}(\partial M(g_{i}),\partial C(\overline{g}_{i}))\rightarrow 0.$

∎

######  Lemma 3.33.

$\overline{\mathcal{P}}_{cp}(M,V)$ is open in
$\overline{\mathcal{P}}_{b}(M,V)$.

###### Proof.

Suppose the converse that there exists $g\in\overline{\mathcal{P}}_{cp}(M,V)$
and a sequence

$\\{g_{i}\\}\subset(\overline{\mathcal{P}}_{b}(M,V)\backslash\overline{\mathcal{P}}_{cp}(M,V))$

converging to $g$ in $\overline{\mathcal{P}}_{b}(M,V)$. By $d_{i}$ and $d$ we
denote the induced intrinsic metrics on $\partial M(g_{i})$ and $\partial
M(g)$. As ${\mathcal{I}}_{V}$ is continuous, the metrics $d_{i}$ converge to
$d$ in the weak Lipschitz sense. Then Corollary 2.21 implies that the systoles
of $d_{i}$ are uniformly bounded from below by some constant $\Delta>0$.

As $g_{i}\notin\overline{\mathcal{P}}_{cp}(M,V)$, Corollary 3.17 implies that
$\partial M(g_{i})\cap\partial C(\overline{g}_{i})$ contains at least one
component, which is a hyperbolic surface with geodesic boundary. We can pick a
hyperbolic pair of pants $P_{i}$ in this component. The lengths of the
boundaries are bounded from below by $\Delta$. Hence, Corollary 2.8 implies
that there exists $\delta>0$ and a point $p_{i}$ in $P_{i}$ such that the
intrinsic distance from $p_{i}$ to $\partial P_{i}$ is at least $\delta$. We
use this to choose an intrinsic geodesic triangle $T_{i}\subset P_{i}$
containing $p_{i}$ with the areas of all $T_{i}$ uniformly bounded from below
and the diameters uniformly bounded from above.

Consider the embeddings of $\widetilde{M}(g_{i})$ and $\widetilde{M}(g)$ to
$\mathbb{H}^{3}$ as in Corollary 3.28, so $\\{\partial\widetilde{M}(g_{i})\\}$
converges to $\partial\widetilde{M}(g)$ in the Hausdorff sense and
$\\{\partial\widetilde{C}(\overline{g}_{i})\\}$ converges to
$\partial\widetilde{C}(\overline{g})$ in the Hausdorff sense. We can choose
lifts $\widetilde{p}_{i}\in\partial\widetilde{M}(g_{i})$ of $p_{i}$ such that
the sequence $\\{\widetilde{p}_{i}\\}$ is bounded in $\mathbb{H}^{3}$. Hence,
up to passing to a subsequence we can assume that $\widetilde{p}_{i}$ converge
to a point $\widetilde{p}\in\partial\widetilde{M}(g)$. Now let
$\widetilde{T}_{i}\subset(\partial\widetilde{M}(g_{i})\cap\partial\widetilde{C}(\overline{g}_{i}))$
be the lift of $T_{i}$ containing $\widetilde{p}_{i}$. As $\widetilde{T}_{i}$
are compact subsets of $\mathbb{H}^{3}$ staying in a bounded domain, up to
taking a subsequence they must converge in the Hausdorff sense to a compact
connected subset $\widetilde{T}$ of
$\partial\widetilde{M}(g)\cap\partial\widetilde{C}(\overline{g})$. The latter
consists only of disjoint geodesic lines, hence $\widetilde{T}$ is a compact
geodesic segment (possibly a point). However, Alexandrov showed that because
$\widetilde{T}_{i}$ and $\widetilde{T}$ belong to complete convex surfaces
$\partial\widetilde{M}(g_{i})$ and $\partial\widetilde{M}(g)$, the areas of
$\widetilde{T}_{i}$ must converge to the area of $\widetilde{T}$, which is
zero. In the Euclidean case this is [4, Chapter X.2, Theorem 2], the changes
for the hyperbolic case are discussed in [4, Chapter XII.1]. This contradicts
to the uniform lower bound on the areas of $\widetilde{T}_{i}$. ∎

Note that Lemmas 3.32 and 3.33 particularly imply that $\mathcal{P}_{sp}(M,V)$
and $\mathcal{P}_{cp}(M,V)$ are open in $\mathcal{P}_{b}(M,V)$.

###  3.4 Differentiability of the realization map

Our current aim is to study the differentiability properties of the
realization map $\mathcal{I}_{V}$. In full generality this is a complicated
task. We will restrict ourselves to controllably polyhedral metrics, and will
prove that $\mathcal{I}_{V}$ is $C^{1}$ at strictly polyhedral metrics and
differentiable at peculiar metrics.

####  3.4.1 Differentiability for strictly polyhedral metrics

######  Lemma 3.34.

The realization map $\mathcal{I}_{V}$ is $C^{1}$ over $\mathcal{P}_{sp}(M,V)$.

###### Proof.

Pick $g\in\mathcal{P}_{sp}(M,V)$ and its neighborhood $U$ in
$\mathcal{P}_{sp}(M,V)$. Lift $g$ to
$(\rho,\widetilde{f})\in\mathcal{R}(\pi_{1}(M),G)\times(\mathbb{H}^{3})^{*V}$.
Provided $U$ is small enough, there exists a neighborhood $\widetilde{U}$ of
$(\rho,\widetilde{f})$ projecting onto $U$ such that the action of
$\pi_{1}(M)$ sends $\widetilde{U}$ to disjoint domains. The action of $G$ on
$\widetilde{U}$ defines a smooth submersion $\widetilde{U}\rightarrow U$. We
will show that $\mathcal{I}_{V}$ is $C^{1}$ as a function over
$\widetilde{U}$.

If there is an edge of $\partial\widetilde{M}(g)$ between
$\widetilde{v},\widetilde{w}\in\widetilde{V}(g)$, one can see that they are
connected by an edge for all $(\rho^{\prime},\widetilde{f}^{\prime})$ in
$\widetilde{U}$ provided that $\widetilde{U}$ is sufficiently small. Indeed,
the key is that the distance from any point of the edge to
$\widetilde{C}(\overline{g})$ is positive and that the limit sets of
converging representations converge in the Hausdorff sense. If there exists a
sequence $(\rho_{i},\widetilde{f}_{i})$ converging to $(\rho,\widetilde{f})$
such that $\widetilde{v}$ and $\widetilde{w}$ are not connected by an edge in
all $\partial\widetilde{M}(g_{i})$, then there exists a sequence of 4-tuples
of points $(p^{1}_{i},p^{2}_{i},p^{3}_{i},p^{4}_{i})$, where all $p^{j}_{i}$
are from
$(\Lambda(\rho)\cup\widetilde{V}(g))\backslash\\{\widetilde{v},\widetilde{w}\\}$,
such that the distance from the convex hull of each 4-tuple to the midpoint of
the segment between $\widetilde{v}$ and $\widetilde{w}$ tends to zero. By
passing to a subsequence we assume that each $p^{j}_{i}$ converge to
$p^{j}\in\Lambda(\rho)$, which means that the distance from the midpoint to
$\widetilde{C}(\overline{g})$ is zero, which is a contradiction. Note that the
length of such an edge is a smooth function in $\widetilde{U}$.

As $g\in\mathcal{P}_{sp}(M,V)$, all edges of $\partial\widetilde{M}(g)$ are
between pairs of vertices and all faces are convex compact hyperbolic
polygons. In particular, the face decomposition of all
$(\rho^{\prime},\widetilde{f}^{\prime})$ in a sufficiently small neighborhood
of $(\rho,\widetilde{f})$ is a subdivision of the face decomposition of
$\partial\widetilde{M}(g)$. Let $d$ be the induced intrinsic metric on
$\partial M(g)$. Consider a geodesic triangulation $\mathcal{T}$ of $\partial
M(g)$ subdividing the face decomposition. As $\mathcal{T}$ is realized in
$(d,V)$, we have $d\in\mathcal{D}(V,\mathcal{T})$. The lengths of all edges of
$\mathcal{T}$ that are edges of $g$ are smooth functions around $g$. We need
to show that the lengths of the other edges of $\mathcal{T}$ are
$C^{1}$-functions.

Consider a non-triangular face $Q$ of $\partial\widetilde{M}(g)$, let $V_{Q}$
be the set of its vertices and $e$ be a geodesic segment in the relative
interior of $Q$ corresponding to an edge of $\mathcal{T}$ that is not an edge
of $g$. When we move the vertices of $V_{Q}$, ${\rm conv}(V_{Q})$ may become a
convex polyhedron. If we move them sufficiently close to the initial
positions, all boundary edges of $Q$ remain among the edges of ${\rm
conv}(V_{Q})$. They divide $\partial{\rm conv}(V_{Q})$ into two parts. One of
them, determined with the help of the orientation of $Q$ with respect to
$\widetilde{M}(g)$, is considered as a deformation of $Q$. We call this part
_upper_. We also require that we perturb the vertices slightly enough so that
the intrinsic angles of all the vertices in the upper part stay less than
$\pi$. The segment $e$ may become a broken line (i.e., after we move the
vertices, we identify $e$ with the intrinsic shortest arc in the upper part
between the vertices of $Q$). Consider a small neighborhood
$U_{Q}\subset(\mathbb{H}^{3})^{V_{Q}}$ of the initial position of $V_{Q}$ for
which the boundary edges of $Q$ remain to be in the boundary of ${\rm
conv}(V_{Q})$ and for which the intrinsic angles of all the vertices in the
upper part stay less than $\pi$. We can consider the length $l_{e}$ as a
continuous function over $U_{Q}$. To finish the proof of the lemma, it is
enough to show

######  Claim 3.34.1.

The function $l_{e}$ is $C^{1}$ over $U_{Q}$ and its differential in the
initial position coincides with the differential of the
$\mathbb{H}^{3}$-distance between the endpoints of $e$.

Let $\mathcal{T}_{1},\ldots,\mathcal{T}_{r}$ be all triangulations of $Q$.
Pick one of them, $\mathcal{T}_{i}$, and now, as we move the vertices,
consider the deformations of $Q$ as of a surface triangulated with
$\mathcal{T}_{i}$ (i.e., the edges of $\mathcal{T}_{i}$ remain geodesic
segments of $\mathbb{H}^{3}$). It may become a non-convex surface, but near
the initial position the edge $e$ is still defined in the obtained surface as
the intrinsic shortest arc between its endpoints. Its length $l_{i,e}$ is a
smooth function because the edge lengths of $\mathcal{T}_{i}$ change smoothly
and the length of $e$ is a smooth function of them. We assume that $U_{Q}$ is
small enough so that for all such polyhedral surfaces the intrinsic angles of
all vertices stay less than $\pi$. Then $e$ is defined in all these surfaces
over the whole $U_{Q}$. Let $U_{i}$ be the subset of $U_{Q}$ corresponding to
the configurations where $\mathcal{T}_{i}$ subdivides the face decomposition
of the upper part of ${\rm conv}(V_{Q})$ is triangulated with
$\mathcal{T}_{i}$. One can see that this produces a decomposition of $U_{Q}$
into cells $U_{1},\ldots,U_{r}$ with non-empty disjoint interiors and
piecewise-analytic boundaries. The function $l_{e}$ is equal to $l_{i,e}$ over
$U_{i}$. In particular, it is smooth in the interiors of $U_{i}$. We need to
check that if $U_{i}$, $U_{j}$ have a common boundary, then the first
derivatives of $l_{i,e}$, $l_{j,e}$ coincide there.

Let $\Pi$ be the plane containing $Q$, we orient it arbitrarily. Set a local
coordinate system in $U_{Q}$: for each $v\in V_{Q}$ choose two coordinates
that are coordinates of the projections of $v$ to $\Pi$ and the third
coordinate is the oriented distance from $v$ to $\Pi$. Call the first two
coordinates _horizontal_ and the third _vertical_. Note that in the initial
moment all the horizontal derivatives of all $l_{i,e}$ coincide and are equal
to the respective derivatives of the distance between the endpoints of $e$.

We now claim that all the vertical derivatives are zero. Indeed, let $ABC$ be
a hyperbolic triangle with the corresponding sides equal to $a,b,c$ and with
the right angle at $A$. Then we have

$\cosh(a)=\cosh(b)\cosh(c).$

Hence, for fixed $b>0$ as we move the vertex $B$ along the line $AB$ we have
$\partial a/\partial c=0$ at $c=0$. This means that when we consider the
bending of the polyhedral surface triangulated along $\mathcal{T}_{i}$, the
vertical derivatives of all the edge lengths of $\mathcal{T}_{i}$ are zero at
the initial moment. Hence, so are the vertical derivatives of $l_{i,e}$. This
implies that all first derivatives of all $l_{i,e}$ coincide at the initial
moment. The same argument implies that all first derivatives of any two
$l_{i,e}$ and $l_{j,e}$ coincide at any point of the common boundary of
$U_{i}$ and $U_{j}$. This shows that $l_{e}$ is $C^{1}$ in $U_{Q}$. In turn
this finishes the proof. ∎

####  3.4.2 Deformations of hyperbolic strips

In order to investigate the differentiability of the realization map for
controllably polyhedral metrics, we need first to restrict ourselves to the
case of closed solid hyperbolic tori with convex polyhedral boundary.

Only in this subsection by $M$ we denote the closed solid torus and by $N$ we
denote its interior. It is easy to see that the space $\mathcal{CH}(N)$ of
convex cocompact hyperbolic metrics on $N$ up to isotopy is in one-to-one
correspondence with the set of loxodromic isometries of $\mathbb{H}^{3}$ up to
conjugation. Any such isometry $\gamma$ is determined up to conjugacy by two
parameters: the shift $a>0$ along the geodesic line
$\widetilde{\psi}\subset\mathbb{H}^{3}$ fixed by $\gamma$ and the rotation
angle $\alpha\in\mathbb{R}/2\pi\mathbb{Z}$ around $\widetilde{\psi}$, measured
clockwise to the shift direction. We make the convention that
$\widetilde{\psi}$ is oriented in the shift direction. Let $\Gamma<{\rm
Iso}^{+}(\mathbb{H}^{3})$ be the cyclic subgroup generated by $\gamma$. The
quotient hyperbolic manifold $\mathbb{H}^{3}/\Gamma$ is convex cocompact and
homeomorphic to $N$. Its convex core is the closed geodesic $\psi$ that is the
image of $\widetilde{\psi}$. Note that for every
$\overline{g}\in\mathcal{CH}(N)$ the hyperbolic manifold $N(\overline{g})$ has
a connected two-dimensional isometry group. Hence, for $V\neq\emptyset$ the
space $\mathcal{CH}(N,V)$ has dimension $3n$, and so is the dimension of
$\mathcal{P}_{b}(M,V)$. It is evident that every bent hyperbolic metric on $M$
is controllably polyhedral, so
$\mathcal{P}_{b}(M,V)=\mathcal{P}_{p}(M,V)=\mathcal{P}_{cp}(M,V)$.

Let $g_{0}\in\mathcal{P}_{cp}(M,V)$. We are interested in the case when
$g_{0}$ is peculiar. Clearly, this is equivalent to the situation when all
points of $\widetilde{V}(g_{0})\subset\mathbb{H}^{3}$ belong to a closed half-
space containing $\widetilde{\psi}$ in its boundary plane. Particularly,
$\alpha_{0}=0$, where $\alpha_{0}$ is the rotation parameter for $g_{0}$.
Also, if $\alpha_{0}=0$ and $\overline{\rm conv}(V)$ is homeomorphic to $M$,
one can see that $n=|V|\geq 2$. By $\Sigma$ we denote the union of faces whose
closures contain $\psi$ (there are either two or one of them), and by
$\widetilde{\Sigma}\subset\partial\widetilde{M}(g)$ denote its universal
cover. In this section we study deformations of the induced metric on $\Sigma$
when we vary $g_{0}$, so we expel from $V$ all vertices except the vertices of
$\Sigma$. Of special interest are the deformations when $\alpha$ becomes
nonzero. Our aim is to show that the induced metric is differentiable at
$g_{0}$ and that the partial derivative of the metric by $\alpha$ is zero. Fix
an intrinsically geodesic triangulation $\mathcal{T}$ of $\Sigma$. There are
two types of edges in $\mathcal{T}$: some connect vertices from different
components of $\partial\Sigma$ and some connect vertices from the same
component. In order to consider less cases we assume that all interior edges
of $\Sigma$ are of the first type. It is easy to see that such $\mathcal{T}$
exists. Let $e$ be an interior edge of $\mathcal{T}$.

At the initial moment all vertices of $\widetilde{\Sigma}$ belong to two half-
planes intersecting at $\widetilde{\psi}$, which we denote by $\Pi^{1}$ and
$\Pi^{2}$. We denote the vertices with the lifts belonging to $\Pi^{1}$ by
$v^{1}_{1},\ldots,v^{1}_{n_{1}}$ and to $\Pi^{2}$ by
$v^{2}_{1},\ldots,v^{2}_{n_{2}},$ where $n_{1}+n_{2}=n$. For each $v^{j}_{i}$,
where $j=1,2$, we fix a lift
$\widetilde{v}^{j}_{i}\in\partial\widetilde{\Sigma}$. By $u^{j}_{i}$ we denote
the projection of $\widetilde{v}^{j}_{i}$ to $\widetilde{\psi}$. By
$x^{j}_{i,0}$ we denote the distance from $\widetilde{v}^{j}_{i}$ to
$\widetilde{\psi}$. By $y^{j}_{i,0}$ we denote the oriented distance from
$u^{1}_{1}$ to $u^{j}_{i}$. We assume that $v^{1}_{1}$ and $v^{2}_{1}$ are the
endpoints of $e$ and consider its lift $\widetilde{e}$, which is the unique
intrinsic shortest arc in $\widetilde{\Sigma}$ between $\widetilde{v}^{1}_{1}$
and $\widetilde{v}^{2}_{1}$. By possibly changing the labels, we assume that
$\widetilde{\psi}$ is oriented in the direction from $u^{1}_{1}$ to
$u^{2}_{1}$. By $\phi_{0}\geq\pi$ we denote the angle from $\Pi^{1}$ to
$\Pi^{2}$ clockwise.

Note that when we vary $g_{0}$ so that $\alpha$ becomes nonzero, the strip
$\Sigma$ becomes triangulated. There are infinitely many triangulations of
$\Sigma$ realized by metrics in any neighborhood of $g_{0}$. This is the
reason, why we can not really apply the methods of Section 3.4.1 to show the
differentiability of the boundary metric, and have to look for another way.

Take $g\in\mathcal{P}_{cp}(M,V)$ sufficiently close to $g_{0}$ so that all
edges of $g_{0}$ except possibly $\psi$ remain to be edges of $g$. Then the
boundary edges of $\Sigma$ divide $\partial M(g)$ into two strips. One of them
is a deformation of $\Sigma$ and we will continue to denote it $\Sigma$. We
assume that $g$ is sufficiently close to $g_{0}$ so that the intrinsic angles
of the vertices in $\Sigma$ remain to be less than $\pi$. We choose the same
lifts of all $v^{j}_{i}$, so $u^{j}_{i}$ remain defined also for $g$, and we
define $x^{j}_{i}$ and $y^{j}_{i}$ in a similar fashion to $x^{j}_{i,0}$ and
$y^{j}_{i,0}$. Hence, $x^{j}_{i}$ and $y^{j}_{i}$ are smooth functions in
$\mathcal{P}_{cp}(M,V)$. By $\Pi^{j}_{i}$ we denote the half-plane spanned by
$\widetilde{\psi}$ and $\widetilde{v}^{j}_{i}$. By $\phi^{j}_{i}$ we denote
the angle between the segments $u^{j}_{i}\widetilde{v}^{j}_{i}$ and the half-
plane $\Pi^{1}_{1}$. Since for $g_{0}$ we have $\phi^{1}_{i,0}=0$,
$\phi^{2}_{i,0}=\phi_{0}$, we consider $\phi^{1}_{i}$ as a real number valuing
in a neighborhood of 0, and $\phi^{2}_{i}$ as a real number valuing in a
neighborhood of $\phi_{0}$. One can see that the $3n$ variables

$x^{1}_{1},\ldots,x^{1}_{n_{1}},x^{2}_{1},\ldots,x^{2}_{n_{2}},y^{1}_{2},\ldots,y^{1}_{n_{1}},y^{2}_{1},\ldots,y^{2}_{n_{2}},\phi^{1}_{2},\ldots,\phi^{1}_{n_{1}},\phi^{2}_{1},\ldots,\phi^{2}_{n_{2}},a,\alpha$

provide local coordinates in a neighborhood of $g_{0}$ in
$\mathcal{P}_{cp}(M,V)$. Particularly we use these coordinates to consider a
neighborhood of $g_{0}$ as a domain in a $(3n)$-dimensional real vector space,
and by $||.||$ we denote the standard Euclidean norm there.

Since now and until the end of this section we are going to work only in the
universal cover embedded in $\mathbb{H}^{3}$. To simplify the reading, until
the end of this section we drop tilde in $\widetilde{M}(g)$,
$\widetilde{\psi}$, $\widetilde{\Sigma}$, $\widetilde{v}^{j}_{i}$, writing
just $M(g)$, $\psi$, $\Sigma$, $v^{j}_{i}$ instead.

We will continue to denote the intrinsic shortest arc in $\Sigma$ with the
metric induced by $g$ between $v^{1}_{1}$ and $v^{2}_{1}$ by $e$. By $l_{e}$
we denote its length, so we consider $l_{e}$ as a function in a neighborhood
of $g_{0}$, particularly a function of our $3n$ variables. The results of
Section 3.4.1 apply also to the situation we have now provided that
$\alpha\neq 0$. Hence, $l_{e}$ is $C^{1}$ when $\alpha\neq 0$. By $e^{\prime}$
we denote the intrinsic shortest arc between $v^{1}_{1}$ and $v^{2}_{1}$ in
$\Pi^{1}_{1}\cup\Pi^{2}_{1}$. By $l^{\prime}_{e}$ we denote the length of
$e^{\prime}$. The function $l^{\prime}_{e}$ is a smooth function of three
variables: $x^{1}_{1}$, $x^{2}_{1}$, $y^{2}_{1}$. Our aim is to show

######  Lemma 3.35.

For every $0<\mu<1$ the difference $l_{e}-l^{\prime}_{e}$ is $O(||\Delta
g||^{1+\mu})$ at $g_{0}$.

Here $\Delta g=g-g_{0}$ is an abstract vector in our coordinates.

######  Corollary 3.36.

The function $l_{e}$ is differentiable at $g_{0}$. Moreover, its differential
coincides with the differential of the function $l^{\prime}_{e}$. In
particular, its derivatives with respect to $\phi^{j}_{i}$ and to $\alpha$ are
zero.

We have to consider separately the cases $\phi_{0}>\pi$ and $\phi_{0}=\pi$. In
the case $\phi_{0}>\pi$ we assume that the deformation is small enough so that
all differences $\phi^{2}_{i_{2}}-\phi^{1}_{i_{1}}$ are greater than $\pi$.
Then the realization of $e$ in $\partial M(g)$ belongs to the concave part of
$\mathbb{H}^{3}\backslash(\Pi^{1}_{1}\cup\Pi^{2}_{1})$, so by the
Busemann–Feller lemma we have $l^{\prime}_{e}\leq l_{e}$ and the non-trivial
part of Lemma 3.35 is to provide an upper bound on $l_{e}-l^{\prime}_{e}$. The
case $\phi=\pi$ is slightly more involved, but the upper bound still remains
the main issue of the proof.

Let $p\in\partial M(g)$ and $q$ be its projection to $\psi$. By
$\phi_{p}\in\mathbb{R}/2\pi\mathbb{Z}$ we denote the angle between the segment
$pq$ and $\Pi^{1}_{1}$ clockwise. By $t_{p}$ we denote the signed distance
$u^{1}_{1}q$. This provides a parametrization of $\partial M(g)$. However, we
are more interested in the coordinate $\phi_{p}$ extended to the universal
cover of $\partial M(g)$ (recall that $\partial M(g)$ is topologically a
cylinder). That is, we consider the universal cover of $\partial M$, fix a
lift $\widehat{v}^{1}_{1}$ of $v^{1}_{1}$ and for $p\in\partial M(g)$ by
$\theta_{p}\in\mathbb{R}$ we denote the extended oriented angle from
$\widehat{v}^{1}_{1}$ to the lift $\widehat{p}$ of $p$ such that $\widehat{p}$
belongs to the same lift of $\Sigma$ or of $\partial M(g)\backslash\Sigma$ as
$\widehat{v}^{1}_{1}$.

Fix $\mu\in\mathbb{R}$ such that $0<\mu<1$ and suppose that $\phi_{0}>\pi$.
Set $\mu^{\prime}=(1+\mu)/2.$ Consider in $\Sigma$ the curve
$e^{\prime\prime}$ consisting of the following three arcs. Denote the
intersection point of $e^{\prime}$ with $\psi$ by $u$. Let $p_{1}\in\Sigma$ be
a point such that
(1) the plane spanned by $v^{1}_{1}$, $u$ and $p_{1}$ is orthogonal to
$\Pi^{1}_{1}$;
(2) $\theta_{p_{1}}=\max_{1\leq i\leq
n_{1}}\phi^{1}_{i}+|\alpha|^{\mu^{\prime}}$
Similarly, let $p_{2}$ be the point such that
(1) the plane spanned by $v^{2}_{1}$, $u$ and $p_{2}$ is orthogonal to
$\Pi^{2}_{1}$;
(2) $\theta_{p_{2}}=\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu^{\prime}}$.
Now let $e^{\prime\prime}_{1}$ be the curve connecting $v^{1}_{1}$ with
$p_{1}$ in the intersection of $\partial M(g)$ with the plane
$v^{1}_{1}up_{1}$ in the clockwise direction. Similarly, let
$e^{\prime\prime}_{2}$ be the curve connecting $v^{2}_{1}$ with $p_{2}$ in the
intersection of $\partial M(g)$ with the plane $v^{2}_{1}up_{2}$ in the
counterclockwise direction. Finally, let $e^{\prime\prime}_{3}$ be the curve
connecting $p_{1}$ with $p_{2}$ in the intersection of $\partial M(g)$ with
the plane spanned by $u,p_{1},p_{2}$ in the clockwise direction. Altogether
the arcs $e^{\prime\prime}_{1}$, $e^{\prime\prime}_{3}$,
$e^{\prime\prime}_{2}$ form a curve in $\Sigma$ connecting $v^{1}_{1}$ with
$v^{2}_{1}$. By $l^{\prime\prime}_{e}$ we denote its length. We have
$l^{\prime\prime}_{e}\geq l_{e}\geq l^{\prime}_{e}$. Thus, Lemma 3.35 for
$\phi_{0}>\pi$ follows from

######  Lemma 3.37.

Let $\phi_{0}>\pi$. For every $0<\mu<1$ the difference
$l^{\prime\prime}_{e}-l^{\prime}_{e}$ is $O(||\Delta g||^{1+\mu})$ at $g_{0}$.

Now consider the case $\phi_{0}=\pi$. In this case we propose a different
construction of the auxiliary curve $e^{\prime\prime}$. Emit a geodesic ray
from $u$ such that it is orthogonal to $\psi$, directed towards $\Sigma$, and
the plane spanned by this ray and $\psi$ bisects the dihedral angle between
$\Pi^{1}_{1}$ and $\Pi^{2}_{1}$. Let $p_{0}$ be the intersection point of this
ray with $\Sigma$. Let $e^{\prime\prime}_{1}$ be the curve connecting
$v^{1}_{1}$ with $p_{0}$ in the intersection of $\partial M(g)$ with the plane
$v^{1}_{1}up_{0}$ in the clockwise direction, and $e^{\prime\prime}_{2}$ be
the curve connecting $p_{0}$ with $v^{2}_{1}$ in the intersection of $\partial
M(g)$ with the plane $v^{2}_{1}up_{0}$. Altogether the arcs
$e^{\prime\prime}_{1}$ and $e^{\prime\prime}_{2}$ form a curve in $\Sigma$
connecting $v^{1}_{1}$ with $v^{2}_{1}$, which we again denote by
$e^{\prime\prime}$. By $l^{\prime\prime}_{e}$ we denote its length. We have
$l^{\prime\prime}_{e}\geq l_{e}$. We will similarly show

######  Lemma 3.38.

Let $\phi_{0}=\pi$ and $\phi^{2}_{1}\geq\pi$. For every $0<\mu<1$ the
difference $l^{\prime\prime}_{e}-l^{\prime}_{e}$ is $O(||\Delta g||^{1+\mu})$
at $g_{0}$.

In the case $\phi^{2}_{1}\geq\pi$, by the Busemann–Feller lemma we have
$l_{e}\geq l^{\prime}_{e}$ and Lemma 3.35 follows from Lemma 3.38. When
$\phi^{2}_{1}<\pi$, a simple additional argument is required, which we provide
below.

Denote the hyperbolic metric of $\mathbb{H}^{3}$ by $d_{\mathbb{H}^{3}}$. To
prove our claims we need to make the following estimates on the geometry of
$\Sigma$.

######  Lemma 3.39.

Let $\phi_{0}>\pi$, $\alpha\neq 0$. For every $0<\mu<1$ and every $T>0$ we
have

$\sup\\{d_{\mathbb{H}^{3}}(p,\psi):p\in\Sigma,-T\leq t_{p}\leq T,\max_{1\leq
i\leq n_{1}}\phi^{1}_{i}+|\alpha|^{\mu}\leq\theta_{p}\leq\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu}\\}=O(||\Delta g||^{2}).$

######  Lemma 3.40.

Let $\alpha\neq 0$. For every $0<\mu<1$ and every $T>0$ we have

$\sup\\{d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1}):p\in\Sigma,-T\leq t_{p}\leq
T,0\leq\theta_{p}\leq\frac{\pi}{2}\\}=O(||\Delta g||^{\mu}),$
$\sup\\{d_{\mathbb{H}^{3}}(p,\Pi^{2}_{1}):p\in\Sigma,-T\leq t_{p}\leq
T,\phi^{2}_{1}-\frac{\pi}{2}\leq\theta_{p}\leq\phi^{2}_{1}\\}=O(||\Delta
g||^{\mu}).$

We postpone proofs of Lemmas 3.39–3.40 and first deduce Lemmas 3.37 and 3.38
from them. To this purpose we also need few less involved estimates

######  Lemma 3.41.

Fix $0<\mu\leq 1$ and let $\xi>0$ be a parameter. Consider the points
$p,p^{\prime}\in\mathbb{H}^{2}$ connected by a geodesic segment $\psi$. Let
$\chi$ and $\chi^{\prime}$ be geodesic rays from $p$ and $p^{\prime}$ at the
same side of $\psi$, and $\sigma$ be a convex arc starting at $p$ tangent to
$\chi$ and ending at a point $p_{0}\in\chi^{\prime}$. By $r_{0}$ denote the
foot of the perpendicular from $p_{0}$ to $\psi$. Assume that
(1) the angle between $\psi$ and $\chi$ is $O(\xi^{\mu})$;
(2) the angle between $\psi$ and $\chi^{\prime}$ is $O(\xi^{\mu})$;
(3) the length of $\psi$ is $O(1)$.
Here all asymptotics are as $\xi$ tends to zero. Then

${\rm len}(\sigma)-pr_{0}=O(\xi^{2\mu}).$

Here and in what follows we will abuse the notation and sometimes write
$pr_{0}$ for the length of the geodesic segment $pr_{0}$.

###### Proof.

We have ${\rm len}(\sigma)-pr_{0}\geq 0$. Clearly, the rays $\chi$ and
$\chi^{\prime}$ intersect provided that $\xi$ is sufficiently small, denote
the intersection point by $p^{\prime\prime}$. Let $\sigma^{\prime\prime}$ be
the broken line consisting of the segments $pp^{\prime\prime}$ and
$p^{\prime\prime}p_{0}$. By the Busemann–Feller lemma we have ${\rm
len}(\sigma^{\prime\prime})\geq{\rm len}(\sigma)$. Thus, it is enough to show
that ${\rm len}(\sigma^{\prime\prime})-pr_{0}=O(\xi^{2\mu}).$

Let $r$ be the foot of the perpendicular from $p^{\prime\prime}$ to $\psi$. We
have

$\tanh(pr)=\tanh(pp^{\prime\prime})\cos(\angle p^{\prime\prime}pr).$

As $\xi\rightarrow 0$ we get from (1)

$1-\cos(\angle p^{\prime\prime}pr)=O(\xi^{2\mu}).$

Also clearly $pp^{\prime\prime}=O(1)$, $pr=O(1)$. Thus, we can estimate

$0\leq
pp^{\prime\prime}-pr\leq\sinh(pp^{\prime\prime}-pr)=\cosh(pp^{\prime\prime})\cosh(pr)(\tanh(pp^{\prime\prime})-\tanh(pr))=\\\
=O(1)\tanh(pp^{\prime\prime})(1-\cos(\angle p^{\prime\prime}pr))=O(\xi^{2}).$
(3.1)

Now for the trapezoid $p^{\prime\prime}p_{0}r_{0}r$ we have from Lemma 2.3

$\frac{\sinh(p^{\prime\prime}p_{0})}{\sinh(rr_{0})}=\frac{\cosh(p^{\prime\prime}r)}{\sin({\angle}p^{\prime\prime}p_{0}r_{0})}.$
(3.2)

From

$\tan({\angle}p_{0}p^{\prime}r_{0})\cosh(p_{0}p^{\prime})=\cot({\angle}p^{\prime}p_{0}r_{0})=\tan\left({\angle}p^{\prime\prime}p_{0}r_{0}-\frac{\pi}{2}\right)$

we get $0\leq{\angle}p^{\prime\prime}p_{0}r_{0}-\frac{\pi}{2}=O(\xi^{\mu}).$
Note that

$\cosh(p^{\prime\prime}r)-1=O(\xi^{2\mu}),~{}~{}~{}~{}~{}1-\cos(\angle
p_{0}p^{\prime}r_{0})=O(\xi^{2\mu}).$

Using this and (3.2) we obtain

$0\leq p^{\prime\prime}p_{0}-rr_{0}\leq
2\sinh\left(\frac{p^{\prime\prime}p_{0}-rr_{0}}{2}\right)\leq
2\sinh\left(\frac{p^{\prime\prime}p_{0}-rr_{0}}{2}\right)\cosh\left(\frac{p^{\prime\prime}p_{0}+rr_{0}}{2}\right)=\\\
=\sinh(p^{\prime\prime}p_{0})-\sinh(rr_{0})=\sinh(p^{\prime\prime}p_{0})\left(1-\cos\left({\angle}p^{\prime\prime}p_{0}r_{0}-\frac{\pi}{2}\right)\right)+\\\
+\sinh(rr_{0})(\cosh(p^{\prime\prime}r)-1)=O(\xi^{2\mu}).$ (3.3)

Altogether (3.1) and (3.3) give us

$0\leq{\rm
len}(\sigma^{\prime\prime})-pr_{0}=pp^{\prime\prime}+p^{\prime}p^{\prime\prime}-pr-
rr_{0}=O(\xi^{2\mu}).$

∎

######  Lemma 3.42.

Fix $0<\mu<1$ and let $\xi>0$ be a parameter. Let $\psi$ be a geodesic line in
$\mathbb{H}^{3}$, $C$ be the surface consisting of all points at distance $x$
from $\psi$ where $x=O(\xi^{2})$ as $\xi$ tends to zero. Consider two points
$p,p^{\prime}\in C$, let $q,q^{\prime}$ be their projections to $\psi$ and
$\sigma\subset C$ be an intrinsic shortest arc connecting them. Assume that
$qq^{\prime}=O(1)$. Then

$0\leq{\rm len}(\sigma)-qq^{\prime}=O(\xi^{2}).$

###### Proof.

Let $p^{\prime\prime}$ be the point at the intersection of $C$, the plane
spanned by $p$ and $\psi$ and of the plane orthogonal to $\psi$ at
$q^{\prime}$. Let $\sigma^{\prime\prime}\subset C$ be the intrinsic shortest
arc connecting $p$ to $p^{\prime\prime}$ and $\sigma^{\prime}\subset C$ be the
intrinsic shortest arc connecting $p^{\prime}$ to $p^{\prime\prime}$. We then
have ${\rm len}(\sigma^{\prime\prime})=qq^{\prime}\cosh(x)$. Thus, $0\leq{\rm
len}(\sigma^{\prime\prime})-qq^{\prime}=O(\xi^{2})$. Also $\sigma^{\prime}$ is
an arc of a circle centered at $q^{\prime}$ with radius $x$, hence ${\rm
len}(\sigma^{\prime})=O(\xi^{2})$. Finally,

${\rm len}(\sigma)-qq^{\prime}\leq{\rm len}(\sigma^{\prime})+{\rm
len}(\sigma^{\prime\prime})-qq^{\prime}=O(\xi^{2}).$

∎

Now we are ready to prove Lemma 3.37.

###### Proof.

Suppose that $\alpha\neq 0$. Consider the arc $e^{\prime\prime}_{1}$. It
belongs to the orthogonal plane to $\Pi^{1}_{1}$ passing through $v^{1}_{1}$
and $u$. Let $r_{1}$ be the foot of the perpendicular from $p_{1}$ to
$v^{1}_{1}u$ and $q_{1}$ be the foot of the perpendicular from $p_{1}$ to
$\psi$. We want to apply Lemma 3.41 to $e^{\prime\prime}_{1}$ and to the
segment $v^{1}_{1}u$. Considering the spherical link at the point $u$ we get

$\tan({\angle}p_{1}uv^{1}_{1})=\tan(\theta_{p_{1}})\sin({\angle}q_{1}uv^{1}_{1}).$

Thus, from $\theta_{p_{1}}=O(||\Delta g||^{\mu^{\prime}})$ we get
${\angle}p_{1}uv^{1}_{1}=O(||\Delta g||^{\mu^{\prime}})$. It remains to show
that the angle between the tangent ray to $e^{\prime\prime}_{1}$ and
$v^{1}_{1}u$ at $v^{1}_{1}$ is $O(||\Delta g||^{\mu^{\prime}})$. Denote this
angle by $\eta_{1}$.

Let $v,w$ be the neighbouring vertices of $v^{1}_{1}$ in $\partial\Sigma$. By
$z_{1}$ denote the intrinsic distance from $v^{1}_{1}$ to the intrinsic
geodesic segment $vw$ in $\Sigma$. Let $p^{\prime}_{1}$ be the intersection
point of $e^{\prime\prime}_{2}$ with $vw$. Note that the part of
$e^{\prime\prime}_{1}$ between $v^{1}_{1}$ and $p^{\prime}_{1}$ is a geodesic
segment in $\mathbb{H}^{3}$ and $v^{1}_{1}p^{\prime}_{1}\geq z_{1}$. Due to
Lemma 3.40, we have
$d_{\mathbb{H}^{3}}(p^{\prime}_{1},\Pi^{1}_{1})=O(|g|^{\mu^{\prime}})$. Thus,
from

$\sinh(d_{\mathbb{H}^{3}}(p^{\prime}_{1},\Pi^{1}_{1}))=\sin(\eta_{1})\sinh(v^{1}_{1}p^{\prime}_{1})$

we get $\eta_{1}=O(||\Delta g||^{\mu^{\prime}})$.

Now we can apply Lemma 3.41. We obtain

${\rm len}(e^{\prime\prime}_{1})-v^{1}_{1}r_{1}=O(||\Delta
g||^{2\mu^{\prime}})=O(||\Delta g||^{1+\mu}).$

We deal just the same with the arc $e^{\prime\prime}_{2}$ of
$e^{\prime\prime}$ between $v^{2}_{1}$ and $p_{2}$. Let $r_{2}$ be the foot of
the perpendicular from $p_{2}$ to $v^{2}_{1}u$. We get

${\rm len}(e^{\prime\prime}_{2})-v^{2}_{1}r_{2}=O(||\Delta g||^{1+\mu}).$

Now we need to bound ${\rm len}(e^{\prime\prime}_{3})-r_{1}u-r_{2}u$. By
$u_{1}$, $u_{2}$ denote the projections of $r_{1}$, $r_{2}$ to $\psi$. We have
$r_{1}u\geq u_{1}u$, $r_{2}u\geq u_{2}u$. Lemma 3.40 shows that

$\sup_{p\in e^{\prime\prime}_{3}}d_{\mathbb{H}^{3}}(p,\psi)=O(||\Delta
g||^{2}).$

Hence we can choose a radius $x=O(||\Delta g||^{2})$ such that
$e^{\prime\prime}_{3}$ belongs to the set of points at distance at most $x$
from $\psi$. Let $C$ be the boundary of this set, $p^{\prime\prime}_{1}$ and
$p^{\prime\prime}_{2}$ be the intersections of $C$ with the rays $u_{1}p_{1}$,
$u_{2}p_{2}$, let $\sigma\subset C$ be the intrinsic shortest arc connecting
$p^{\prime\prime}_{1}$ and $p^{\prime\prime}_{2}$. Lemma 3.42 implies that
${\rm len}(\sigma)-u_{1}u_{2}=O(||\Delta g||^{2})$. The Busemann–Feller lemma
shows that

${\rm len}(e^{\prime\prime}_{3})\leq p_{1}p^{\prime\prime}_{1}+{\rm
len}(\sigma)+p^{\prime\prime}_{2}p_{2}={\rm len}(\sigma)+O(||\Delta g||^{2}).$

Thus, ${\rm len}(e^{\prime\prime}_{3})-u_{1}u_{2}=O(||\Delta g||^{2}).$ Hence,
also ${\rm len}(e^{\prime\prime}_{3})-r_{1}u-r_{2}u=O(||\Delta g||^{2}).$

Summing this up we get

$l^{\prime\prime}_{e}-l^{\prime}_{e}=O(||\Delta g||^{1+\mu}).$

Now suppose that $\alpha=0$. Then the arc $e^{\prime\prime}_{3}$ is the union
of two geodesic segments, $p_{1}u$ and $p_{2}u$. Denote similarly the points
$p^{\prime}_{1}$, $p^{\prime}_{2}$ and the angles $\eta_{1}$, $\eta_{2}$. We
have

$d_{\mathbb{H}^{3}}(p^{\prime}_{1},\Pi^{1}_{1})\leq\sinh(d_{\mathbb{H}^{3}}(p^{\prime}_{1},\Pi^{1}_{1}))\leq\max\\{\sin(|\phi^{1}_{i}|)\\}\max\\{\sinh(x^{1}_{i})\\}=O(||\Delta
g||).$

Hence, also $\eta_{1}=O(||\Delta g||)$ and from Lemma 3.41 we get

${\rm len}(e^{\prime\prime}_{1})+p_{1}u-v^{1}_{1}u=O(||\Delta g||^{2}).$

Similarly,

${\rm len}(e^{\prime\prime}_{2})+p_{2}u-v^{2}_{1}u=O(||\Delta g||^{2})$

and altogether

$l^{\prime\prime}_{e}-l^{\prime}_{e}={\rm
len}(e^{\prime\prime}_{1})+p_{1}u+p_{2}u+{\rm
len}(e^{\prime\prime}_{2})-v^{1}_{1}u-v^{2}_{1}u=O(||\Delta g||^{2}),$

which finishes the proof. ∎

For Lemma 3.38 we also require

######  Lemma 3.43.

Fix $0<\mu\leq 1$ and let $\xi>0$ be a parameter. Let $p_{1}p_{0}p_{2}q$ be a
hyperbolic quadrilateral with ${\angle}p_{1}qp_{2}\geq\pi$, but with the other
angles $<\pi$, such that
(1) the angle ${\angle}p_{1}qp_{2}-\pi=O(\xi)$;
(2) the angles ${\angle}p_{0}p_{1}q$ and ${\angle}p_{0}p_{2}q$ are
$O(\xi^{\mu})$;
(3) the lengths $p_{1}q$ and $p_{2}q$ are $O(1)$.
Here all asymptotics are as $\xi$ tends to zero. Let $\sigma$ be a convex arc
connecting $p_{1}$ and $p_{2}$ such that $q$ belongs to the convex side. Then

${\rm len}(\sigma)-p_{1}q-p_{2}q=O(\xi^{2\mu}).$

###### Proof.

We have ${\rm len}(\sigma)-p_{1}q-p_{2}q\geq 0$. Let $\sigma^{\prime}$ be the
broken line consisting of the segments $p_{1}p_{0}$ and $p_{2}p_{0}$. By the
Busemann–Feller lemma we have ${\rm len}(\sigma^{\prime})\geq{\rm
len}(\sigma)$. Thus, it is enough to show

${\rm
len}(\sigma^{\prime})-p_{1}q-p_{2}q=p_{1}p_{0}+p_{2}p_{0}-p_{1}q-p_{2}q=O(\xi^{2\mu}).$

Let the ray $p_{1}q$ intersect the segment $p_{0}p_{2}$ in a point
$p^{\prime}_{0}$. It is easy to check that

${\angle}p_{0}p^{\prime}_{0}p_{1}=O(\xi^{\mu}).$

Applying Lemma 3.41 to the triangle $p_{1}p_{0}p^{\prime}_{0}$ we get

$p_{1}p_{0}+p_{0}p^{\prime}_{0}-p^{\prime}_{0}p_{1}=O(\xi^{2\mu}).$

Applying Lemma 3.41 to the triangle $p_{2}qp^{\prime}_{0}$ we obtain

$qp^{\prime}_{0}+p^{\prime}_{0}p_{2}-p_{2}q=O(\xi^{2\mu}).$

Altogether this gives

$p_{1}p_{0}+p_{2}p_{0}-p_{1}q-p_{2}q=O(\xi^{2\mu}).$

∎

Now we prove Lemma 3.38.

###### Proof.

Suppose that $\alpha\neq 0$. Let $\eta_{1}$ be the angle between the tangent
ray to $e^{\prime\prime}_{1}$ and $v^{1}_{1}u$ at $v^{1}_{1}$, $\eta_{2}$ be
the angle between the tangent ray to $e^{\prime\prime}_{2}$ and $v^{2}_{1}u$
at $v^{2}_{1}$. Similarly to the proof of Lemma 3.37, from Lemma 3.40 we get
that $\eta_{1}$ and $\eta_{2}$ are $O(||\Delta g||^{\mu^{\prime}})$. Consider
the union of the half-planes of the planes $v^{1}_{1}p_{0}u$ and
$v^{2}_{1}p_{0}u$ containing the arc $e^{\prime\prime}_{1}$. Develop them to
$\mathbb{H}^{2}$. Clearly, when $||\Delta g||$ is sufficiently small, the
tangent rays to $e^{\prime\prime}_{2}$ at its endpoints intersect in
$\mathbb{H}^{2}$. Let $p^{\prime}_{0}$ be the intersection point. Our result
follows from applying Lemma 3.43 to the quadrilateral
$v^{1}_{1}p^{\prime}_{0}v^{2}_{1}u$ and the arc $e^{\prime\prime}_{2}$.

For $\alpha=0$ we argue like in the proof of Lemma 3.37 to obtain that
$\eta_{1}$ and $\eta_{2}$ are $O(||\Delta g||)$. Then Lemma 3.43 gives us

$l^{\prime\prime}_{e}-l^{\prime}_{e}=O(||\Delta g||^{2}),$

which finishes the proof. ∎

We can finish the proof of Lemma 3.35.

###### Proof of Lemma 3.35..

Due to the discussion above, Lemma 3.35 follows from Lemmas 3.37 and 3.38
except the case $\phi_{0}=0$, $\phi^{2}_{1}<\pi$, so we deal with this case
here. Suppose that $\alpha\neq 0$. Let $\Pi$ be the plane containing the half-
plane $\Pi^{1}_{1}$ and $v_{0}$ be the orthogonal projection of $v^{2}_{1}$ to
$\Pi$. By $e_{0}$ we denote the union of the segments $v^{1}_{1}u$ and
$uv_{0}$, by $l_{0}$ we denote its length. We act now similarly to the proof
of Lemma 3.38. By $\eta_{1}$, $\eta_{2}$ we denote the angles between the
tangent rays to $e^{\prime\prime}_{2}$ at its endpoints and between the
segments connecting the endpoints with $u$. We get that $\eta_{1}$, $\eta_{2}$
are $O(||\Delta g||^{\mu^{\prime}})$. Since the dihedral angle between
$\Pi^{2}_{1}$ and $\Pi$ is $O(||\Delta g||)$, it is easy to see that when
$||\Delta g||$ is small enough, the extension of the tangent ray to
$e^{\prime\prime}_{2}$ at $v^{2}_{1}$ in the converse direction intersects
$\Pi$ and the intersection angle is $O(||\Delta g||^{\mu^{\prime}})$. Applying
Lemma 3.41 we get

$l^{\prime\prime}_{e}-l_{0}=O(||\Delta g||^{2\mu^{\prime}}).$

By $l^{\prime}_{0}$ we denote the length $v^{1}_{1}v_{0}$. It is easy to check
that since the dihedral angle between $\Pi^{2}_{1}$ and $\Pi$ is $O(||\Delta
g||)$, we get $l^{\prime}_{0}-l_{0}=O(||\Delta g||^{2})$ and
$l^{\prime}_{e}-l^{\prime}_{0}=O(||\Delta g||^{2})$. Thus,
$l^{\prime\prime}_{e}-l^{\prime}_{0}=O(||\Delta g||^{2\mu^{\prime}}).$ Since
$l^{\prime\prime}_{e}\geq l_{e}\geq l^{\prime}_{0}$, we obtain
$l_{e}-l^{\prime}_{0}=O(||\Delta g||^{2\mu^{\prime}})$, and, finally,

$l_{e}-l^{\prime}_{e}=O(||\Delta g||^{2\mu^{\prime}}).$

For $\alpha=0$, similarly to the proofs of Lemmas 3.37, 3.38 we can obtain
that $\eta_{1}$, $\eta_{2}$ are $O(||\Delta g||)$. The remaining part is
identical to the argument above. ∎

Now we start to prove Lemmas 3.39–3.40. To this purpose we introduce an
auxiliary surface. Since now we assume $\alpha\neq 0$. Let $\Gamma^{*}<{\rm
Iso}^{+}(\mathbb{H}^{3})$ be the smooth 1-parametric subgroup consisting of
isometries with the oriented shift $t\in\mathbb{R}$ along $\psi$ in the
positive direction and rotating by $\frac{\alpha t}{a}$ clockwise. Note that
$\Gamma<\Gamma^{*}$. By $\tau^{j}_{i}$ we denote the $\Gamma^{*}$-orbit of
$v^{j}_{i}$, which is a smooth curve, by $M^{*}(g)$ we denote the closed
convex hull of all $\tau^{j}_{i}$. We have $M(g)\subset M^{*}(g)$.

######  Lemma 3.44.

If $g$ is sufficiently close to $g_{0}$, then the boundary $\partial M^{*}(g)$
contains a curve $\tau^{1}_{i_{1}}$ for some $i_{1}$ and a curve
$\tau^{2}_{i_{2}}$ for some $i_{2}$.

###### Proof.

Clearly $\partial M^{*}(g)$ must contain at least one curve $\tau^{j}_{i}$.
Suppose that there exists a sequence $g_{k}$ converging to $g$ in
$\mathcal{P}_{b}(M,V)$ such that $\partial M^{*}(g)$ does not contain curves
of the type $\tau^{2}_{i}$. Similarly to Corollary 3.28, one can see that
$\partial M^{*}(g_{k})$ must converge in the Hausdorff sense to $\partial
M^{*}(g)$. However, it is easy to see that $\overline{\rm
conv}(\\{\tau^{1}_{i}\\})$ converge in the Hausdorff sense to a subset of
$\Pi^{1}$ bounded by $\psi$ and by a curve $\tau^{1}_{i_{0}}\subset\Pi^{1}$ at
the greatest distance from $\psi$ in $g_{0}$. ∎

The union of the curves $\tau^{j}_{i}$ belonging to $\partial M^{*}(g)$ divide
$\partial M^{*}(g)$ into strips. When $g$ is sufficiently close to $g_{0}$,
there are exactly two strips bounded by a curve $\tau^{1}_{i_{1}}$ for some
$i_{1}$ and a curve $\tau^{2}_{i_{2}}$ for some $i_{2}$. We choose the one,
for which the curve $\tau^{2}_{i_{2}}$ is clockwise with respect to the curve
$\tau^{1}_{i_{1}}$. We denote this strip by $\Sigma^{*}$. From now on we
denote the boundary curves of $\Sigma^{*}$ just by $\tau_{1}$ and $\tau_{2}$.
Similarly, we denote $v^{1}_{i_{1}}$, $v^{2}_{i_{2}}$, $x^{1}_{i_{1}}$,
$x^{2}_{i_{2}}$, $y^{1}_{i_{1}}$, $y^{2}_{i_{2}}$, $\phi^{1}_{i_{1}}$,
$\phi^{2}_{i_{2}}$ just by $v_{1}$, $v_{2}$, $x_{1}$, $x_{2}$, $y_{1}$,
$y_{2}$, $\phi_{1}$, $\phi_{2}$. Define $y:=y_{2}-y_{1}$,
$\phi:=\phi_{2}-\phi_{1}$.

The surface $\Sigma^{*}$ is invariant with respect to $\Gamma^{*}$. Each point
$p\in\Sigma^{*}\backslash(\tau_{1}\cup\tau_{2})$ either belongs to a segment
contained in $\Sigma^{*}$ with vertices on $\tau_{1}$, $\tau_{2}$, or belongs
to a triangle contained in $\Sigma^{*}$ with vertices on $\tau_{1}$,
$\tau_{2}$. As there is no plane invariant with respect to $\Gamma^{*}$, the
second option is not possible. Thus, $\Sigma^{*}$ is a ruled surface
decomposing into the union of segments so that the decomposition is
$\Gamma^{*}$-invariant.

The main step is to prove analogues of Lemmas 3.39 and 3.40 for the surface
$\Sigma^{*}$.

######  Lemma 3.45.

Let $\phi_{0}>\pi$. For every $0<\mu<1$ and every $T>0$ we have

$\sup\\{d_{\mathbb{H}^{3}}(p,\psi):p\in\Sigma^{*},-T\leq t_{p}\leq
T,\max_{1\leq i\leq
n_{1}}\phi^{1}_{i}+|\alpha|^{\mu}\leq\theta_{p}\leq\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu}\\}=O(||\Delta g||^{2}).$

######  Lemma 3.46.

For every $0<\mu<1$ and every $T>0$ we have

$\sup\\{d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1}):p\in\Sigma^{*},-T\leq t_{p}\leq
T,\max_{1\leq i\leq
n_{1}}\phi^{1}_{i}+|\alpha|^{\mu}\leq\theta_{p}\leq\frac{\pi}{2}\\}=O(||\Delta
g||^{\mu}),$ $\sup\\{d_{\mathbb{H}^{3}}(p,\Pi^{2}_{1}):p\in\Sigma^{*},-T\leq
t_{p}\leq T,\phi^{2}_{1}-\frac{\pi}{2}\leq\theta_{p}\leq\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu}\\}=O(||\Delta g||^{\mu}).$

Note that there is a slight difference in the bounds on $\theta_{p}$ between
Lemma 3.40 and Lemma 3.46. This is done on purpose to simplify our argument.
Since $M(g)\subset M^{*}(g)$, Lemmas 3.39 and 3.40 follow from Lemmas 3.45 and
3.46 quite easily:

###### Proof of Lemma 3.39..

Note that if $p\in\tau^{j}_{i}$, then

$\theta_{p}=\phi^{j}_{i}+\frac{\alpha}{a}(t_{p}-y^{j}_{i}).$

Thus, for every $T>0$ we have

$\sup\\{\theta_{p}:p\in\tau^{1}_{i},1\leq i\leq n_{1},-T\leq t_{p}\leq
T\\}\leq\max_{1\leq i\leq n_{1}}\phi^{1}_{i}+O(|\alpha|),$
$\inf\\{\theta_{p}:p\in\tau^{2}_{i},1\leq i\leq n_{2},-T\leq t_{p}\leq
T\\}\geq\min_{1\leq i\leq n_{2}}\phi^{2}_{i}+O(|\alpha|).$

Let $p\in\Sigma$ and $p^{*}\in\partial M^{*}(g)$ be the point with the same
coordinates $(\theta_{p},t_{p})$. Due to the bounds above, for sufficiently
small $\alpha$, if $p\in\Sigma$, $-T\leq t_{p}\leq T$ and

$\max_{1\leq i\leq
n_{1}}\phi^{1}_{i}+|\alpha|^{\mu}\leq\theta_{p}\leq\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu},$

then $p^{*}\in\Sigma^{*}$. Thus, Lemma 3.39 follows from Lemma 3.45. ∎

###### Proof of Lemma 3.40..

As in the proof of Lemma 3.39, for $p\in\Sigma$ let $p^{*}\in\partial
M^{*}(g)$ be the point with the same coordinates $(\theta_{p},t_{p})$. For
sufficiently small $\alpha$, if $p\in\Sigma$, $-T\leq t_{p}\leq T$ and

$\max_{1\leq i\leq
n_{1}}\phi^{1}_{i}+|\alpha|^{\mu}\leq\theta_{p}\leq\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu},$

then $p^{*}\in\Sigma^{*}$, and for such $p$ Lemma 3.40 follows from Lemma
3.46.

But for $p\in\Sigma$ satisfying $0\leq\theta_{p}\leq\max_{1\leq i\leq
n_{1}}\phi^{1}_{i}+|\alpha|^{\mu}$ we have

$d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1})\leq\sinh(d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1}))=\sin(\theta_{p})\sinh(d_{\mathbb{H}^{3}}(p,\psi))\leq\theta_{p}\sinh(\max(x^{j}_{i}))=O(||\Delta
g||^{\mu}).$

Similarly for $p$ such that $\min_{1\leq i\leq
n_{2}}\phi^{2}_{i}-|\alpha|^{\mu}\leq\theta_{p}\leq\phi^{2}_{1}$ we have

$d_{\mathbb{H}^{3}}(p,\Pi^{2}_{1})\leq\sinh(d_{\mathbb{H}^{3}}(p,\Pi^{2}_{1}))=\sin(\phi^{2}_{1}-\theta_{p})\sinh(d_{\mathbb{H}^{3}}(p,\psi))\leq$
$\leq(\phi^{2}_{1}-\theta_{p})\sinh(\max(x^{j}_{i}))=O(||\Delta g||^{\mu}).$

This finishes the proof. ∎

Now we are going to prove Lemmas 3.45–3.46. We will need the following
preliminaries

######  Lemma 3.47.

Let $p_{1}p_{2}q_{1}q_{2}$ be a hyperbolic quadrilateral, possibly self-
intersecting, such that
${\angle}p_{1}q_{1}q_{2}={\angle}p_{2}q_{2}q_{1}=\pi/2$. Consider a point
$p\in p_{1}p_{2}$, let $q\in q_{1}q_{2}$ be its orthogonal projection to the
line $q_{1}q_{2}$. Then

$pq\leq 2\max_{j=1,2}\\{p_{j}q_{j}\exp(-qq_{j})\\}.$

###### Proof.

Let $q^{\prime}$ be the projective intersection point of the lines
$p_{1}p_{2}$ and $q_{1}q_{2}$, possibly ideal or hyperideal. Assume that $q$
lies between $q^{\prime}$ and $q_{1}$. We will show that

$pq\leq 2p_{1}q_{1}\exp(-qq_{1}).$

If $q^{\prime}$ is not hyperideal, we can rotate the line $p_{1}p_{2}$ around
the point $p_{1}$ until we make $q^{\prime}$ hyperideal so that $pq$ increases
and $q$ stays between $q^{\prime}$ and $q_{1}$. Here we mean that the point
$q$ stays fixed and the point $p$ is defined as the intersection point of the
new line with the perpendicular to $q_{1}q_{2}$ at $q$. Hence, we can assume
that $q^{\prime}$ is hyperideal, so the lines $p_{1}p_{2}$ and $q_{1}q_{2}$
are ultra-parallel. Let $p_{0}$, $q_{0}$ be the respective closest points on
these lines. By construction, the point $q$ lies between $q_{0}$ and $q_{1}$.
From Lemmas 2.2 and 2.3 we get

$\tanh(p_{0}q_{0})=\frac{\tanh(p_{1}q_{1})}{\cosh(q_{0}q_{1})}.$

And also

$\tanh(pq)=\tanh(p_{0}q_{0})\cosh(qq_{0})=\frac{\tanh(p_{1}q_{1})\cosh(qq_{0})}{\cosh(q_{0}q_{1})}=$
$=\tanh(p_{1}q_{1})\exp(-qq_{1})\frac{1+\exp(-2qq_{0})}{1+\exp(-2q_{0}q_{1})}\leq
2\tanh(p_{1}q_{1})\exp(-qq_{1}).$

It remains to note that $pq\leq p_{1}q_{1}$, thus
$pq\leq\frac{p_{1}q_{1}}{\tanh(p_{1}q_{1})}\tanh(pq).$ ∎

######  Lemma 3.48.

Let $\alpha\neq 0$, let $(\phi-\pi)a-\alpha y>0$ and let $p_{1}p_{2}$ be a
generatrix of $\Sigma^{*}$, where $p_{1}\in\tau_{1}$, $p_{2}\in\tau_{2}$. Then
$\operatorname{sgn}(t_{p_{1}}-t_{p_{2}})=\operatorname{sgn}(\alpha)$,
$0<\theta_{p_{2}}-\theta_{p_{1}}<\pi$ and

$\left|\frac{t_{p_{1}}-t_{p_{2}}}{\theta_{p_{2}}-\theta_{p_{1}}}\right|\geq\left|\frac{(\phi-\pi)a-\alpha
y}{\alpha\pi}\right|.$

###### Proof.

It is clear that $-\pi<\theta_{p_{2}}-\theta_{p_{1}}<\pi$. Let
$p^{\prime}_{1}\in\tau_{1}$ be a point with
$\theta_{p^{\prime}_{1}}=\theta_{p_{1}}-\pi$. Then we have
$\operatorname{sgn}(t_{p^{\prime}_{1}}-t_{p_{2}})=\operatorname{sgn}(\alpha)$.
Hence, the same holds for $\operatorname{sgn}(t_{p_{1}}-t_{p_{2}})$.

We now show that $\theta_{p_{2}}>\theta_{p_{1}}$. Consider the equidistant
surfaces from $\psi$. The induced intrinsic metrics on such surfaces are
locally Euclidean. Consider now the supporting plane $\Pi$ to $\Sigma^{*}$
containing the generatrix $p_{1}p_{2}$. Its intersection with each equidistant
surface to $\psi$ is an ellipse in the intrinsic metric of this surface. In
$\Pi$ all such intersections are concentric figures around a point
$p_{0}\in\Pi$, which is the closest point from $\Pi$ to $\psi$. For each such
ellipse we call its “left side” all the points $p$ with
$\theta_{p}<\theta_{p_{0}}$ and its “right side” the set of points with
$\theta_{p}>\theta_{p_{0}}$. It is easy to see that by the choice of
$\Sigma^{*}$ the curve $\tau_{1}$ is tangent to the corresponding ellipse in
its left side, and $\tau_{2}$ tangents the corresponding ellipse in its right
side. Hence, $\theta_{p_{1}}>\theta_{p_{0}}>\theta_{p_{2}}$.

Due to the $\Gamma^{*}$-invariance of $\Sigma^{*}$, we can assume that
$p_{2}=v_{2}$, so $t_{p_{2}}=y_{2}$ and $\theta_{p_{2}}=\phi_{2}$. As above,
let $p^{\prime}_{1}\in\tau_{1}$ be the point with
$\theta_{p_{2}}-\theta_{p_{1}^{\prime}}=\pi$. We get
$\theta_{p^{\prime}_{1}}=\phi_{2}-\pi$. Hence,
$t_{p_{1}^{\prime}}-t_{p_{2}}=\frac{(\phi-\pi)a}{\alpha}-y$. The chord
$p^{\prime}_{1}p_{2}$ is tangent to $\psi$, and in order for the chord
$p_{1}p_{2}$ to be in the right location with respect to $\psi$, the following
bound must hold

$\left|\frac{t_{p_{1}}-t_{p_{2}}}{\theta_{p_{2}}-\theta_{p_{1}}}\right|\geq\left|\frac{t_{p^{\prime}_{1}}-t_{p_{2}}}{\theta_{p_{2}}-\theta_{p^{\prime}_{1}}}\right|.$

Our last claim follows. ∎

######  Lemma 3.49.

Let $\alpha\neq 0$, let $(\phi-\pi)a-\alpha y>0$ and let $p\in\Sigma^{*}$ be
such that $-T\leq t_{p}\leq T$, $p_{1}\in\tau_{1}$, $p_{2}\in\tau_{2}$ be the
endpoints of the generatrix containing $p$. Then there are constants
$c_{1},c_{2}>0,$ $c_{3},c_{4}\geq 0$ such that in a neighborhood of $g_{0}$ we
have

$|t_{p_{1}}-t_{p}|\geq\frac{c_{1}(\theta_{p}-\phi_{1})}{|\alpha|}-c_{2},$
$|t_{p}-t_{p_{2}}|\geq\frac{c_{3}(\phi_{2}-\theta_{p})}{|\alpha|}-c_{4}.$

###### Proof.

We have $\theta_{p_{1}}=\phi_{1}+\frac{\alpha(t_{p_{1}}-y_{1})}{a}$. On the
other hand, from Lemma 3.48 we get

$|t_{p_{1}}-t_{p}|\geq(\theta_{p}-\theta_{p_{1}})\frac{(\phi-\pi)a-\alpha
y}{\pi|\alpha|}.$

Substituting the first into the second and rearranging we get

$|t_{p_{1}}-t_{p}|\frac{\phi a-\alpha y}{\pi
a}\geq(\theta_{p}-\phi_{1})\frac{(\phi-\pi)a-\alpha
y}{\pi|\alpha|}-\operatorname{sgn}(\alpha)(t_{p}-y_{1})\frac{(\phi-\pi)a-\alpha
y}{\pi a},$

from which we obtain

$|t_{p_{1}}-t_{p}|\geq\frac{a(\theta_{p}-\phi_{1})}{|\alpha|}\cdot\frac{(\phi-\pi)a-\alpha
y}{\phi a-\alpha y}-(T+y_{1})\cdot\frac{(\phi-\pi)a-\alpha y}{\phi a-\alpha
y}.$

The first claim follows.

Similarly we have

$\theta_{p_{2}}=\phi_{2}+\frac{\alpha(t_{p_{2}}-y_{2})}{a}$

and from Lemma 3.48

$|t_{p}-t_{p_{2}}|\geq(\theta_{p_{2}}-\theta_{p})\frac{(\phi-\pi)a-\alpha
y}{\pi|\alpha|}.$

From this we get

$|t_{p}-t_{p_{2}}|\frac{\phi a-\alpha y}{\pi
a}\geq(\phi_{2}-\theta_{p})\frac{(\phi-\pi)a-\alpha
y}{\pi|\alpha|}+\operatorname{sgn}(\alpha)(t_{p}-y_{2})\frac{(\phi-\pi)a-\alpha
y}{\pi a}$

and in turn

$|t_{p}-t_{p_{2}}|\geq\frac{a(\phi_{2}-\theta_{p})}{|\alpha|}\cdot\frac{(\phi-\pi)a-\alpha
y}{\phi a-\alpha y}-(T+y_{2})\cdot\frac{(\phi-\pi)a-\alpha y}{\phi a-\alpha
y}.$

The second claim follows. ∎

Now we can prove Lemma 3.45.

###### Proof of Lemma 3.45..

We may assume that $||\Delta g||$ is small enough so that $\phi>\pi$ and

$(\phi-\pi)a-\alpha y>0.$

Let $p$ be a point satisfying the conditions of the lemma. It belongs to a
generatrix of $\Sigma^{*}$, with endpoints $p_{1}\in\tau_{1}$,
$p_{2}\in\tau_{2}$. Let $q_{1},q_{2},q$ be the orthogonal projections of
$p_{1},p_{2},p$ to $\psi$. Due to the restrictions on $\theta_{p}$, from Lemma
3.49 we get that in a neighborhood of $g_{0}$, there exist constants
$c_{1},c_{2}>0$ such that

$q_{1}q=|t_{p_{1}}-t_{p}|\geq\frac{c_{1}}{|\alpha|^{1-\mu}},$
$qq_{2}=|t_{p}-t_{p_{2}}|\geq\frac{c_{2}}{|\alpha|^{1-\mu}}.$

Consider the tetrahedron $p_{1}p_{2}q_{2}q_{1}$. Rotate the segment
$p_{2}q_{2}$ in the plane orthogonal to $q_{1}q_{2}$. We keep $q$ fixed and by
$p$ we mean the point at $p_{1}p_{2}$ such that its orthogonal projection to
$q_{1}q_{2}$ is $q$. We see that the length $pq$ maximizes when the
tetrahedron degenerates to a planar trapezoid. Indeed, the trajectory of $p$
is a circle in the plane orthogonal to $q_{1}q_{2}$ at $q$ centered at the
intersection of this plane with the line $p_{1}q_{2}$. The farthest point from
this circle to $q$ corresponds exactly to the situation when $pq$ becomes
coplanar to $p_{1}q_{2}$ and lying on the same side of $\psi$.

Let $p^{\prime}_{1}p^{\prime}_{2}q^{\prime}_{2}q^{\prime}_{1}$ be the
trapezoid with $p^{\prime}_{1}q^{\prime}_{1}=p_{1}q_{1}$,
$p^{\prime}_{2}q^{\prime}_{2}=p_{2}q_{2}$ and
$q_{1}q_{2}=q^{\prime}_{1}q^{\prime}_{2}$, $p^{\prime}$ be the point at the
segment $p^{\prime}_{1}p^{\prime}_{2}$ with the orthogonal projection $q$ to
the line $q^{\prime}_{1}q^{\prime}_{2}$ such that
$q^{\prime}q^{\prime}_{1}=qq_{1}$, $q^{\prime}q^{\prime}_{2}=qq_{2}$. From the
argument above we have $pq\leq p^{\prime}q^{\prime}$. Applying Lemma 3.47 we
get

$p^{\prime}q^{\prime}\leq
2\max_{j=1,2}\left\\{x_{j}\exp\left(-\frac{c_{j}}{|\alpha|^{1-\mu}}\right)\right\\}\leq
2\max_{j=1,2}\left\\{\frac{x_{j}}{c_{j}^{\frac{2}{1-\mu}}}\right\\}|\alpha|^{2}=O(||\Delta
g||^{2}).$

∎

###### Proof of Lemma 3.46..

Consider the first claim of the lemma. Let $p$ be a point satisfying the
conditions of the claim. We treat separately the cases $\phi_{0}>\pi$ and
$\phi_{0}=\pi$. Suppose that $\phi_{0}>\pi$. We may assume that $||\Delta g||$
is small enough so that $\phi>\pi$ and

$(\phi-\pi)a-\alpha y>0.$

We have

$d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1})\leq\sinh(d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1}))=\sin(\theta_{p})\sinh(d_{\mathbb{H}^{3}}(p,\psi))\leq$
$\leq\max_{j=1,2}\left\\{\frac{\sinh(x_{j})}{x_{j}}\right\\}\theta_{p}d_{\mathbb{H}^{3}}(p,\psi).$

The point $p$ belongs to a generatrix of $\Sigma^{*}$, with endpoints
$p_{1}\in\tau_{1}$, $p_{2}\in\tau_{2}$. Let $q_{1},q_{2},q$ be the orthogonal
projections of $p_{1},p_{2},p$ to $\psi$. From Lemma 3.49 we get that in a
neighborhood of $g_{0}$ there exist constants $c_{1},c_{2}>0$ such that

$q_{1}q=|t_{p_{1}}-t_{p}|\geq\frac{c_{1}}{|\alpha|^{1-\mu}},$
$qq_{2}=|t_{p}-t_{p_{2}}|\geq\frac{c_{2}}{|\alpha|^{1-\mu}}.$

Thus, similarly to the proof of Lemma 3.45, from Lemma 3.41 in this case we
can bound

$d_{\mathbb{H}^{3}}(p,\psi)=O(||\Delta
g||^{2}),~{}~{}~{}~{}~{}d_{\mathbb{H}^{3}}(p,\Pi^{1}_{1})=O(||\Delta
g||^{2}).$

Now we deal with the case $\phi_{0}=\pi$. Let $\Pi$ be the plane containing
the half-plane $\Pi^{1}_{1}$, and $r_{1}$, $r_{2}$, $r$ be the orthogonal
projections of $p_{1}$, $p_{2}$, $p$ to $\Pi$. From Lemma 3.47 we get

$d_{\mathbb{H}^{3}}(p,\Pi)=pr\leq 2\max_{j=1,2}\\{p_{j}r_{j}\exp(-rr_{j})\\}.$
(3.4)

Suppose that $rr_{j}\geq|\alpha|^{\mu-1}$. Then

$p_{j}r_{j}\exp(-rr_{j})\leq x_{j}|\alpha|^{2}=O(|\alpha|^{2}).$

Suppose that $rr_{j}\leq|\alpha|^{\mu-1}$. Then also
$qq_{j}\leq|\alpha|^{\mu-1}$ and

$|t_{p_{j}}-y_{j}|\leq T+qq_{j}+|y_{j}|\leq T+|\alpha|^{\mu-1}+|y_{j}|\leq
c|\alpha|^{\mu-1}$

for some constant $c>1$. We have

$\theta_{p_{j}}=\phi_{j}+\frac{\alpha(t_{p_{j}}-y_{j})}{a}.$

In the case $j=1$ we get

$|\theta_{p_{1}}|\leq|\phi_{1}|+\frac{c|\alpha|^{\mu}}{a}=O(||\Delta
g||^{\mu}),$
$p_{1}r_{1}=d_{\mathbb{H}^{3}}(p_{1},\Pi)\leq\sinh(p_{1}r_{1})=\sin(|\theta_{p_{1}}|)\sinh(p_{1}q_{1})\leq$
$\leq|\theta_{p_{1}}|\sinh(x_{1})=O(||\Delta g||^{\mu}).$

In the case $j=2$ we get

$|\theta_{p_{2}}-\pi|\leq|\phi_{2}-\pi|+\frac{c|\alpha|^{\mu}}{a}=O(||\Delta
g||^{\mu}),$
$p_{2}r_{2}=d_{\mathbb{H}^{3}}(p_{2},\Pi)\leq\sinh(p_{2}r_{2})=\sin(|\theta_{p_{2}}-\pi|)\sinh(p_{2}q_{2})\leq$
$\leq|\theta_{p_{2}}-\pi|\sinh(x_{2})=O(||\Delta g||^{\mu}).$

In any case we see that

$p_{j}r_{j}\exp(-qq_{j})=O(||\Delta g||^{\mu}).$

Due to (3.4), this finishes the proof. The second claim of the lemma is
obtained similarly. ∎

####  3.4.3 Differentiability for controllably polyhedral metrics

We use the results of the previous section to show

######  Lemma 3.50.

Let $g\in\mathcal{P}_{cp}(M,V)$. Then $\mathcal{I}_{V}$ is differentiable at
$g$.

###### Proof.

We start as in the proof of Lemma 3.34. Pick $g\in\mathcal{P}_{cp}(M,V)$, lift
it to
$(\rho,\widetilde{f})\in\mathcal{R}(\pi_{1}(M),G)\times(\mathbb{H}^{3})^{*V}$,
choose a neighborhood $\widetilde{U}$ of $(\rho,\widetilde{f})$ submersing
smoothly onto a neighborhood $U$ of $g$ in $\mathcal{P}_{cp}(M,V)$ and
consider $\mathcal{I}_{V}$ as a function over $\widetilde{U}$. Let $d$ be the
induced intrinsic metric on $\partial M(g)$.

Let $\psi_{1},\ldots,\psi_{r}$ be all components of $\partial M(g)\cap\partial
C(\overline{g})$. If $\psi_{i}$ is an edge of $g$, then by $\Sigma_{i}$ we
denote the union of $\psi_{i}$ and the two faces adjacent to $\psi_{i}$. If
$\psi_{i}$ is not an edge of $g$, then by $\Sigma_{i}$ we denote the face
containing $\psi_{i}$. Pick any triangulation $\mathcal{T}$ realized in
$(d,V)$ such that (1) every edge of $g$ that is not one of $\psi_{i}$ is an
edge of $\mathcal{T}$; (2) all $\Sigma_{i}$ are subdivided so that all the
interior edges of $\mathcal{T}$ in $\Sigma_{i}$ connect two different boundary
components of $\Sigma_{i}$. Then if $U$ is small enough, we get
$\mathcal{I}_{V}(U)\subset\mathcal{D}_{c}(V,\mathcal{T})$.

The lengths of the edges of $\mathcal{T}$ that are edges of $g$ are smooth
functions around $g$. For each face of $g$ that is a convex hyperbolic
polygon, $\mathcal{T}$ subdivides it, and the lengths of the edges of
$\mathcal{T}$ in the interior of the faces are $C^{1}$-functions around $g$ by
the same argument as in Lemma 3.34. The only remaining edges are the edges of
$\mathcal{T}$ belonging to interiors of $\Sigma_{i}$. For every such edge
Corollary 3.36 shows that its length is a differentiable function at $g$. This
finishes the proof. ∎

####  3.4.4 Differentiability for convex cores

Let $V=\emptyset$, so for each $g\in\mathcal{P}_{b}(M,V)$ the set $M(g)$
coincides with $C(\overline{g})$. The set $\mathcal{P}_{b}(M,V)$ is identified
with a subset of $\mathcal{CH}(N)$ corresponding to non-Fuchsian metrics on
$N$ (recall that it is the whole $\mathcal{CH}(N)$ except the case when $M$ is
an interval bundle over a surface). The map $\mathcal{I}_{V}$ coincides with a
map $\mathcal{I}$ defined in the introduction, which naturally extends to the
whole $\mathcal{CH}(N)$ when there are Fuchsian metrics. It was shown by
Bonahon in [14]

######  Lemma 3.51.

The map $\mathcal{I}$ is continuously differentiable over $\mathcal{CH}(N)$.

We will need some additional knowledge on the differential of $\mathcal{I}$.
We recall from the introduction that there is another map
$\mathcal{I}^{*}:\mathcal{CH}(N)\rightarrow\mathcal{ML}(\partial M)$ sending a
convex cocompact metric on $N$ to the bending lamination of its convex core.
The space $\mathcal{ML}(\partial M)$ does not carry a natural smooth
structure, but carries a PL-structure. In [14] Bonahon proposed a natural
relaxation of the notion of differentiability for PL-structures. Let
$U\subset\mathbb{R}^{n}$ be a domain and $f:U\rightarrow\mathbb{R}^{k}$ be a
map. _The tangent map_ to $f$ at $x\in U$, which we denote $df_{x}$, is a map
$\mathbb{R}^{n}\rightarrow\mathbb{R}^{k}$ such that for every continuous curve
$\tau:[0,\varepsilon]\rightarrow U$ with $\tau(0)=x$, differentiable at $0$
with $\tau^{\prime}(0)=v$, the curve $f\circ\tau$ is differentiable at zero
and $df_{x}(v)=(f\circ\tau)^{\prime}(0)$. We say that $f$ is _tangentiable_ at
$x$ if it admits a tangent map at $x$, and $f$ is tangetiable over $U$ if it
is tangentiable at every $x\in U$.

By the same construction as for differential manifolds, PL-manifolds admit
tangent spaces at each point, which, however, lack the addition structure on
the tangent vectors, and allow only the multiplication by non-negative
scalars. For an $n$-dimensional PL-manifold, the tangent space at a point is
isomorphic to $\mathbb{R}^{n}$ endowed with such a structure. One can
naturally speak about tangentiable maps between PL-manifolds. For a bit more
information we refer to [14, Section 1]. Bonahon proved in [14] that the map
$\mathcal{I}^{*}$ is tangentiable.

Let us recall from Section 2.4 that $\mathcal{CH}(N)$ can be thought as a
covering space over a smooth open subset of the character space
$\mathcal{X}(\pi_{1}(M),G)$, where $G={\rm Iso}^{+}(\mathbb{H}^{3})$. We need
to slightly expand this construction. Let $S_{i}$, ${i=1\ldots r}$, be the
components of $\partial M$. Define $\mathcal{R}(\pi_{1}(S_{i}),G)$ and
$\mathcal{X}(\pi_{1}(S_{i}),G)$ as in Section 2.4 and denote
$\prod_{i=1}^{r}\mathcal{X}(\pi_{1}(S_{i}),G)$ by $\mathcal{X}(\partial M,G)$.
Let $[\rho_{i}]\in\mathcal{X}(\pi_{1}(S_{i}),G)$ be the characters induced by
a representation $\rho_{\overline{g}}\in\mathcal{R}(\pi_{1}(M),G)$ for
$\overline{g}\in\mathcal{CH}(N)$. It is known that even when the boundary of
$M$ is compressible, $[\rho_{i}]$ belongs to the smooth part of
$\mathcal{X}(\pi_{1}(S_{i}),G)$, which is locally a complex manifold of
complex dimension $-3\chi(S_{i})$, see, e.g., [40]. Hence, also
$\mathcal{X}(\partial M,G)$ is locally a complex manifold around the induced
characters, and the smooth part of $\mathcal{X}(\pi_{1}(M),G)$ is immersed to
$\mathcal{X}(\partial M,G)$ as a complex submanifold of half dimension.

Thurston proposed a well-known parametrization of the space of complex
projective structures, which defines a local homeomorphism

$\mathcal{J}:\mathcal{T}(\partial M)\times\mathcal{ML}(\partial
M)\rightarrow\mathcal{X}(\partial M,G),$

see details in [45]. The points of $\mathcal{X}(\partial M,G)$ coming from
convex cocompact metrics on $M$ are in the image of $\mathcal{J}$, and
$\mathcal{I}\times\mathcal{I}^{*}$ coincides with a local inverse of
$\mathcal{J}$ on such points. Bonahon showed in [14] that $\mathcal{J}$ is
tangentiable and that its tangent map is an isomorphism of the tangent spaces
at each point. Moreover, he proposed a convenient description of the tangent
map.

We need to introduce few definitions from another paper [13] of Bonahon. An
_(abstract) pleated surface_ with topological type $S$ (for a closed connected
surface $S$) is a pair $(\widetilde{\iota},\rho)$, where
$\widetilde{\iota}:\widetilde{S}\rightarrow\mathbb{H}^{3}$ is a map from the
universal cover $\widetilde{S}$ of $S$ and $\rho:\pi_{1}(S)\rightarrow G$ is a
homomorphism, such that
(1) for every $\gamma\in\pi_{1}(S)$ we have
$\widetilde{\iota}\circ\gamma=\rho(\gamma)\circ\widetilde{\iota}$;
(2) the pull-back of the intrinsic metric on
$\widetilde{\iota}(\widetilde{S})$ induces a hyperbolic metric $d$ on $S$;
(3) there exists a geodesic lamination $\lambda$ in metric $d$ such that
$\widetilde{\iota}$ sends each edge of its preimage $\widetilde{\lambda}$ in
$\widetilde{S}$ to a geodesic in $\mathbb{H}^{3}$ and $\widetilde{\iota}$ is
totally geodesic on the complement to $\widetilde{\lambda}$.

This notion is different from the common notion of a pleating surface in that
the holonomy representation $\rho$ is not required to be faithful or to act
freely or properly discontinuously on $\mathbb{H}^{3}$.

Now let us step back and consider a geodesic lamination on a hyperbolic
surface $(S,d)$. An _$A$ -valued transverse cocycle_ for $\lambda$, where $A$
is an abelian group, is an $A$-valued transverse finitely additive measure for
$\lambda$. By $\mathcal{H}_{A}(S,\lambda)$ we denote the space of transverse
cocycles for $\lambda$. For $A=\mathbb{R}$ this is a finite-dimensional vector
space. In particular, measured laminations and tangent vectors to
$\mathcal{ML}(\partial M)$ with the supports contained in $\lambda$ are
$\mathbb{R}$-valued transverse cocycles for $\lambda$. (For the description of
tangent vectors to $\mathcal{ML}(\partial M)$ we refer to [12, 68].) In [13]
Bonahon showed that one can associate to a pleated surface an
$\mathbb{R}/2\pi\mathbb{Z}$-valued transverse cocycle measuring the bending of
the surface. Moreover, he described a converse process of constructing a
pleated surface from a hyperbolic surface $(S,d)$ together with an
$\mathbb{R}/2\pi\mathbb{Z}$-valued transverse cocycle for $\lambda$. This
determines a map

$\mathcal{T}(S)\times\mathcal{H}_{\mathbb{R}}(S,\lambda)\rightarrow\mathcal{X}(S,G).$

(Here we associate to $\mathbb{R}$-valued transverse cocycles their
$\mathbb{R}/2\pi\mathbb{Z}$-reductions.) The space
$\mathcal{T}(S)\times\mathcal{H}_{\mathbb{R}}(S,\lambda)$ can be endowed with
a complex structure such that
$iT\mathcal{T}(S)=T\mathcal{H}_{\mathbb{R}}(S,\lambda)$, where
$T\mathcal{T}(S)$ and $T\mathcal{H}_{\mathbb{R}}(S,\lambda)$ are considered as
sub-bundles of the tangent bundle to
$\mathcal{T}(S)\times\mathcal{H}_{\mathbb{R}}(S,\lambda)$. Bonahon showed in
[13] that when $\lambda$ is maximal, the map above is a local biholomorphic
homeomorphism. We consider the product of these maps for different components
of $\partial M$

$\mathcal{J}_{\lambda}:\mathcal{T}(\partial
M)\times\mathcal{H}_{\mathbb{R}}(\partial
M,\lambda)\rightarrow\mathcal{X}(\partial M,G).$

In [14] Bonahon proved

######  Lemma 3.52.

Let $(d,\mu)\in\mathcal{T}(\partial M)\times\mathcal{ML}(\partial M)$,
$\dot{d}\in T_{d}\mathcal{T}(\partial M)$, $\dot{\mu}\in
T_{\mu}\mathcal{ML}(\partial M)$ and let $\lambda$ be a maximal geodesic
lamination containing the supports of $\mu$ and $\dot{\mu}$. Then

$(d\mathcal{J})_{(d,\mu)}(\dot{d},\dot{\mu})=(d\mathcal{J}_{\lambda})_{(d,\mu)}(\dot{d},\dot{\mu}).$

##  4 Infinitesimal rigidity

This section is devoted to a proof of the following infinitesimal rigidity
result.

######  Lemma 4.1.

Let either $g\in\mathcal{P}_{cp}(M,V)$ or $V=\emptyset$ and
$g\in\mathcal{P}_{p}(M,V)$. Then $d\mathcal{I}_{V}$ has the full rank at $g$.

We now show that it implies the corresponding local rigidity results

######  Corollary 4.2.

Let $g\in{\mathcal{P}}_{cp}(M,V)$. Then $\mathcal{I}_{V}$ is a local
homeomorphism at $g$.

###### Proof.

The manifolds $\mathcal{P}_{b}(M,V)$ and $\mathcal{D}_{c}(\partial M,V)$ have
the same dimension $3(n-k)$ and $\mathcal{P}_{cp}(M,V)$ is an open subset of
the former. Since $\mathcal{I}_{V}$ is differentiable over
$\mathcal{P}_{cp}(M,V)$ and the differential is everywhere non-degenerate, by
the inverse function theorem for differentiable maps (see, e.g., [74]) we get
that $\mathcal{I}_{V}$ is locally injective. By the Brower invariance of the
domain theorem, $\mathcal{I}_{V}$ is a local homeomorphism around $g$. ∎

######  Corollary 4.3.

Let $V=\emptyset$ and $g\in{\mathcal{P}}_{p}(M,V)$. Then $\mathcal{I}_{V}$ is
a local $C^{1}$-diffeomorphism at $g$.

###### Proof.

The proof is almost the same, but here we use that $\mathcal{I}_{V}$ is
$C^{1}$ over $\mathcal{P}_{b}(M,V)$. Hence, it is non-degenerate in a
neighborhood of $g$ and we can use the standard inverse function theorem to
make the desired conclusion. ∎

Particularly, we get the first claim of Part (1) of Corollary C. We can also
deduce Part (2).

######  Corollary 4.4.

Let $\overline{g}\in\mathcal{CH}(N)$ have polyhedral convex core. Then the
tangent map $d\mathcal{I}^{*}_{\overline{g}^{\prime}}$ is an isomorphism for
every $\overline{g}^{\prime}$ sufficiently close to $\overline{g}$.

###### Proof.

Due to the invariance of domain, it is enough to show that
$d\mathcal{I}^{*}_{\overline{g}^{\prime}}$ is injective. We will rely only on
that $d\mathcal{I}_{\overline{g}^{\prime}}$ is non-degenerate, which is true
in a neighborhood of $\overline{g}$ since $\mathcal{I}$ is $C^{1}$. Thus, it
is enough to prove the lemma for $\overline{g}^{\prime}=\overline{g}$. In what
follows we won’t be using the subscript ${}_{\overline{g}}$ for tangent maps
meaning that this is the case.

Denote $\mathcal{I}\times\mathcal{I}^{*}(\overline{g})$ by $(d,\mu)$. First
let us show that $d\mathcal{I}^{*}$ is non-degenerate. Suppose the converse,
that there exists $\dot{\overline{g}}\neq 0$ such that
$d\mathcal{I}^{*}(\dot{\overline{g}})=0$. Define

$P:=d\mathcal{J}(T_{d}\mathcal{T}(\partial M)\times\\{0\\})\subset
T_{\overline{g}}\mathcal{X}(\partial M,G).$

Due to Lemma 3.52, $P$ is a linear subspace and $iP$ is transversal to $P$. We
have $\dot{\overline{g}}\in P$. Let $\mathcal{J}^{-1}$ be a local inverse for
$\mathcal{J}$ sending $\overline{g}$ to $(d,\mu)$. Then
$d\mathcal{J}^{-1}(i\dot{\overline{g}})\in\\{0\\}\times
T_{\mu}\mathcal{ML}(\partial M)$. As $\mathcal{X}(\pi_{1}(M),G)$ is a complex
submanifold of $\mathcal{X}(\partial M,G)$, we have $i\dot{\overline{g}}\in
T_{\overline{g}}\mathcal{X}(\pi_{1}(M),G)$. We get
$d\mathcal{I}(i\dot{\overline{g}})=0$, which contradicts to Corollary 4.3.

Now suppose that for $\dot{\overline{g}}_{1},\dot{\overline{g}}_{2}\in
T_{\overline{g}}\mathcal{CH}(N)$ we have
$d\mathcal{I}^{*}(\dot{\overline{g}}_{1})=d\mathcal{I}^{*}(\dot{\overline{g}}_{2})=\dot{\mu}$.
Let $\lambda$ be a maximal geodesic lamination containing the supports of
$\mu$ and $\dot{\mu}$, and let $\mathcal{J}_{\lambda}^{-1}$ be a local inverse
for the map $\mathcal{J}_{\lambda}$ sending $\overline{g}$ to $(d,\mu)$. Due
to Lemma 3.52, we have
$d\mathcal{J}_{\lambda}^{-1}(\dot{\overline{g}}_{1})=(\dot{d}_{1},\dot{\mu})$,
$d\mathcal{J}_{\lambda}^{-1}(\dot{\overline{g}}_{2})=(\dot{d}_{2},\dot{\mu})$
for $\dot{d}_{1},\dot{d}_{2}\in T_{d}\mathcal{T}(\partial M)$. Hence,
$d\mathcal{J}_{\lambda}^{-1}(\dot{\overline{g}}_{1}-\dot{\overline{g}}_{2})=(\dot{d}_{1}-\dot{d}_{2},0)$.
Due to Lemma 3.52, we get
$d\mathcal{I}^{*}(\dot{\overline{g}}_{1}-\dot{\overline{g}}_{2})=0$, which
contradicts to our first claim. ∎

In order to pass from local rigidity to global rigidity we will need the
following

######  Proposition 4.5.

Let $g_{0}\in\mathcal{P}_{b}(M,V)$ be such that $\mathcal{I}_{V}$ is a local
homeomorphism at $g_{0}$. Let $V\subseteq W$,
$g\in\overline{\mathcal{P}}_{b}(M,W)$ be a lift of $g_{0}$ and let
$d:=\mathcal{I}_{W}(g)$. Then $g$ is isolated in $\mathcal{I}_{W}^{-1}(d)$.

###### Proof.

If $V=W$, then the claim is straightforward, so suppose that $V\subsetneq W$.
By $\overline{\mathcal{P}}_{b}(M,V,W)$ we denote the subset of
$\overline{\mathcal{P}}_{b}(M,W)$ of metrics $g^{\prime}$ with
$V(g^{\prime})=V$, by $\overline{\mathcal{D}}_{c}(M,V,W)$ we denote the subset
of $\overline{\mathcal{D}}_{c}(M,W)$ of metrics $d$ with $V(d)=V$. Consider
the commutative diagram

${\overline{\mathcal{P}}_{b}(M,V,W)}$${\overline{\mathcal{D}}_{c}(M,V,W)}$${{\mathcal{P}}_{cp}(M,V)}$${{\mathcal{D}}_{cp}(M,V)}$$\scriptstyle{\mathcal{I}_{W}}$$\scriptstyle{\mathcal{I}_{V}}$

Since the bottom arrow is a local homeomorphism around $g_{0}$, which is the
vertical image of $g$, and by definition $\mathcal{I}_{W}$ is a homeomorphism
when restricted to a fiber of the vertical map, it is easy to deduce that the
restriction of $\mathcal{I}_{W}$ to $\overline{\mathcal{P}}_{b}(M,V,W)$ is
locally injective. Because
$\mathcal{I}_{W}^{-1}(d)\subset\overline{\mathcal{P}}_{b}(M,V,W)$, the claim
follows. ∎

###  4.1 Infinitesimal substitution

Let $g\in\mathcal{P}_{p}(M,V)$ and $\Psi$ be the set of all edges
$\psi\subset\partial M(g)$ that are simple closed geodesics (note that there
are finitely many of them). We orient all these geodesics arbitrarily. Let
$V^{\prime}$ be a finite set in bijection with $\Psi$. Consider a smooth
section $\sigma$ of the bundle
$\mathcal{CH}(N,V^{\prime})\rightarrow\mathcal{CH}(N)$ such that for every
$\overline{g}\in\mathcal{CH}(N)$ and every $v\in V^{\prime}$ the point marked
by $v$ in $N(\overline{g})$ belongs to the closed geodesic homotopic to
$\psi$, where $\psi$ is the element of $\Psi$ corresponding to $v$. It is not
hard to see that such a section indeed can be chosen smooth since for every
$\gamma\in\pi_{1}(M)$ the axis fixed by $\rho(\gamma)$ varies smoothly as
$\rho$ varies in $\mathcal{R}(\pi_{1}(M),G)$. By $W$ we denote the union
$V\cup V^{\prime}$.

By Lemmas 3.23 and 3.24 there exists a finite cover $M^{c}(g^{c})$ of $M(g)$
admitting a geodesic triangulation $\mathcal{T}$ with vertices in $W^{c}$,
where $W^{c}$ is the preimage of $W$. Taking a cover provides a smooth
immersion of $\mathcal{P}_{b}(M,V)$ to $\mathcal{P}_{b}(M^{c},V^{c})$, and if
we take $\dot{g}\in T_{g}\mathcal{P}_{b}(M,V)$ and consider the corresponding
vector $\dot{g}^{c}\in T_{g^{c}}\mathcal{P}_{b}(M^{c},V^{c})$, then
$d\mathcal{I}_{V}(\dot{g})=0$ is equivalent to
$d\mathcal{I}_{V^{c}}(\dot{g}^{c})=0$. Hence, if we show that
$d\mathcal{I}_{V^{c}}(\dot{g}^{c})$ implies $\dot{g}^{c}=0$, we also obtain
that $d\mathcal{I}_{V}(\dot{g})$ implies $\dot{g}=0$. We extend the section
$\sigma$ to a smooth section $\sigma^{c}$ of
$\mathcal{CH}(N^{c},W^{c})\rightarrow\mathcal{CH}(N^{c})$ with the same
property that every point marked by an element of $W^{c}\backslash V^{c}$
belongs to the corresponding geodesic. Until the end of this section we will
work with $M^{c}(g^{c})$, so to simplify the reading we now omit to write c
and assume that $M(g)$ has a large triangulation $\mathcal{T}$ with vertices
in $W$ (now there can be several new vertices at each $\psi\in\Psi$, but we
continue to denote by $V^{\prime}$ the difference $W\backslash V$). We write
$E(\mathcal{T})$ for the set of edges of $\mathcal{T}$ and divide them in two
groups: $E_{in}(\mathcal{T})$ are the interior edges and $E_{bd}(\mathcal{T})$
are the boundary edges.

We take the direct product $\mathcal{P}_{b}(M,V)\times\mathbb{R}^{V^{\prime}}$
and immerse it in $\mathcal{CH}(N,W)$ as follows. Denote the elements of of
$\mathbb{R}^{V^{\prime}}$ by $z$ and their components by $z_{v}$ for $v\in
V^{\prime}$. For $\overline{g}\in\mathcal{CH}(N)$ denote
$\sigma(\overline{g})\in\mathcal{CH}(N,V^{\prime})$ by
$[g_{\sigma(\overline{g})},f_{\sigma(\overline{g})}].$ For every
$([g,f],z)\in\mathcal{P}_{b}(M,V)\times\mathbb{R}^{V^{\prime}}$ we map it to
$[g^{\prime},f^{\prime}]\in\mathcal{CH}(N,W)$ such that
$[g^{\prime},f^{\prime}]$ is a lift of $[g,f]$ and for every $v\in V^{\prime}$
the marked point $f^{\prime}(v)$ is the point on the corresponding geodesic at
oriented distance $t_{v}$ from $f_{\sigma(\overline{g})}(v)$. We identify
$\mathcal{P}_{b}(M,V)\times\mathbb{R}^{V^{\prime}}$ with its image and write
for simplicity its elements as $(g,z)$. For each
$(g,z)\in\mathcal{P}_{b}(M,V)\times\mathbb{R}^{V^{\prime}}$ and for each edge
$e$ of $\mathcal{T}$ we consider the length of the unique geodesic segment in
$M(g)$ with the same endpoints as $e$ and in the same homotopy class (with
fixed endpoints). This defines a smooth map

$l:\mathcal{P}_{b}(M,V)\times\mathbb{R}^{V^{\prime}}\rightarrow\mathbb{R}^{E(\mathcal{T})}.$

We now return to our initial metric $g$ and consider the differential $dl$ of
the above map at the point $(g,0)$.

######  Lemma 4.6.

If there exists $\dot{z}\in T_{0}\mathbb{R}^{V^{\prime}}$ such that
$dl(\dot{g},\dot{z})=0$, then $\dot{g}=0$.

###### Proof.

It is easy to see that $dl(\dot{g},\dot{z})=0$ implies that the induced
deformation of $\overline{g}$ is trivial. Indeed, pick arbitrary $v\in W$.
Every element $\gamma\in\pi_{1}(M)$ can be represented by a simplicial loop
based at $v$ in the 1-skeleton of $\mathcal{T}$. Pick a cyclic sequence of
tetrahedra along this loop based on a tetrahedron $T$ containing $v$ so that
any two subsequent tetrahedra have a 2-face in common, and develop this
sequence in $\mathbb{H}^{3}$. Let
$\widetilde{v}_{1},\widetilde{v}_{2}\in\mathbb{H}^{3}$ be the lifts of $v$
that are the endpoints of the lifted loop, $\widetilde{T}_{1}$ and
$\widetilde{T}_{2}$ be the starting and the ending lifts of $T$ so that
$\widetilde{T}_{i}$ contains $\widetilde{v}_{i}$. The image
$\rho_{\overline{g}}(\gamma)$ can be reconstructed by its action on any point
in $\mathbb{H}^{3}$ together with a basis in the tangent space of this point.
Take $\widetilde{v}_{1}$ and the basis given by the edges of
$\widetilde{T}_{1}$. As $dl(\dot{g},\dot{z})=0$, we get that
$\widetilde{v}_{2}$ and $\widetilde{T}_{2}$ infinitesimally do not move and so
the induced change on $\rho_{\overline{g}}(\gamma)$ is trivial.

Hence, $\dot{g}$ is vertical with respect to the fibration
$\mathcal{CH}(N,V)\rightarrow\mathcal{CH}(N)$. Thus, it represents an
infinitesimal change of the locations of the points of $V$. Consider the
universal cover $\widetilde{M}(g)\subset\mathbb{H}^{3}$. Then $\dot{g}$ can be
represented by a vector field on $\widetilde{V}=\widetilde{V}(g)$ equivariant
with respect to $\rho_{\overline{g}}$. We extend this vector field to the full
preimage $\widetilde{W}$ of $W$ using $\dot{z}$. As $dl(\dot{g},\dot{z})=0$,
it extends to a Killing field on each 2-face of $\mathcal{T}$ belonging to the
boundary of $\widetilde{M}(g)$ such that the Killing fields coincide on the
edges. Moreover, as $dl(\dot{g},\dot{z})=0$, this vector field preserves the
link of each vertex, thus it is the same Killing field on every connected
component of $\partial\widetilde{M}(g)$. Because it needs to be equivariant,
it must be zero. Hence, $\dot{g}=0$. ∎

We now show

######  Lemma 4.7.

Let $g\in\mathcal{P}_{cp}(M,V)$. If $d\mathcal{I}_{V}(\dot{g})=0$, then there
exists $\dot{z}\in T_{0}\mathbb{R}^{V^{\prime}}$ such that for each $e\in
E_{bd}(\mathcal{T})$ we have $dl_{e}(\dot{g},\dot{z})=0$

###### Proof.

If both endpoints of $e$ are in $V$ and $e$ is an edge of $g$, then the
statement is clearly true for any $\dot{z}$. If $e$ is not an edge and belongs
to a simply connected face, the statement is again true for any $\dot{z}$,
which follows from Claim 3.34.1 that the differential of the length of the
realization of $e$ in the intrinsic metric of the boundary coincides with the
differential of the extrinsic distance between its endpoints.

Consider a simple closed geodesic $\psi\in\partial M(g)$, let
$\gamma\in\pi_{1}(M)$ be a generating element for a subgroup of $\pi_{1}(M)$
corresponding to $\psi$, let $F_{1}$ and $F_{2}$ be two faces of $\partial
M(g)$ adjacent to $\psi$ (possibly degenerating to one) and let $\Sigma$ be
their union. Develop the universal cover $\widetilde{\Sigma}$ of $\Sigma$ to
$\mathbb{H}^{3}$ and consider the coordinate system from Subsection 3.4.2. By
$\widetilde{\psi}$ denote the development of $\psi$. Let $g_{t}$,
$t\in[0,\varepsilon]$ for some small $\varepsilon>0$, be a path tangent to
$\dot{g}$. We consider a corresponding path of
$\rho_{\overline{g}_{t}}(\gamma)$ and the trajectories of the vertices
adjacent to $\Sigma$ and develop them in our coordinate system. By
$V^{\prime}_{\psi}$ denote the subset of $V^{\prime}$ belonging to $\psi$. For
each $v\in V^{\prime}_{\psi}$ choose a lift $\widetilde{v}\in\widetilde{\psi}$
and by $T_{\widetilde{\psi}}$ denote the subspace of $\Pi_{v\in
V^{\prime}_{\psi}}T_{\widetilde{v}}\mathbb{H}^{3}$ consisting of tuples of
tangent vectors parallel to $\widetilde{\psi}$. The section $\sigma$
determines trajectories for points $\widetilde{v}$. Considering the
derivatives at $t=0$ we obtain an element of $T_{\widetilde{\psi}}$. It is
easy to see that $T_{0}\mathbb{R}^{V^{\prime}_{\psi}}$ is in one-to-one
correspondence with $T_{\widetilde{\psi}}$ (more exactly, there is a natural
non-degenerate affine map between them). Let $\dot{z}_{\psi}\in
T_{0}\mathbb{R}^{V^{\prime}_{\psi}}$ be the preimage of the zero vector.

The deformation $\dot{g}$ induces the deformation of the initial coordinates
such that the deformation of the induced metric on $\Sigma$ is zero. Corollary
3.36 shows that if we set all $\dot{\phi}^{j}_{i}=0$ and $\dot{\alpha}=0$,
then the obtained deformation of $\Sigma$ is still zero. But this implies that
all $\dot{x}^{j}_{i}=0$, all $\dot{y}^{j}_{i}=0$ and $\dot{a}=0$. Together
with zero vectors chosen at each $\widetilde{v}$, this induces zero
infinitesimal deformation on every geodesic segment of $\mathcal{T}$ in
$\Sigma$. Let $\dot{z}\in T_{0}\mathbb{R}^{V^{\prime}}$ be the vector obtained
from gluing all $\dot{z}_{\psi}$. We have $dl_{e}(\dot{g},\dot{z})=0$ for all
$e\in E_{bd}(\mathcal{T})$. ∎

######  Lemma 4.8.

Let $V=\emptyset$ and $g\in\mathcal{P}_{p}(M,V)$. If
$d\mathcal{I}_{V}(\dot{g})=0$, then there exists $\dot{z}\in
T_{0}\mathbb{R}^{V^{\prime}}$ such that for each $e\in E_{bd}(\mathcal{T})$ we
have $dl_{e}(\dot{g},\dot{z})=0$.

###### Proof.

As described in Section 3.4.4, there is an immersion from
$\mathcal{P}_{b}(M,V)$ to the smooth part of $\mathcal{X}(\partial M,G)$ and
we identify $g$ with its image. Let $S_{i}$, $i=1\ldots r$, be the components
of $\partial M$. By $V^{\prime}_{i}$ we denote the subset of $V^{\prime}$
corresponding to geodesics belonging to $S_{i}$ in $M(g)$. As in Section
3.1.2, we define the fibrations

$\mathcal{X}(\pi_{1}(S_{i}),G,V^{\prime}_{i})\rightarrow\mathcal{X}(\pi_{1}(S_{i}),G)$

and by $\mathcal{X}(\partial M,G,V^{\prime})$ we denote the product
$\prod_{i=1}^{r}\mathcal{X}(\pi_{1}(S_{i}),G,V^{\prime}_{i})$, which fibers
over $\mathcal{X}(\partial M,G)$. Let $U_{0}$ be a small enough neighborhood
of $g$ in $\mathcal{X}(\partial M,G)$ and $U^{\prime}_{0}$ be its preimage in
$\mathcal{X}(\partial M,G,V^{\prime})$. We consider a smooth section
$\sigma^{\prime}$ of the fibration $U^{\prime}_{0}\rightarrow U_{0}$ extending
the section $\sigma$ such that again the marked points belong to the
respective geodesics. We consider the product
$U_{0}\times\mathbb{R}^{V^{\prime}}$ and use $\sigma^{\prime}$ to immerse it
in $U^{\prime}_{0}$ in the same way as above we immersed
$\mathcal{P}_{b}(M,V)\times\mathbb{R}^{V^{\prime}}$ in $\mathcal{CH}(N,W)$. We
also define a map
$l_{bd}:U_{0}\times\mathbb{R}^{V^{\prime}}\rightarrow\mathbb{R}^{E_{bd}(\mathcal{T})}$
in the same way as we defined it above, though now we care only about the
boundary edges of $\mathcal{T}$.

We now use the maps $\mathcal{J}$ and $\mathcal{J}_{\lambda}$ from Section
3.4.4. Let $\mathcal{J}^{-1}$ be a local inverse to $\mathcal{J}$ defined on
$U_{0}$, provided that $U_{0}$ is sufficiently small, and coinciding with
$\mathcal{I}\times\mathcal{I}^{*}$ on $U_{0}\cap\mathcal{X}(\pi_{1}(M),G)$.
Define $(d,\mu):=\mathcal{J}^{-1}(g)\in\mathcal{T}(\partial
M)\times\mathcal{ML}(\partial M)$. As $d\mathcal{J}$ at $(d,\mu)$ is an
isomorphism of the tangent spaces and $d\mathcal{I}_{V}(\dot{g})=0$, we have

$d\mathcal{J}^{-1}(\dot{g})\in\\{0\\}\times T_{\mu}\mathcal{ML}(\partial M).$

We are going to prove that for every $\dot{\mu}\in
T_{\mu}\mathcal{ML}(\partial M)$ there exists $\dot{z}\in
T_{0}\mathbb{R}^{V^{\prime}}$ such that
$dl_{bd}(d\mathcal{J}(0,\dot{\mu}),\dot{z})=0$. This will imply the claim of
the lemma.

We first suppose that the support of $\dot{\mu}$ is rational. Let $\lambda$ be
a maximal rational geodesic lamination containing the supports of $\mu$ and
$\dot{\mu}$. Consider a local inverse $\mathcal{J}_{\lambda}^{-1}$ to the map
$\mathcal{J}_{\lambda}$ defined over $U_{0}$ and sending $g$ to $(d,\mu)$. We
identify the vector space $\mathcal{H}_{\mathbb{R}}(\partial M,\lambda)$ with
its tangent space and consider a curve
$\mu+t\dot{\mu}\in\mathcal{H}_{\mathbb{R}}(\partial M,\lambda)$ for
$t\in[0,\varepsilon]$ and some $\varepsilon>0$. Take a component $S_{i}$
endowed with the hyperbolic metric $d$ induced from $M(g)$, we consider it now
as abstract hyperbolic surface triangulated by the restriction of
$\mathcal{T}$ to $S_{i}$.

Look at the pleated surfaces $(\widetilde{\iota}_{t},\rho_{t})$ in
$\mathbb{H}^{3}$ obtained from $(S_{i},d)$ by pairs $(d,\mu+t\dot{\mu})$. The
starting surface is a boundary component of
$\partial\widetilde{C}(\overline{g})$. Let us fix a base point on it, together
with a basis in the tangent space to the base point. We can choose
$(\widetilde{\iota}_{t},\rho_{t})$ so that they fix the base point and the
chosen basis. Since $\lambda$ is rational, it is easy to understand what
happens with it: the surface starts to bend along the geodesics in
$\widetilde{\iota}_{0}(\widetilde{\lambda})$, which is a discrete set, with
the speeds of the bending angles prescribed by $\dot{\mu}$.

Pick an edge $e$ of $\mathcal{T}$ in $S_{i}$, let $\widetilde{e}$ be its lift
to $(\widetilde{S}_{i},d)$ and $p_{0},\ldots,p_{s}$ be the intersection points
of $\widetilde{e}$ with $\widetilde{\lambda}$ including the endpoints. The
curves $\widetilde{\iota}_{t}(p_{j})$ are differentiable at zero. We claim
that the derivative of the hyperbolic distance between
$\widetilde{\iota}_{t}(p_{1})$ and $\widetilde{\iota}_{t}(p_{s})$ is zero at
$t=0$. Indeed, let $x_{j}$ be the tangent vector at
$\widetilde{\iota}_{0}(p_{j})$ to its trajectory, and $x^{k}_{j}$ be the
result of the parallel translation of $x_{j}$ to
$\widetilde{\iota}_{0}(p_{k})$ along $\widetilde{\iota}_{0}(\widetilde{e})$
with respect to the metric connection of $\mathbb{H}^{3}$. Since the length of
each segment between $\widetilde{\iota}_{0}(p_{j})$ and
$\widetilde{\iota}_{0}(p_{j+1})$ does not change infinitesimally, we have that
$x^{j+1}_{j}-x_{j}$ is orthogonal to $\widetilde{\iota}_{0}(\widetilde{e})$.
Hence $x^{s}_{1}-x_{s}$ is orthogonal to
$\widetilde{\iota}_{0}(\widetilde{e})$ and the claim follows.

The trajectories $\widetilde{\iota}_{t}(p_{1})$ and
$\widetilde{\iota}_{t}(p_{s})$ determine tangent vectors
$\dot{z}_{v_{1}},\dot{z}_{v_{2}}\in T_{0}\mathbb{R}^{V^{\prime}}$ in the
components corresponding to the endpoints $v_{1}$ and $v_{2}$ of $e$. The
obtained vectors are independent on the choice of $\widetilde{e}$ or $e$. By
determining the tangent vector for every element of $V^{\prime}$ we obtain a
tangent vector $\dot{z}\in T_{0}\mathbb{R}^{V^{\prime}}$. For each edge $e\in
E_{bd}(\mathcal{T})$ the deformation
$dl_{bd,e}(d\mathcal{J}_{\lambda}(0,\dot{\mu}),\dot{z})$ coincides with the
deformation of the distance between $\widetilde{\iota}_{0}(p_{1})$ and
$\widetilde{\iota}_{0}(p_{s})$ in the construction above. Hence,
$dl_{bd}(d\mathcal{J}_{\lambda}(0,\dot{\mu}),\dot{z})=0$. By Lemma 3.52 we get
$dl_{bd}(d\mathcal{J}(0,\dot{\mu}),\dot{z})=0$ as desired.

If the support of $\dot{\mu}$ is irrational, we can approximate it by tangent
vectors $\dot{\mu}_{j}$ whose supports are rational. By the previous argument,
for every $j$ there exists $z_{j}\in T_{0}\mathbb{R}^{V^{\prime}}$ such that
$dl_{bd}(d\mathcal{J}(0,\dot{\mu}_{j}),\dot{z}_{j})=0$. This means that for
every $\dot{z}\in T_{0}\mathbb{R}^{V^{\prime}}$ the vector
$dl_{bd}(d\mathcal{J}(0,\dot{\mu}_{j}),\dot{z})$ belongs to the subspace
$dl_{bd}(\\{0\\}\times T_{0}\mathbb{R}^{V^{\prime}})$. By continuity of the
tangent map, the same is true for
$dl_{bd}(d\mathcal{J}(0,\dot{\mu}),\dot{z})$. Hence, there exists $\dot{z}\in
T_{0}\mathbb{R}^{V^{\prime}}$ such that
$dl_{bd}(d\mathcal{J}(0,\dot{\mu}),\dot{z})=0$. This finishes the proof. ∎

There is a neighborhood of $l=l(g,0)$ in $\mathbb{R}^{E(\mathcal{T})}$ where
the edge lengths induce a hyperbolic cone-3-manifold structure on $M$
triangulated with $\mathcal{T}$. Recall that for each edge $e\in
E(\mathcal{T})$ by $\nu_{e}$ we denote the total dihedral angle of $e$.
Clearly, every $\dot{g}\in T_{g}\mathcal{P}(M,V)$ and $\dot{z}\in
T_{0}\mathbb{R}^{V^{\prime}}$ induce a deformation
$\dot{l}=dl(\dot{g},\dot{z})$ such that for every $e\in E_{in}(\mathcal{T})$
the infinitesimal change $d\nu_{e}(\dot{l},\dot{z})=0$. Now our aim is to show

######  Lemma 4.9.

Let $\dot{l}\in T_{l}\mathbb{R}^{E(\mathcal{T})}$ be such that for each $e\in
E_{bd}(\mathcal{T})$ we have $\dot{l}_{e}=0$ and for each $e\in
E_{in}(\mathcal{T})$ we have $d\nu_{e}(\dot{l})=0$. Then $\dot{l}=0$.

It is clear that the combination of Lemmas 4.6–4.9 imply Lemma 4.1. We prove
Lemma 4.9 in the next subsections.

###  4.2 Affine automorphic vector fields

Consider the Klein model of $\mathbb{H}^{3}$ and denote by $AF$ the set of
affine vector fields restricted to $\mathbb{H}^{3}$. For every non-degenerate
tetrahedron $T$ with a vertex set $V(T)$ there is a canonical isomorphism

$J_{T}:\prod_{v\in V(T)}T_{v}\mathbb{H}^{3}\rightarrow AF,$

mapping four tangent vectors at each vertex of $T$ to the unique affine vector
field coinciding with these vectors at $V(T)$. This isomorphism is preserved
under isometries of $\mathbb{H}^{3}$. Note that $AF$ has dimension 12 and
contains the 6-dimensional subspace $KF$ of Killing fields.

We look at the action of $AF$ on the edge lengths of $T$. Consider the 6-tuple
of its edge lengths as an element $l\in\mathbb{R}^{E(T)}$. There is a natural
linear map $AF\rightarrow T_{l}\mathbb{R}^{E(T)}$ sending a vector field to
the induced infinitesimal change of the edge lengths. As hyperbolic tetrahedra
are infinitesimally rigid, the kernel of this map exactly equals to $KF$,
hence the map is surjective and produces an isomorphism

$T_{l}\mathbb{R}^{E(T)}\cong AF/KF.$ (4.1)

Consider now a (possibly non-convex) polyhedron $P\subset\mathbb{H}^{3}$ with
a geodesic triangulation $\mathcal{T}$ (without new vertices). Denote all the
tetrahedra of $\mathcal{T}$ by $T_{1},\ldots,T_{r}$. Pick an open cover of $P$
by sets $U_{1},\ldots,U_{r}$ such that for each $i$ we have $T_{i}\subset
U_{i}$ and $U_{i}$ does not contain vertices of $P$ except the vertices of
$T_{i}$. Choose a partition of unity over $P$ with respect to $U_{i}$, i.e.,
smooth functions $\lambda_{i}:\mathbb{H}^{3}\rightarrow[0,1]$, $i=1,\ldots,r$,
such that the support of $\lambda_{i}$ is in $U_{i}$ and for every $p\in P$ we
have $\sum_{i}\lambda_{i}(p)=1$. We denote the set of pairs
$(U_{i},\lambda_{i})$ by $\mathcal{U}$. By $AF(P,\mathcal{U})$ we denote the
space of vector fields of the form $\sum_{i}\lambda_{i}X_{i}$, where $X_{i}\in
AF$ and if $T_{i}$ and $T_{j}$ are two tetrahedra adjacent by a 2-face, then
the restriction of $X_{i}$ to the plane containing this face coincides with
the restriction of $X_{j}$. We call such fields _affine fields on $P$ with
respect to $\mathcal{U}$_. The meaning of this construction is that we
construct a vector field on $P$ by assigning an affine field $X_{i}$ to each
tetrahedron $T_{i}$ so that the assigned fields coincide on the common faces.
However, the resulting field will be continuous, but not smooth, so we smooth
it out with the help of a partition of unity. We do it in a way that the
values of the smoothed vector field does not change at the vertices of $P$.

If $V$ is the set of vertices of $P$, there is a natural isomorphism

$J_{P}:\prod_{v\in V}T_{v}\mathbb{H}^{3}\rightarrow AF(P,\mathcal{U})$

defined by

$J_{P}(Y)=\sum_{i}\lambda_{i}J_{T_{i}}(Y|_{V(T_{i})}),$

where $Y\in\prod_{v\in V}T_{v}\mathbb{H}^{3}$ and $V(T_{i})$ is the vertex set
of $T_{i}$. Note that for every $v\in V$ and $Y\in\prod_{v\in
V}T_{v}\mathbb{H}^{3}$ we have $J_{P}(Y)|_{v}=Y|_{v}$. There is the
6-dimensional subspace $KF(P,\mathcal{U})\subset AF(P,\mathcal{U})$ consisting
of the fields $\sum_{i}\lambda_{i}X_{i}$ such that all $X_{i}$ are equal to
the same Killing field $X$, so the restriction of an element of
$KF(P,\mathcal{U})$ to $P$ is exactly $X$.

Similarly to the consideration above, we regard the tuple of edge lengths of
$\mathcal{T}$ as a tuple $l\in\mathbb{R}^{E(\mathcal{T})}$ and look at the
natural map $AF(P,\mathcal{U})\rightarrow T_{l}\mathbb{R}^{E(\mathcal{T})}$.
Again, its kernel is equal $KF(P,\mathcal{U})$. We now consider cone
structures on $P$ induced by deformations of $l$ in
$\mathbb{R}^{E(\mathcal{T})}$. Denote the set of the interior edges of
$\mathcal{T}$ by $E_{in}(\mathcal{T})$. For every $e\in E_{in}(\mathcal{T})$
the total dihedral angle $\nu_{e}$ is an analytic function in a neighborhood
of $l$ in $\mathbb{R}^{E(\mathcal{T})}$. We denote

$T^{P}_{l}\mathbb{R}^{E(\mathcal{T})}=\left\\{\bigcap_{e\in
E_{in}(\mathcal{T})}\ker d\nu_{e}\right\\}.$

Clearly, the image of $AF(P,\mathcal{U})$ belongs to
$T^{P}_{l}\mathbb{R}^{E(\mathcal{T})}$. We now prove

######  Lemma 4.10.

$T^{P}_{l}\mathbb{R}^{E(\mathcal{T})}\cong
AF(P,\mathcal{T})/KF(P,\mathcal{U}).$

###### Proof.

We need to show that $T^{P}_{l}\mathbb{R}^{E(\mathcal{T})}$ is the image of
our map. Pick $\dot{l}\in T_{l}^{P}\mathbb{R}^{E(\mathcal{T})}$. Start from an
arbitrary tetrahedron $T_{1}$ of $\mathcal{T}$. As described above, there is
always an affine field inducing the restriction of $\dot{l}$ to $T_{1}$, we
pick one of them. Now let $T_{2}$ be a tetrahedron adjacent to $T_{1}$ by a
2-face. We try to find a tangent vector at the fourth vertex $v$ of $T_{2}$
such that the induced infinitesimal lengths-change is the restriction of
$\dot{l}$ to $T_{2}$. Consider the hyperboloid model of $\mathbb{H}^{3}$ in
$\mathbb{R}^{3,1}$. Denote the other three vertices of $T_{2}$ by $v_{i}$,
$i=1,\ldots,3$, and denote the edge from $v_{i}$ to $v$ by $e_{i}$. We have
the system of equations

$\langle\dot{v},v_{i}\rangle+\langle\dot{v}_{i},v\rangle=-\dot{l}_{e_{i}}\cosh
l_{e_{i}},~{}~{}~{}~{}~{}i=1,\ldots,3.$

Here we identify $v_{i}$, $v$ with their radius-vectors in $\mathbb{R}^{3,1}$,
$\dot{v}_{i}\in T_{v_{i}}\mathbb{H}^{3}$ are the chosen tangent vectors to
$v_{i}$ and $\dot{v}$ is a vector that we want to find in
$T_{v}\mathbb{H}^{3}$. Suppose that there is no solution. Then from linearity
there exists a non-trivial solution to the system

$\langle\dot{v},v_{i}\rangle=0,~{}~{}~{}~{}~{}i=1,\ldots,3.$

Due to the infinitesimal rigidity of tetrahedra, such a solution $\dot{v}\in
T_{v}\mathbb{H}^{3}$ together with the assignment of zero vectors in
$T_{v_{i}}\mathbb{H}^{3}$ is the restriction of a global Killing field of
$\mathbb{H}^{3}$, but clearly such a non-zero Killing field does not exist.

We continue this procedure. It remains to check what happens when we close a
cycle of tetrahedra around an interior edge $e$ of $\mathcal{T}$. Let $T$ be
the last tetrahedron to close the chain and $e^{\prime}$ be its last edge. The
previous choices of tangent vectors at the vertices of $T$ automatically
induce an affine vector field on $T$, which induces an infinitesimal change of
the length $l_{e^{\prime}}$. We need to see that it coincides with the one
from $\dot{l}$. But this follows from $\frac{\partial\nu_{e}}{\partial
l_{e^{\prime}}}\neq 0$, as this implies that for any assignment of
infinitesimal lengths-changes to all edges of tetrahedra adjacent to $e$
except $e^{\prime}$ there exists a unique choice of $\dot{l}_{e^{\prime}}$
such that $\dot{\nu}_{e}=0$. This finishes the proof. ∎

Now let $(K,\Sigma,\mathcal{T})$ be a closed triangulated hyperbolic
cone-3-manifold, $L$ be its subset obtained by deleting from $K$ the singular
locus $\Sigma$. By $E_{0}(\mathcal{T})$ denote the set of edges not belonging
to $\Sigma$, so with $\nu_{e}=2\pi$. Let $T_{1},\ldots,T_{r}$ be the
tetrahedra of $\mathcal{T}$. We need to show that there exists some special
open cover of $L$.

######  Lemma 4.11.

Let $\mathcal{T}$ be large. There exists an open cover of $L$ by sets
$U_{1},\ldots,U_{r}$ such that

  * •

each $U_{i}$ is simply-connected;

  * •

$U_{i}\supset(T_{i}\cap L)$;

  * •

the closure of each $U_{i}$ in $K$ does not contain vertices of $\mathcal{T}$
except the vertices of $T_{i}$.

###### Proof.

Pick a tetrahedron $T_{i}$ and develop it to $\mathbb{H}^{3}$, we denote the
image by $\widehat{T}$ and the set of its vertices by $\widehat{V}$. Let us
call the edges of $\widehat{T}$ coming from an edge in $E_{0}(\mathcal{T})$
_important_. For each important edge of $\widehat{T}$ develop all tetrahedra
adjacent to the preimage of this edge. For every 2-face of $\widehat{T}$ not
adjacent to important edges develop also the other tetrahedron adjacent to its
preimage. Enumerate the obtained tetrahedra except $\widehat{T}$ by
$\widehat{T}_{1},\ldots,\widehat{T}_{s}$ and denote their set by
$\widehat{\mathcal{T}}$. Note that as $\mathcal{T}$ is large, among the
tetrahedra of $\widehat{\mathcal{T}}$ there is no other copy of $T_{i}$, but
some other tetrahedra of $\mathcal{T}$ could be developed more than once.
Denote the barycenter of $\widehat{T}_{j}$ by $p_{j}$. Also for each 2-face of
$\widehat{T}_{j}$ that is not a 2-face of $\widehat{T}$, but contains an
important edge, mark the barycenter of this face and denote the set of marked
points by $\widehat{S}$. In each $\widehat{T}_{j}$ by $\widehat{U}_{j}$ we
denote the interior of the convex hull of $p_{j}$,
$\widehat{S}\cap\widehat{T}_{j}$ and $\widehat{V}\cap T_{j}$. We are now
constructing the set $\widehat{U}$ as follows

  * •

add all $\widehat{U}_{j}$ to $\widehat{U}$;

  * •

for each face $F$ that is not a face of $\widehat{T}$, but is adjacent to an
important edge, we add to $\widehat{U}$ the relative interior of the triangle
formed by $\widehat{S}\cap F$ and $\widehat{V}\cap F$;

  * •

we add to $\widehat{U}$ the relative interior of every important edge;

  * •

we add to $\widehat{U}$ the interior of $\widehat{T}$.

By $U_{i}\subset L$ we denote the preimage of the set $\widehat{U}$. It is not
hard to see from the construction that $U_{i}$ is simply-connected and open in
$L$. Doing this for every $T_{i}$ we obtain an open cover of $L$ satisfying
the desired properties. ∎

Take an open cover $U_{1},\ldots,U_{r}$ of $L$ from Lemma 4.11 and fix a
partition of unity of $L$ with respect to this cover. Denote the corresponding
partition functions by $\lambda_{i}:L\rightarrow[0,1]$. We denote the set of
pairs $(U_{i},\lambda_{i})$ by $\mathcal{U}$, but we will also abuse the
notation and sometimes will say just $U_{i}\in\mathcal{U}$. Consider a
developing map $dev:\widetilde{L}\rightarrow\mathbb{H}^{3}$. Let
$\widetilde{\mathcal{U}}$ be the partition of unity of $\widetilde{L}$ that is
the lift of $\mathcal{U}$. By $I$ we denote a countable set indexing elements
of $\mathcal{U}$. By $AF(\widetilde{L},\widetilde{\mathcal{U}})$ we denote the
space of vector fields on $\widetilde{L}$ of the form $\sum_{i\in
I}\widetilde{\lambda}_{i}X_{i}$ where $\widetilde{\lambda}_{i}$ are the
partition functions and $X_{i}$ are pull-backs on $\widetilde{L}$ of affine
vector fields on $\mathbb{H}^{3}$ via $dev$ such that for every two tetrahedra
$\widetilde{T}_{i}$ and $\widetilde{T}_{j}$ in $\widetilde{L}$ adjacent by a
2-face the fields $X_{i}$ and $X_{j}$ coincide on this face. We call such
fields _affine vector fields on $\widetilde{L}$ with respect to
$\mathcal{U}$_. It is easy to see that
$AF(\widetilde{L},\widetilde{\mathcal{U}})$ is independent on the choice of
$dev$. By $KF(\widetilde{L})$ we denote the space of the Killing fields on
$\widetilde{L}$. Note that all vector fields in $KF(\widetilde{L})$ are pull-
backs of Killing fields on $\mathbb{H}^{3}$ via $dev$ and
$KF(\widetilde{L})\subset AF(\widetilde{L},\mathcal{U})$ as any such field can
be represented as $\sum_{i\in I}\widetilde{\lambda}_{i}X_{i}$ where all
$X_{i}$ are the same Killing field on $\widetilde{L}$.

As previously, we consider the tuple of edge lengths of $\mathcal{T}$ as
$l\in\mathbb{R}^{E(\mathcal{T})}$ and by
$T^{L}_{l}\mathbb{R}^{E(\mathcal{T})}$ we denote the intersection of kernels
of $d\nu_{e}$ for all edges $e\in E_{0}(\mathcal{T})$. We want to establish an
isomorphism of $T^{L}_{l}\mathbb{R}^{E(\mathcal{T})}$ with a subspace of
$AF(\widetilde{L},\widetilde{\mathcal{U}})$ modulo $KF(\widetilde{L})$. We
need to understand which vector fields from $AF(\widetilde{L},\mathcal{U})$
well-define variations of the edge lengths of $\mathcal{T}$. These are so-
called _automorphic_ ones.

This notion applies to a slightly more general context and is heavily
connected with variations of a hyperbolic structure on manifolds. For the
moment, let $L$ be any open hyperbolic 3-manifold and $\mathcal{HM}(L)$ be the
set of hyperbolic metrics on $L$ up to isotopies and thickenings. (For the
notion of thickening, see, e.g., [24, Chapter I.1.6]. This is a simple
equivalence relation for incomplete hyperbolic metrics on open manifolds.) We
note that the space of Killing vector fields on $\widetilde{L}$ always has
dimension exactly 6: it is easy to see it from any developing map to
$\mathbb{H}^{3}$. Consider a smooth path of hyperbolic metrics on $L$. It
gives rise to a smooth path of developing maps from $\widetilde{L}$ to
$\mathbb{H}^{3}$ defined up to a path of global isometries of
$\mathbb{H}^{3}$. In the first order it produces a vector field $X$ on
$\widetilde{L}$ defined up to Killing fields. It is easy to see that $X$
satisfies the _automorphic property_ : for every $\gamma\in\pi_{1}(L)$
considered as a deck transformation
$\gamma:\widetilde{L}\rightarrow\widetilde{L}$ the vector field
$X-d\gamma^{-1}(X)$ is a Killing field (various for various $\gamma$). If the
path of metrics is trivial, e.g., coming from the pull-back of an isotopy,
then $X$ is the sum of an equivariant vector field and a Killing vector field,
i.e., the vector field above is the same for all $\gamma$.

Now we return to our setting and by $AF_{a}(\widetilde{L},\mathcal{U})$ denote
the space of automorphic affine vector fields on $\widetilde{L}$ with respect
to $\mathcal{U}$. Note that $KF(\widetilde{L})\subset
AF_{a}(\widetilde{L},\mathcal{U})$. However, we are going to see that the
space $AF_{a}(\widetilde{L},\mathcal{U})$ is larger and to prove

######  Lemma 4.12.

$T^{L}_{l}\mathbb{R}^{E(\mathcal{T})}\cong
AF_{a}(\widetilde{L},\mathcal{U})/KF(\widetilde{L}).$

###### Proof.

Let $X\in AF_{a}(\widetilde{L},\mathcal{U})$ and take $\gamma\in\pi_{1}(L)$.
As $X-d\gamma^{-1}(X)$ is a Killing field, we see that for every tetrahedron
$T$ of $\mathcal{T}$ and every two its lifts $\widetilde{T}_{1}$ and
$\widetilde{T}_{2}$ in $\widetilde{L}$ that differ by $\gamma$, the edge-
length change induced by $X$ is the same. By construction, $X$ does not change
infinitesimally the dihedral angles of the edges from $E_{0}(\mathcal{T})$.
Hence, all lengths changes of tetrahedra in $\widetilde{L}$ project to a
length change $\dot{l}\in T^{L}_{l}\mathbb{R}^{E(\mathcal{T})}$. This defines
a linear map

$AF_{a}(\widetilde{L},\mathcal{T})\rightarrow
T^{L}_{l}\mathbb{R}^{E(\mathcal{T})}$

with $KF(\widetilde{L})$ equal to its kernel.

Conversely, let $\dot{l}\in\mathbb{R}^{E(\mathcal{T})}$. Similarly to the
proof of Lemma 4.10, we produce inductively an affine vector field $X$ on
$\widetilde{L}$ with respect to $\mathcal{U}$ inducing $\dot{l}$. We need to
show that it is automorphic.

Pick a tetrahedron $\widetilde{T}_{1}$ of $\mathcal{T}$ in $\widetilde{L}$ and
$\gamma\in\pi_{1}(L)$. Let $\widetilde{T}^{\prime}_{1}$ be the tetrahedron
$\gamma(\widetilde{T}_{1})$, and let $X_{1}$ and $X^{\prime}_{1}$ be the
components of $X$ corresponding to the tetrahedra $\widetilde{T}_{1}$ and
$\widetilde{T}^{\prime}_{1}$ respectively. As $X_{1}$ and
$d\gamma^{-1}(X^{\prime}_{1})$ produce the same length change on the
tetrahedron $\widetilde{T}_{1}$, we see that the difference
$X_{1}-d\gamma^{-1}(X^{\prime}_{1})$ is a Killing field on the tetrahedron
$\widetilde{T}_{1}$.

We need to show that for any other tetrahedron $\widetilde{T}_{2}$ if
$\widetilde{T}^{\prime}_{2}:=\gamma(\widetilde{T}_{2})$ and the corresponding
components of $X$ are $X_{2}$ and $X^{\prime}_{2}$, then the vector field
$X_{2}-d\gamma^{-1}(X^{\prime}_{2})$ is the same Killing field as
$X_{1}-d\gamma^{-1}(X^{\prime}_{1})$. Assume that $\widetilde{T}_{2}$ is
adjacent to $\widetilde{T}_{1}$ by a 2-face and that a developing map
$dev:\widetilde{L}\rightarrow\mathbb{H}^{3}$ is fixed, so we treat the
components of $X$ as elements of $AF$. The deformations of the edge lengths of
$\widetilde{T}_{1}$ and $\widetilde{T}_{2}$ induced by $\dot{l}$ together with
the isomorphism (4.1) define respectively two affine 6-dimensional spaces
$A_{1},A_{2}\subset AF$ parallel to $KF$ and a map

$A_{21}:A_{2}\rightarrow A_{1}$

mapping an affine field $X_{2}\in A_{2}$ to the unique affine field $X_{1}\in
A_{1}$ such that the restrictions of $X_{1}$ and $X_{2}$ to the common face of
$\widetilde{T}_{1}$ and $\widetilde{T}_{2}$ coincide. We extend $A_{21}$
linearly to the linear span of $A_{2}$. The latter contains $KF$ and $A_{21}$
preserves $KF$ pointwise.

Similarly we get affine subspaces $A^{\prime}_{1},A^{\prime}_{2}\subset AF$
defined by the restriction of $\dot{l}$ to the edge lengths of
$\widetilde{T}^{\prime}_{1}$ and $\widetilde{T}^{\prime}_{2}$ and get an
affine map

$A^{\prime}_{21}:A^{\prime}_{2}\rightarrow A^{\prime}_{1},$

which we extend to the linear spans of $A^{\prime}_{1}$, $A^{\prime}_{2}$. We
have

$A^{\prime}_{21}=d\gamma\circ A_{21}\circ d\gamma^{-1},$
$d\gamma(X_{1})-X^{\prime}_{1}=d\gamma\circ
A_{21}(X_{2})-A^{\prime}_{21}(X^{\prime}_{2})=d\gamma\circ
A_{21}(X_{2}-d\gamma^{-1}(X^{\prime}_{2})).$
$X_{1}-d\gamma^{-1}(X^{\prime}_{1})=A_{21}(X_{2}-d\gamma^{-1}(X^{\prime}_{2})).$

It follows that $X_{1}-d\gamma^{-1}(X^{\prime}_{1})$ and
$X_{2}-d\gamma^{-1}(X^{\prime}_{2})$ is the same Killing field. By induction
we show that for every component $X_{i}$ of $X$, $X_{i}-d\gamma^{-1}(X_{i})$
is the same Killing field on the whole $\widetilde{L}$. This finishes the
proof. ∎

We now describe affine equivariant vector fields on $\widetilde{L}$ with
respect to $\mathcal{U}$.

######  Lemma 4.13.

Let $X\in AF(\widetilde{L},\mathcal{U})$ be equivariant and $\widetilde{T}$ be
a tetrahedron of $\mathcal{T}$ in $\widetilde{L}$. Consider a developing map
$dev:\widetilde{T}\rightarrow\mathbb{H}^{3}$, identify $\widetilde{T}$ with
the image and extend $X$ to the vertices. Then $X$ is zero at each vertex that
has valence greater than 2 in $\Sigma$. For vertices of valence 2, $X$ is
parallel to (the $dev$-image) of the singular locus.

###### Proof.

Indeed, suppose that for a vertex $v$ of $\widetilde{T}$ having valence at
least 3 in $\Sigma$ the extension of $X$ is non-zero. Then there exists an
axis of $\Sigma$ that, after developing to $\mathbb{H}^{3}$, is not parallel
to $X$ at $v$. Consider a chain of tetrahedra adjacent by 2-faces, starting at
$\widetilde{T}$ and looping around this axis. We develop it to
$\mathbb{H}^{3}$. Due to the equivariance, the extension of $X$ to $v$ in the
last tetrahedron of the chain should differ from the extension in
$\widetilde{T}$. On the other hand, since all tetrahedra in the chain are
adjacent by 2-faces, these extensions must coincide. This is a contradiction.

This argument similarly shows that if $v$ has valence 2 in $\Sigma$, then the
extension of $X$ must be parallel to (the ${\rm dev}$-image of) the singular
axis of $\Sigma$ passing through $X$. ∎

###  4.3 Bundle-valued differential forms

Recall that $G=PSL(2,\mathbb{C})$, $\widetilde{G}=SL(2,\mathbb{C})$. By
$\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$ we denote the Lie algebra of
$\widetilde{G}$. By a result of Culler [29], every holonomy map
$\pi_{1}(L)\rightarrow G$ lifts to a holonomy map
$\pi_{1}(L)\rightarrow\widetilde{G}$. Denote the latter by $\rho$. Consider
the trivial bundle $\widetilde{E}\rightarrow\widetilde{L}$, where
$\widetilde{E}=\widetilde{L}\times\mathfrak{g}$. It is naturally equipped with
a flat connection coming from the trivialization. Consider the action of
$\pi_{1}(L)$ on $\widetilde{E}$ where the action on $\widetilde{L}$ is given
by the deck transformations, and the action on $\mathfrak{g}$ is given via
$Ad\circ\rho$, where $Ad$ is the adjoint representation of $\widetilde{G}$ on
$\mathfrak{g}$. By taking the quotient, we obtain a flat bundle $E\rightarrow
L$, which is the bundle of germs of Killing fields on $L$. Let
$\Omega^{*}(L,E)$ be the de Rham complex of $E$-valued differential forms on
$L$, where the exterior derivative $d$ is induced by the flat connection on
$E$. This complex is naturally isomorphic to the complex
$\Omega^{*}(\widetilde{L},\widetilde{E},Ad\circ\rho)$ consisting of
$\widetilde{E}$-valued differential forms on $\widetilde{L}$ that are
equivariant with respect to the action $Ad\circ\rho$. This construction goes
back to Weil [93], see a general exposition in the book [73] of Raghunathan,
and papers [43, 94, 95] for the applications to the infinitesimal theory of
hyperbolic cone-3-manifolds.

Let $e$ be an edge of $\Sigma$ and $\gamma_{e}$ be a meridian around $e$. In
[95, Theorem 3.15] Weiss showed that

######  Theorem 4.14.

Let $\omega\in\Omega^{1}(L,E)$ belong to
$\ker(d\operatorname{tr}(\gamma_{e}))$ for all $e\in\Sigma$. Then $\omega$ is
cohomologous to zero.

For an open hyperbolic 3-manifold $L$ there are two main ways to describe an
infinitesimal deformation of the hyperbolic structure: via the first
cohomology group $H^{1}(L,E)$ and via the automorphic vector fields. Following
[43] we now sketch why these ways are equivalent.

At each point $p\in L$ the fiber $E_{p}\cong\mathfrak{g}$ has a natural
decomposition

$E_{p}=E_{1,p}\oplus E_{2,p}$

into the subspace $E_{1,p}$ of infinitesimal translations at $p$ and the
subspace $E_{2,p}$ of infinitesimal rotations around $p$. This produces the
decomposition $E=E_{1}\oplus E_{2}$ of $E$ into two respective sub-bundles. We
remark that this decomposition is not preserved by the flat connection of $E$.
The subspace $E_{1,p}$ can be naturally identified with the tangent space
$T_{p}L$ by associating to each tangent vector the infinitesimal translation
in its direction, hence the sub-bundle $E_{1}$ can be naturally identified
with the tangent bundle $TL$, and we will use this identification freely. The
Lie algebra $\mathfrak{g}=\mathfrak{sl}(2,\mathbb{C})$ has the standard
complex structure, which induces a complex structure on $E$ such that
$E_{2}=iE_{1}$. For $X\in T_{p}L\cong E_{1,p}$ the element $iX\in E_{2,p}$ is
the infinitesimal rotation with the axis parallel to $X$ and having the same
norm. Here the norm on $E_{1,p}$ comes from the metric on $TL$ and the norm on
$E_{1,p}$ comes from the usual metric on $\mathfrak{so}(3)$.

Lift a 1-form $\omega\in\Omega^{1}(L,E)$ to an equivariant 1-form
$\widetilde{\omega}\in\Omega^{1}(\widetilde{L},\widetilde{E},Ad\circ\rho)$. If
$\omega$ is closed, so is $\widetilde{\omega}$, and, as $\widetilde{L}$ is
simply connected, $\widetilde{\omega}$ is exact. Hence,
$\widetilde{\omega}=ds$, where $s$ is a section of $\widetilde{E}$. However,
$s$ may be non-equivariant. But the notion of automorphicity extends to
sections of $\widetilde{E}$ and $s$ is automorphic, see, e.g., [43, Lemma
2.1]. Here a section is called automorphic if for every $\gamma\in\pi_{1}(L)$
the section $s-\gamma^{-1}_{*}s$ is closed. It follows conversely that for an
automorphic section $s$ the 1-form $ds$ is equivariant. Using the complex
structure on $\widetilde{E}$, we can consider every section of $\widetilde{E}$
as a pair of vector fields. Clearly, $s$ is automorphic (resp. equivariant or
constant) if and only if the both corresponding vector fields are automorphic
(resp. equivariant or Killing). It is evident that $s$ is the sum of an
equivariant section and a constant section if and only if $\omega$ is exact,
so it is cohomologous to zero.

Now we need to learn the passage from an automorphic vector field $X$ on
$\widetilde{L}$ to a closed $E$-valued 1-form. Let $D$ be the exterior
derivative on $\Omega^{*}(\widetilde{L},T\widetilde{L})$ induced by the
Levi–Civita connection for the metric on $\widetilde{L}$. Thus, $DX$ is a
$(T\widetilde{L})$-valued 1-form. Any $(T\widetilde{L})$-valued 1-form can be
viewed as an element of $Hom(T\widetilde{L},T\widetilde{L})$. We decompose it
into the symmetric and the skew-symmetric parts using the metric on
$\widetilde{L}$. As $T_{p}\widetilde{L}$ is a 3-dimensional vector space with
a positively-definite metric, it is canonically isomorphic to the skew-
symmetric subspace of $Hom(T_{p}M,T_{p}M)$. Indeed, if
$\\{e_{1},e_{2},e_{3}\\}$ is an orthonormal basis of $T_{p}M$ and
$\\{e^{1},e^{2},e^{3}\\}$ is the dual basis, then the isomorphism is defined
by sending $e_{i}$ to $e_{j}\otimes e^{h}-e_{h}\otimes e^{j}$, where $(i,j,h)$
is a cyclic permutation of $(1,2,3)$. This defines a bundle-isomorphism of
$T\widetilde{L}$ and $Hom(T\widetilde{L},T\widetilde{L})$. We denote by
$\operatorname{curl}(X)$ the vector field obtained from the skew-symmetric
part of $DX$ under this isomorphism. By $s_{X}:=X-i\operatorname{curl}(X)$ we
denote the _canonical lift_ of $X$, which is a section of $\widetilde{E}$.
Hodgson–Kerckhoff showed

######  Lemma 4.15 ([43], Lemma 2.3).

The canonical lift of an automorphic vector field is an automorphic section.
Every cohomology class in $H^{1}(L,E)$ can be represented by a 1-form $\omega$
such that the associated local section
$s:\widetilde{L}\rightarrow\widetilde{E}$ is the canonical lift of the vector
field determined by its real part.

The following lemma follows from the discussion above.

######  Lemma 4.16.

Assume that for an automorphic vector field $X$ the 1-form
$\omega\in\Omega^{1}(L,E)$ obtained from the projection of the 1-form $ds_{X}$
is cohomologous to zero. Then $X$ is the sum of an equivariant and a Killing
vector field.

###  4.4 Proof of the infinitesimal rigidity

We now have all the ingredients to prove Lemma 4.1. As it was showed in
Subsection 4.1, it is enough to prove Lemma 4.9.

###### Proof of Lemma 4.9..

Let $K$ be the closed hyperbolic cone-3-manifold obtained from $M(g)$ by
reflecting it along the boundary. As usually, by $\Sigma$ we denote the
singular locus and by $L$ we denote $K\backslash\Sigma$. The triangulation
$\mathcal{T}$ extends to a triangulation $\mathcal{T}^{D}$ of $K$. Note that
due to Lemma 3.24, $\mathcal{T}$ is large, and one can see that
$\mathcal{T}^{D}$ is also large. By $\mathcal{U}$ we denote a partition of
unity with respect to an open cover from Lemma 4.11. We can make $\mathcal{U}$
symmetric with respect to the antipodal involution of $K$ switching the copies
of $M(g)$. Consider an infinitesimal deformation $\dot{l}$ of the edge lengths
of $\mathcal{T}$ as in the statement of Lemma 4.9. We associate to it a
deformation $\dot{l}^{D}$ of the edges of $\mathcal{T}^{D}$ as follows. For
every edge $e\in E(\mathcal{T}^{D})$ coming from the boundary of $M(g)$ we set
$\dot{l}^{D}_{e}=0$. Next, there are two isometric copies of $M(g)$ in $K$. In
one copy, for every edge $e_{1}\in E(\mathcal{T}^{D})$ coming from an interior
edge $e\in E(\mathcal{T})$ we set $\dot{l}^{D}_{e_{1}}=\dot{l}_{e}$. For the
second copy and an edge $e_{2}\in E(\mathcal{T}^{D})$ coming from the same
interior edge $e\in E(\mathcal{T})$ we set $\dot{l}^{D}_{e_{1}}=-\dot{l}_{e}$.
It is trivial to see now that for every edge $e\in E(\mathcal{T}^{D})$ we have
$d\nu_{e}(\dot{l}^{D})=0$.

From Lemma 4.12 we associate to $\dot{l}^{D}$ an affine automorphic vector
field $X$ on $\widetilde{L}$ with respect to $\mathcal{U}$. Using Subsection
4.3 we consider its canonical lift
$s_{X}:\widetilde{L}\rightarrow\widetilde{E}$ and the 1-form
$\omega\in\Omega^{1}(L,E)$ obtained from the equivariant 1-form
$\widetilde{\omega}=ds_{X}$.

We now remark that for every meridian $\gamma_{e}$ around an edge $e$ of
$\Sigma$ we have $\omega\in\ker(d\operatorname{tr}(\gamma_{e}))$. Indeed, we