# Prescribed Curvature Problem for Discrete Conformality on Convex Spherical
Cone-Metrics

Ivan Izmestiev, Roman Prosanov, Tianqi Wu The research of R.P. is funded by
the Austrian Science Fund (FWF) ESPRIT grant ESP-12-N.The research of T.W. is
funded by the U.S. National Science Foundation 1760471.

###### Abstract

Let $S$ be the 2-sphere and $V\subset S$ be a finite set of at least three
points. We show that for each function $\kappa:V\rightarrow(0,2\pi)$
satisfying elementary necessary conditions, in each discrete conformal class
of spherical cone-metrics there exists a unique metric realizing $\kappa$ as
its discrete curvature. This can be seen as a discrete version of a result of
Luo and Tian.

##  1 Introduction

###  1.1 Prescribing curvature on surfaces

A natural question in differential geometry of surfaces is: _which smooth
functions on a surface $S$ can appear as the Gaussian curvature of a
Riemannian metric $g$ on $S$?_ For simplicity we restrict the exposition here
to closed surfaces. A basic necessary condition is given by the Gauss–Bonnet
theorem, which states that

$\int_{S}\kappa_{g}d{\rm area}_{g}=2\pi\chi(S).$ (1.1)

Here $\kappa_{g}$ is the curvature of $g$ and $d{\rm area}_{g}$ is its area
form. It implies that a potential function $\kappa$ somewhere must attain a
value of the same sign as the Euler characteristic $\chi(S)$.

An early insight was that this problem has too much flexibility, and actually
becomes more tractable when we consider not all Riemannian metrics, but
restrict ourselves to some _conformal class_.

######  Definition 1.1.

Two Riemannian metrics $g$ and $g^{\prime}$ on $S$ are _simply conformally
equivalent_ if for a smooth function $u:S\rightarrow\mathbb{R}$ and a
diffeomorphism $\phi:S\rightarrow S$ we have $g^{\prime}=e^{2u}\phi^{*}(g).$

They are called _conformally equivalent_ if $\phi$ is the identity.

Note that some authors say conformal equivalence for the former definition,
and say pointwise conformal equivalence for the latter.

The prescribed curvature problem in a given conformal class becomes equivalent
to a resolution of a non-linear elliptic partial differential equation. When
one works in a given simple conformal class, one may also greatly change the
potential curvature function acting on it by diffeomorphisms.

A fundamental special case is given by the celebrated Koebe–Poincaré
uniformization theorem [7], which states in our language that every Riemannian
metric is simply conformally equivalent to a unique up to scaling metric of
constant curvature. Due to (1.1), the sign of the curvature is the same as the
sign of the Euler characteristic. Moreover, the claim can be strengthened to
the conformal equivalence except the uniqueness claim in the case of 2-sphere.
The 2-sphere is the only surface, for which the conformal group is bigger than
the isometry group of a constant curvature metric, so the uniqueness in a
conformal class holds up to Möbius transformation.

The prescribed curvature problem in a given simple conformal class for
surfaces was fully solved by Kazdan and Warner [21, 22], who showed that the
sign condition dictated by (1.1) is the sufficient condition. They also gave
necessary and sufficient conditions for a resolution of this problem in a
given conformal class for non-positive Euler characteristic. The case of the
conformal class for the standard 2-sphere is known as _the Nirenberg problem_
and remains to be unresolved in general despite many partial progress, we
refer to [3] as one of the most recent advancements. The uniqueness of a
solution in a conformal class is an interesting problem. We mention that in
the case of a negative Euler characteristic and a negative curvature function,
the solution is unique due to the maximum principle [21].

A significant amount of work was done to understand the prescribed curvature
problem for so-called _cone-metrics_ on surfaces. We do not give a general
intrinsic definition, but they can be thought as gluings of triangles, which
are endowed with a Riemannian metric and have geodesic boundary away from the
vertices. The resulting surface may have isolated singularities at the
vertices if the total angle turns out to be different from $2\pi$. Such a
singular point is called a _cone-point_. The metric is smooth away from cone-
points and one can define its (simple) conformal class in the same way as
above. Cone-points are atomic points of the curvature measure with the value
$2\pi$ minus the total angle. For a cone-metric $d$ with the set of cone-
points $V$ we say that its _discrete curvature_
$\kappa(d)\in(-\infty,2\pi)^{V}$ is the data of the curvatures at the cone-
points.

Of the main interest are cone-metrics of constant curvature outside the cone-
points, which were actively studied by Alexandrov in order to describe
intrinsically the geometry of polyhedral surfaces in 3-dimensional space-
forms, see [2]. Since then they became a central object in the field of
_discrete differential geometry_. After scaling we may assume that the
curvature outside the cone-points is 1, 0 or $-1$. We naturally call such
cone-metrics _spherical, Euclidean_ and _hyperbolic_ respectively.

Most of the study of the prescribed curvature problem for cone-metrics deals
with prescribing the discrete curvature and looking for constant curvature
outside the singularities. If we fix a conformal structure outside of
singularities (but extending over the singularities), then this can be seen as
the uniformization of punctured surfaces with prescribed behaviour at the
punctures. The interest to this setting goes back to Picard (see, e.g., [33]),
who investigated some cases of the prescribed discrete curvature problem in a
conformal class of hyperbolic cone-metrics. A complete result (see [17, 28])
states that if $V\subset S$ is a finite set and
$\kappa:V\rightarrow(-\infty,2\pi)$ is a function such that

$\sum_{v\in V}\kappa_{v}>2\pi\chi(S),$

then in each conformal class on $S\backslash V$, extending to a conformal
structure over $S$, there exists a unique hyperbolic cone-metric realizing
$\kappa$ as its discrete curvature. A similar result for Euclidean cone-
metrics was established by Troyanov [42]. In [43] Troyanov also studied some
cases when the curvature varies away from cone-points.

The curvature prescription problem for spherical cone-metrics exhibits curious
difficulties, especially when $S$ is the 2-sphere. In the latter case it turns
out that the Gauss–Bonnet condition is not the only necessary condition. For
instance, in the case of _convex_ spherical cone-metrics (i.e., when the
discrete curvature $\kappa\in(0,2\pi)^{V}$) Luo and Tian [25] proved the
following result:

######  Theorem 1.2.

Let $S$ be the 2-sphere, $V\subset S$ be a finite set of at least three
points, $S$ be endowed with a conformal structure and $\kappa$ be in
$(0,2\pi)^{V}$. Then there exists a spherical cone-metric $d$ on $(S,V)$ such
that

(a) the conformal structure induced by $d$ on $S\backslash V$ coincides with
the given one, and

(b) $d$ has the discrete curvature $\kappa$,

if and only if

$\sum_{v\in V}\kappa_{v}<4\pi$ (1.2)

and

$\kappa_{v}<\sum_{w\neq v}\kappa_{w}\text{, for any $v\in V$}.$ (1.3)

Further, such $d$ is unique if it exists.

The existence part of Theorem 1.2 was also established in [43]. The current
paper is devoted to a discrete version of Theorem 1.2 formulated in the next
subsection.

In the non-convex case it was not until recently when Mondello and Panov [29]
obtained nearly complete necessary and sufficient conditions on the curvatures
of spherical cone-metrics on the 2-sphere (the study of last remaining cases
was completed by Eremenko [9]). They also treated the case of surfaces of
higher genus in [30], where they showed that the Gauss–Bonnet condition is
actually sufficient.

It is worth to mention that a solution of the prescribed discrete curvature
problem in a simply conformal class of spherical cone-metrics on the 2-sphere
is non-unique in general. It is conjectured, however, that the number of
solutions is finite. We refer to the recent survey of Eremenko [10]. A well-
studied case is when all $\kappa_{v}(d)$ are multiples of $2\pi$, so the
metric is a branched covering of the standard 2-sphere. The answer on the
number of solutions for a generic position of points was given by Scherbak
[39].

###  1.2 Discrete conformality of cone-metrics

The purpose of discrete differential geometry is to discretize the notions
from smooth differential geometry in order to obtain analogues of classical
results. Ideally, the existence results in the discrete setting should imply
the existence results in the smooth setting by means of some kind of
approximation. It frequently happens that the discretization of various
equivalent notions gives rise to non-equivalent discrete objects.

Discretization of conformal geometry has a long history mainly focused on
conformal maps between domains in the complex plane, especially on
establishing a discrete version of the Riemann mapping theorem. The most well-
known approach is via _circle packings_ as proposed by Thurston. However,
circle packings do not work that well for discretizing the notion of conformal
equivalence between Riemannian metrics on surfaces, since it was often unclear
how to construct a circle packing on a general Riemannian surface.

Let $S$ be a closed surface, $d$ be a Euclidean cone-metric on $S$, $V$ be a
finite point set containing all cone-points of $d$ and $\mathcal{T}$ be a
geodesic triangulation of $d$ with vertices at $V$. (In what follows we will
say that $d$ is a Euclidean cone-metric on $(S,V)$ realizing $\mathcal{T}$.)
Then $d$ is uniquely determined by the lengths of the edges of $\mathcal{T}$.
One can attempt to discretize the notion of pointwise conformal equivalence by
considering metrics $d^{\prime}$ defined by the same data $(S,V,\mathcal{T})$
and satisfying

$l_{e}(d^{\prime})=\exp\left(\frac{u_{v}+u_{w}}{2}\right)l_{e}(d),$ (1.4)

for each edge $e$ of $\mathcal{T}$ with endpoints $v$ and $w$, where
$l_{e}(d)$, $l_{e}(d^{\prime})$ are the lengths of $e$ in $d$, $d^{\prime}$
and $u:V\rightarrow\mathbb{R}$ is some function. One can say that such
$d^{\prime}$ is _discretely conformally equivalent_ to $d$ _with respect to
$\mathcal{T}$_. This is motivated by the fact that if a Riemannian metric
tensor changes by $e^{2u}$, then the distances change due to (1.4) up to the
third order.

This definition of discrete conformality first occurred in [38] in an attempt
to discretize some aspects of the theory of relativity. It was rediscovered by
Luo [23] in his development of a combinatorial Yamabe flow. In [5] Bobenko,
Pinkall and Springborn showed a surprising connection of this discrete
conformal change with the change of decorations on some associated hyperbolic
cusp-metric. Finally, in [16] Gu, Luo, Sun and Wu gave a novel definition of
discrete conformality, which extends (1.4), but does not employ an arbitrary
triangulation.

Each Euclidean cone-metric has a canonical _Delaunay decomposition_ into
convex polygons, which is generically a triangulation. The _discrete conformal
class_ in the sense of [16] is generated by conformal changes of the type
(1.4) performed on Delaunay triangulations, and by diagonal flips among
Delaunay triangulations for metrics, which admit several of them. (An exact
definition is given in Definition 2.9.) An evidence of the validity of this
definition is given by the fact that it is coherent with the connection
developed in [5]: all metrics obtained this way have the same associated
hyperbolic cusp-metric. In the present paper we work in the setting of
discrete conformality proposed in [16].

In [16] the authors resolved completely the discrete curvature prescription
problem in the new setting, demonstrating the same behaviour as in the smooth
setting. They showed

######  Theorem 1.3.

Let $S$ be a closed surface, $V\subset S$ be a finite set of points, $d$ be a
Euclidean cone-metric on $(S,V)$ and $\kappa\in(-\infty,2\pi)^{V}$ be a
function satisfying

$\sum_{v\in V}\kappa_{v}=2\pi\chi(S).$

Then there exists a unique up to scaling Euclidean cone-metric $d^{\prime}$ on
$(S,V)$ that is discretely conformal to $d$ and has the discrete curvature
$\kappa$.

In a sequel paper [15] a similar result for hyperbolic cone-metrics was also
established:

######  Theorem 1.4.

Let $S$ be a closed surface, $V\subset S$ be a finite set of points, $d$ be a
hyperbolic cone-metric on $(S,V)$ and $\kappa\in(-\infty,2\pi)^{V}$ be a
function satisfying

$\sum_{v\in V}\kappa_{v}>2\pi\chi(S).$

Then there exists a unique hyperbolic cone-metric $d^{\prime}$ on $(S,V)$ that
is discretely conformal to $d$ and has the discrete curvature $\kappa$.

In particular, the discrete uniformization theorems follow for surfaces of
non-positive Euler characteristic: in each discrete conformal class of
hyperbolic (resp. Euclidean) cone-metrics on a surface of genus at least two
(resp. genus one) there exists a unique metric (resp. unique up to scaling)
without cone-singularities.

In subsequent papers [14, 24, 26, 46] various types of convergence were
examined. It was shown that this version of discrete uniformization allows to
compute conformal maps in the Riemannian mapping theorem and the
uniformization factors for Riemannian surfaces provided that the approximating
triangulations are chosen carefully enough.

As outlined in [5], the discrete uniformization in this setting is connected
with the geometry of convex ideal polyhedral surfaces in the hyperbolic
3-space. More generally, it turns out that this notion of discrete
conformality has a nice geometric interpretation in terms of hyperbolic cone-
polyhedra. This viewpoint was explored in [34], where an alternative
variational proof of Theorem 1.4 was given. It was based on some previous
papers [4, 19, 11, 12], which used cone-polyhedra to obtain results on convex
realizations of cone-metrics and their rigidity. One can see a deep connection
between these topics, which is further developed in the present paper. Another
paper working out this connection is the recent paper [40] of Springborn. Its
results imply, in particular, the discrete uniformization theorem for
spherical cone-metrics on the 2-sphere, although it is not explicitly stated
in the text.

Except the last fact nothing is currently known about discrete conformality of
spherical cone-metrics. By analogy with the smooth setting, one should
naturally expect a more complicated behaviour from this case. In this paper we
make a step further and establish a discrete analogue of Theorem 1.2.

######  Theorem 1.5.

Let $S$ be the 2-sphere, $V\subset S$ be a finite set of at least three
points, $d$ be a spherical cone-metric on $(S,V)$ that admits a triangulation
into convex triangles, and $\kappa\in(0,2\pi)^{V}$. Then there exists a
spherical cone-metric $d^{\prime}$ on $(S,V)$ that is discretely conformal to
$d$ and has the discrete curvature $\kappa$, if and only if (1.2) and (1.3)
hold. Further, such $d^{\prime}$ is unique if it exists.

Theorem 1.5 is proven by a standard continuity method popularized by
Alexandrov, see [2]. We parametrize the discrete conformal class of $d$ by
some domain $U(d)\subset\mathbb{R}^{V}$ and consider the curvature as a
$C^{1}$-map $\kappa:U(d)\rightarrow(-\infty,2\pi)^{V}$. Let $U_{C}(d)\subset
U(d)$ be the subset of convex cone-metrics. We show that, when restricted to
$U_{C}(d)$, the differential of $\kappa$ is non-degenerate, the map is proper
and the domain is connected. Since the target is clearly simply connected, we
obtain that $\kappa|_{U_{C}(d)}$ is a homeomorphism onto its image, which is
the set of $\kappa\in(0,2\pi)^{V}$ satisfying (1.2) and (1.3).

The same method was used in the proofs of Theorems 1.3 and 1.4 in [16, 15]. In
all works the non-degeneracy of the differential is based on the fact that it
is equal to the Hessian of an auxiliary functional $H$. The main difference
lies in the fact that in the hyperbolic and Euclidean settings the respective
functional is strictly concave, which immediately implies the non-degeneracy.
This no longer holds in the spherical case, which brings many complications
and is responsible for some flexibility phenomena specific to this case.

In order to obtain the non-degeneracy in our situation, we give $H$ a
geometric interpretation in terms of ideal cone-polyhedra, similarly as it was
done in [34]. Then we use dimensional reduction arguments inspired by [4, 19].

A very interesting question is what is going on without the convexity
assumption. First of all, if all $\kappa_{v}(u)$ are zero, that is, we have
just the standard sphere with marked points, then $d\kappa$ has a
3-dimensional kernel. It comes from the possibility to move the interior point
inside our cone-polyhedron. The kernel consists of the linear functions on the
associated fan. This actually corresponds to a local non-rigidity of such
“metrics” caused by Möbius transformations that are not isometries. The same
happens with branched coverings of the standard sphere, i.e., the case when
all $\kappa_{v}$ are integer multiples of $2\pi$.

We were unable to construct more non-trivial examples of global or local non-
rigidity, although we conjecture that they exist, similarly to the smooth
setting. We, however, detected an infinitesimal non-rigidity, which is not
caused by the effects above.

Inner angle | $2\pi/5$ | $2.4\pi/5$ | $2.8\pi/5$ | $3.2\pi/5$ | $3.6\pi/5$ | $4\pi/5$ | $4.4\pi/5$
---|---|---|---|---|---|---|---
Eigenvalues | $-2.7751$ | $-8.2113$ | $-16.6180$ | $-29.1759$ | $-48.3430$ | $-80.5748$ | $-149.9030$
$0.0000$ | $-3.3463$ | $-8.6839$ | $-16.7566$ | $-29.1077$ | $-49.7980$ | $-93.9818$
$0.0000$ | $-3.3463$ | $-8.6839$ | $-16.7566$ | $-29.1077$ | $-49.7980$ | $-93.9818$
$0.0000$ | $-3.3463$ | $-8.6839$ | $-16.7566$ | $-29.1077$ | $-49.7980$ | $-93.9818$
$3.2492$ | $2.3498$ | $0.6055$ | $-2.2159$ | $-6.5865$ | $-13.7638$ | $-28.5081$
$3.2492$ | $2.3498$ | $0.6055$ | $-2.2159$ | $-6.5865$ | $-13.7638$ | $-28.5081$
$3.2492$ | $2.3498$ | $0.6055$ | $-2.2159$ | $-6.5865$ | $-13.7638$ | $-28.5081$
$3.2492$ | $2.3498$ | $0.6055$ | $-2.2159$ | $-6.5865$ | $-13.7638$ | $-28.5081$
$3.2492$ | $2.3498$ | $0.6055$ | $-2.2159$ | $-6.5865$ | $-13.7638$ | $-28.5081$
$4.4903$ | $4.5256$ | $4.1538$ | $3.3382$ | $2.0158$ | $0.0000$ | $-3.4994$
$4.4903$ | $4.5256$ | $4.1538$ | $3.3382$ | $2.0158$ | $0.0000$ | $-3.4994$
$4.4903$ | $4.5256$ | $4.1538$ | $3.3382$ | $2.0158$ | $0.0000$ | $-3.4994$

Table 1: Eigenvalues of the Jacobian of $d\kappa$ for regular icosahedron-type
metrics.

Let $(S,V,\mathcal{T})$ be the combinatorial structure of an icosahedron.
Consider a family of spherical cone-metrics $d_{t}$ assigning equal lengths to
all edges. We parametrize them by the value of an inner angle in a triangle of
$\mathcal{T}$. We computed the eigenvalues of the Jacobian matrix of $d\kappa$
for some $d_{t}$, see Table 1. We see the expected 3-dimensional kernel for
the angles $2\pi/5$ and $4\pi/5$. Except this, a 5-dimensional kernel appears
for some angle between $2.8\pi/5$ and $3.2\pi/5$. It is curious if there is a
geometric explanation to this phenomenon.

If there are actual examples of non-uniqueness not coming from Möbius
transformations, we conjecture, similarly to the smooth setting, that the
number of such solutions is finite. Finally, we emphasize that nothing is
currently known about discrete conformality of spherical cone-metrics on
surfaces other than the 2-sphere. This is an interesting direction of future
research.

We also would like to mention the related recent works [31, 13] considering
the prescription problem of the so-called _geodesic curvature_ for circle
patterns in spherical cone-metrics. It is interesting to note that the
functional appearing in that problem is strictly convex, similarly to the
Euclidean and hyperbolic cases.

##  2 Preliminaries

###  2.1 Cone-metrics and Delaunay triangulations

Suppose that $S$ is a closed orientable surface and $V\subset S$ is a finite
point-set.

######  Definition 2.1.

A _spherical (resp. Euclidean) cone-metric_ $d$ on $(S,V)$ is a metric on $S$
locally isometric to the standard sphere (resp. the Euclidean plane) except
points of $V$. At $v\in V$ the metric $d$ is locally isometric to the metric
of a spherical (resp. Euclidean) cone with total angle $\omega_{v}(d)$, which
might be distinct from $2\pi$. A cone-metric $d$ is called _convex_ if for
every $v\in V$ we have $\omega_{v}(d)\leq 2\pi$. Two cone-metrics on $(S,V)$
are considered equivalent if there is an isometry fixing $V$ and isotopic to
the identity with respect to $V$.

The number $2\pi-\omega_{v}(d)$ is called the _curvature_ of $v$ in $d$. We
will denote it by $\kappa_{v}(d)$ in the case of spherical cone-metrics and by
$\delta_{v}(d)$ in the case of Euclidean cone-metrics. In this paper we are
primarily interested in spherical cone-metrics, but at some steps we need also
to use their connection with Euclidean cone-metrics.

By $\mathfrak{M}_{+}(S,V)$ and $\mathfrak{M}_{0}(S,V)$ we denote the sets of
equivalent classes of spherical and Euclidean cone-metrics respectively. In
what follows by a cone-metric we sometimes mean an equivalence class of cone-
metrics.

We say that a spherical triangle is convex if any inner angle is in $(0,\pi)$
or, equivalently, the triangle is contained in an open hemisphere. Under this
assumption, a spherical triangle is determined up to isometry by its edge
lengths, and the space of possible edge lengths is

$\\{(l_{1},l_{2},l_{3})\in(0,\pi)^{3}:l_{1}+l_{2}+l_{3}<2\pi,\text{$(l_{1},l_{2},l_{3})$
satisfies the triangle inequality}\\}.$

In particular, the length of each edge is less than $\pi$.

######  Definition 2.2.

A _polygonal decomposition_ of $(S,V)$ is a collection of simple disjoint arcs
(called _edges_) with endpoints in $V$ that cut $S$ into topological polygons
(called _faces_) without points of $V$ in the interior. We require that each
polygon has at least three vertices after being cut (although some of them may
coincide as actual points of $V$). Two cell decompositions on $(S,V)$ are
considered equivalent if they are isotopic with respect to $V$.

A _triangulation_ of $(S,V)$ is a polygonal decomposition such that all faces
are triangles.

Note that the definition above allows loops and multiple edges. We denote the
set of edges of a triangulation $\mathcal{T}$ by $E(\mathcal{T})$. In what
follows by a polygonal decomposition or by a triangulation we sometimes mean
an equivalence class of them.

Given a cone-metric $d$ on $(S,V)$, we say that a triangulation $\mathcal{T}$
of $(S,V)$ is _realized_ by $d$ if there exists a geodesic triangulation of
$(S,V)$ equivalent to $\mathcal{T}$ (which is then clearly unique). In the
case of spherical cone-metrics we additionally assume that the geodesic
triangulation consists only of convex triangles.

For a triangulation $\mathcal{T}$ of $(S,V)$ by
$\mathfrak{M}_{+}(\mathcal{T})\subset\mathfrak{M}_{+}(S,V)$ and
$\mathfrak{M}_{0}(\mathcal{T})\subset\mathfrak{M}_{0}(S,V)$ we denote the
subsets of cone-metrics realizing $\mathcal{T}$. Any cone-metric in
$\mathfrak{M}_{+}(\mathcal{T})$, $\mathfrak{M}_{0}(\mathcal{T})$ is uniquely
determined by its type (spherical or Euclidean), $\mathcal{T}$ and the edge
lengths. Thus, $\mathfrak{M}_{+}(\mathcal{T})$,
$\mathfrak{M}_{0}(\mathcal{T})$ can be parametrized as open convex polyhedra
in $\mathbb{R}^{E(\mathcal{T})}$:

$\mathfrak{M}_{+}(\mathcal{T})=\\{l\in\mathbb{(}0,\pi)^{E(\mathcal{T})}:\text{$l$
satisfies the triangle inequalities},\\\ \text{the perimeter of each triangle
is $<2\pi$}\\},$
$\mathfrak{M}_{0}(\mathcal{T})=\\{l\in\mathbb{(}0,\infty)^{E(\mathcal{T})}:\text{$l$
satisfies the triangle inequalities}\\}.$

######  Remark 2.3.

Not any spherical cone-metric $d$ on $(S,V)$ has a geodesic triangulation with
vertex set $V$. For example, if $(S,d)$ is the unit sphere and $V$ is
contained in an open hemisphere, then $(S,V,d)$ does not admit a geodesic
triangulation. If $V$ belongs to a great circle, then $(S,V,d)$ admits a
triangulation, but not into convex triangles. On the other hand, it is not
hard to show that if for every point of $S$ there exists a point of $V$ at
distance less than $\pi/2$, then $d$ admits a triangulation into convex
triangles. For this one can adapt arguments from [41, Proposition 3.1], [37,
Section 3].

By $\mathfrak{M}_{+,T}(S,V)\subset\mathfrak{M}_{+}(S,V)$ we denote the subset
of spherical cone-metrics on $(S,V)$ admitting a geodesic triangulation into
convex triangles. Due to the remark above,

$\mathfrak{M}_{+,T}(S,V)\subsetneq\mathfrak{M}_{+}(S,V).$

On the other hand, any Euclidean triangle is considered to be convex, and it
is well-known that any Euclidean cone-metric on $(S,V)$ admits a geodesic
triangulation.

The families $\\{\mathfrak{M}_{+}(\mathcal{T})\\}_{\mathcal{T}}$ and
$\\{\mathfrak{M}_{0}(\mathcal{T})\\}_{\mathcal{T}}$ provide open coverings of
the spaces $\mathfrak{M}_{+,T}(S,V)$ and $\mathfrak{M}_{0}(S,V)$ respectively,
and endow them with the structure of smooth manifolds. The discrete curvatures
$\kappa$ and $\delta$ can be considered as smooth $\mathbb{R}^{V}$-valued
functions on $\mathfrak{M}_{+,T}(S,V)$ and $\mathfrak{M}_{0}(S,V)$
respectively.

We note that there are infinitely many isotopy classes of triangulations of
$(S,V)$. Moreover, a Euclidean cone-metric may realize infinitely many
triangulations: consider, e.g., the surface of the Euclidean unit cube.
However, for spherical metrics the convexity condition does not allow this.

######  Lemma 2.4.

Each metric $d\in\mathfrak{M}_{+,T}(S,V)$ realizes finitely many
triangulations.

###### Proof.

It is enough to show that between any two points $v,w\in V$ there are finitely
many simple geodesic arcs of length $<\pi$. To this purpose we adapt the
argument from [18, Proposition 1].

Note that any two different simple geodesic arcs of length $<\pi$ are non-
isotopic on $S\backslash V$, so they must bound together a region with at
least one point of $V$. Similarly, none of them is isotopic to an arc of
length $\pi$. The family of all simple geodesic arcs from $v$ to $w$ of length
at most $\pi$ is uniformly Lipschitz. Thus, by the Arzelà–Ascoli theorem it is
compact. Hence, if there are infinitely many arcs of length $<2\pi$, we can
extract a convergent subsequence. But we saw that no arc can appear in the
limit, so we get a contradiction. ∎

Of special importance among triangulations are the well-known _Delaunay
triangulations_.

######  Definition 2.5.

We say that a triangulation $\mathcal{T}$ realized by a spherical (resp.
Euclidean) cone-metric $d$ is _Delaunay_ in $d$ if when we develop each pair
of the adjacent triangles to the standard sphere (resp. the Euclidean plane),
then the circumscribed disk of each triangle does not contain the opposite
vertex of the other triangle in the interior.

A Delaunay triangulation is a refinement of a more general _Delaunay
decomposition_.

######  Definition 2.6.

A _Delaunay decomposition_ of a spherical (resp. Euclidean) cone-metric on
$(S,V)$ is a polygonal decomposition such that each cell is isometric to a
circumscribed convex polygon in the standard sphere $\mathbb{S}^{2}$ (resp. in
the Euclidean plane $\mathbb{E}^{2}$) and for each pair of adjacent cells
after we develop them in $\mathbb{S}^{2}$ (resp. $\mathbb{E}^{2}$) the closed
circumscribed disk of each polygon does not contain the vertices of the other
one except the vertices of the adjacent edge. In the spherical case by a
_convex polygon_ we mean that all angles are at most $\pi$ and it is contained
in an open hemisphere.

So in particular, if we take a Delaunay triangulation and erase all the edges
such that adjacent triangles form a cyclic quadrilateral, then we obtain a
Delaunay decomposition. Conversely, any triangulation refining a Delaunay
decomposition is Delaunay. It is well-known that a Delaunay decomposition
exists for every Euclidean cone-metric and is actually unique, see [6, 18]. We
need to check that this is also true for spherical cone-metrics.

######  Lemma 2.7.

Let $d\in\mathfrak{M}_{+,T}(S,V)$. A Delaunay decomposition of $d$ exists and
is unique.

Instead of adapting the arguments from [6], we will prove Lemma 2.7 in Section
2.5 with the help of our main tool: _ideal cone-polyhedra_.

Note that although a Euclidean cone-metric may realize infinitely many
triangulations, it is clear from the uniqueness of the Delaunay decomposition
that it has only finitely many Delaunay triangulations. Moreover, both in
Euclidean and in spherical cases any two Delaunay triangulations can be
connected by a sequence of flips via Delaunay triangulations.

By $\mathfrak{D}_{+}(\mathcal{T})\subset\mathfrak{M}_{+}(\mathcal{T})$,
$\mathfrak{D}_{0}(\mathcal{T})\subset\mathfrak{M}_{0}(\mathcal{T})$ we denote
the spaces of cone-metrics realizing $\mathcal{T}$ as a Delaunay
triangulation. We have the decompositions

$\mathfrak{M}_{+,T}(S,V)=\bigcup_{\mathcal{T}}\mathfrak{D}_{+}(\mathcal{T}),~{}~{}~{}~{}~{}\mathfrak{M}_{0}(S,V)=\bigcup_{\mathcal{T}}\mathfrak{D}_{0}(\mathcal{T}).$

One can see that these decompositions are locally finite, each cell
$\mathfrak{D}_{+}(\mathcal{T})$ or $\mathfrak{D}_{0}(\mathcal{T})$ is closed,
has non-empty interior and piecewise-smooth boundary, and any two cells have
disjoint interiors.

###  2.2 Discretely conformally equivalence

######  Definition 2.8.

We say that $d$ and $d^{\prime}$ in $\mathfrak{M}_{+,T}(S,V)$ are _discretely
conformally equivalent_ if there are a sequence of metrics
$d=d_{1},d_{2},\ldots,d_{m}=d^{\prime}$ and a sequence of triangulations
$\mathcal{T}_{1},\ldots,\mathcal{T}_{m}$ of $(S,V)$ such that

(a) $d_{i}\in\mathfrak{D}_{+}(\mathcal{T}_{i})$, and

(b) if $\mathcal{T}_{i}=\mathcal{T}_{i+1}$, then there is a function
$u:V\rightarrow\mathbb{R}$ such that for each edge $e$ of $\mathcal{T}$ with
endpoints $v$ and $w$ we have

$\sin\left(\frac{l_{e}(d_{i+1})}{2}\right)=\exp\left(\frac{u_{v}+u_{w}}{2}\right)\sin\left(\frac{l_{e}(d_{i})}{2}\right),$
(2.1)

where $l_{e}(d)$ is the length of $e$ in $d$; and

(c) if $\mathcal{T}_{i}\neq\mathcal{T}_{i+1}$, then $d_{i}=d_{i+1}$.

We denote the discrete conformal class of $d\in\mathfrak{M}_{+,T}(S,V)$ by
$\mathfrak{C}_{+}(d)$.

######  Definition 2.9.

We say that $d$ and $d^{\prime}$ in $\mathfrak{M}_{0}(S,V)$ are _discretely
conformally equivalent_ if there are a sequence of metrics
$d=d_{1},d_{2},\ldots,d_{m}=d^{\prime}$ and a sequence of triangulations
$\mathcal{T}_{1},\ldots,\mathcal{T}_{m}$ of $(S,V)$ such that

(a) $d_{i}\in\mathfrak{D}_{0}(\mathcal{T}_{i})$, and

(b) if $\mathcal{T}_{i}=\mathcal{T}_{i+1}$, then there is a function
$u:V\rightarrow\mathbb{R}$ such that for each edge $e$ of $\mathcal{T}$ with
endpoints $v$ and $w$ we have

$l_{e}(d_{i+1})=\exp\left(\frac{u_{v}+u_{w}}{2}\right)l_{e}(d_{i}),$ (2.2)

where $l_{e}(d)$ is the length of $e$ in $d$; and

(c) if $\mathcal{T}_{i}\neq\mathcal{T}_{i+1}$, then $d_{i}=d_{i+1}$.

We denote the discrete conformal class of $d\in\mathfrak{M}_{0}(S,V)$ by
$\mathfrak{C}_{0}(d)$.

Let $d\in\mathfrak{D}_{0}(\mathcal{T})$. Then clearly for any real $\lambda>0$
the metric $d_{\lambda}$ obtained from $d$ by multiplying all the edge lengths
of $\mathcal{T}$ by $\lambda$ belongs to
$\mathfrak{D}_{0}(\mathcal{T})\cap\mathfrak{C}_{0}(d)$. This defines an
equivalence relation on $\mathfrak{C}_{0}(d)$. We denote the set of
equivalence classes by $\widetilde{\mathfrak{C}}_{0}(d)$. It is clear that for
all $\lambda>0$ we have $\delta(d_{\lambda})=\delta(d)$. Thus, we may consider
$\delta$ as a function over $\widetilde{\mathfrak{C}}_{0}(d)$.

Note that due to the Gauss–Bonnet theorem, the image of $\delta$ lies in the
set

$\Delta=\\{\delta\in(-\infty,2\pi)^{V}:\sum_{v\in
V}\delta_{v}=4\pi\\}\subset\mathbb{R}^{V}.$

The main result of [16] is the following.

######  Theorem 2.10.

$\mathfrak{C}_{0}(d)$ is $C^{1}$-diffeomorphic to $\mathbb{R}^{V}$, and the
map

$\delta:\widetilde{\mathfrak{C}_{0}}(d)\rightarrow\Delta$

is a $C^{1}$-diffeomorphism.

Let $\mathcal{T}$ be a triangulation of $(S,V)$. Given
$l\in\mathbb{R}^{E(\mathcal{T})}$, we define
$\sin(l/2)\in\mathbb{R}^{E(\mathcal{T})}$ as $\sin(l/2)_{e}=\sin(l_{e}/2)$ for
each $e\in E(\mathcal{T})$. Let $T$ be a convex spherical triangle and
$l\in(0,\pi)^{E(T)}$ be the 3-tuple of its side lengths. Consider it on the
unit 2-sphere $\mathbb{S}^{2}$ embedded in the Euclidean 3-space. Then the
Euclidean triangle subtending $T$ on $\mathbb{S}^{2}$ has lengths
$2\sin(l/2)$. This means that if $d\in\mathfrak{D}_{+}(\mathcal{T})$ and
$l\in(0,\pi)^{E(\mathcal{T})}$ is the vector of its edge lengths, then
$2\sin(l/2)$ determines a Euclidean cone-metric, which we denote by
$\Phi_{\mathcal{T}}(d)$. This defines a map

$\Phi_{\mathcal{T}}:\mathfrak{D}_{+}(\mathcal{T})\rightarrow\mathfrak{M}_{0}(\mathcal{T}).$

The image of $\Phi_{\mathcal{T}}$ consists exactly of those cone-metrics in
$\mathfrak{M}_{0}(\mathcal{T})$ having the circumradius of each triangle of
$\mathcal{T}$ less than 1.

By this construction, it is easy to see that if $d\in\mathfrak{M}_{+,T}(S,V)$
realizes more than one Delaunay triangulations, the image of $d$ does not
depend on the choice of the map $\Phi_{\mathcal{T}}$. So the maps
$\Phi_{\mathcal{T}}$ give an injective map

$\Phi:\mathfrak{M}_{+,T}(S,V)\rightarrow\mathfrak{M}_{0}(S,V).$

Consider two adjacent triangles in the unit sphere. They satisfy the Delaunay
condition if and only if the dihedral angle between the subtending Euclidean
triangles is at most $\pi$ when measured inwards the sphere. By unbending them
we see that this corresponds to the Delaunay condition between the Euclidean
triangles. This shows

######  Lemma 2.11.

The metric $d\in\mathfrak{D}_{+}(\mathcal{T})$ if and only if
$\Phi(d)\in\mathfrak{D}_{0}(\mathcal{T})$.

From this and the definitions we get

######  Lemma 2.12.

Two metrics $d$ and $d^{\prime}$ are discretely conformally equivalent if and
only if $\Phi(d)$ is discretely conformally equivalent to $\Phi(d^{\prime})$.

Now we show

######  Lemma 2.13.

The space $\mathfrak{C}_{+}(d)$ is a connected manifold of dimension $|V|$
without boundary.

###### Proof.

Since $\mathfrak{C}_{0}(\Phi(d))$ is homeomorphic $\mathbb{R}^{V}$ by Theorem
2.10 and $\Phi$ is continuous and injective, it suffices to show that
$\Phi(\mathfrak{C}_{+}(d))$ is open and connected in
$\mathfrak{C}_{0}(\Phi(d))$. $\Phi(\mathfrak{C}_{+}(d))$ contains all the
metric $d^{\prime}\in\mathfrak{C}_{0}(\Phi(d))$ where the radius of any
circumcircle of the Delaunay decomposition of $d^{\prime}$ is smaller than
$1$. So the openness is obvious. Let
$d^{\prime},d^{\prime\prime}\in\mathfrak{C}_{+}(d)$, we can connect
$\Phi(d^{\prime})$ and $\Phi(d^{\prime\prime})$ by a path
$d_{t}\subset\mathfrak{C}_{0}(\Phi(d))$. As the decomposition of
$\mathfrak{C}_{0}$ induced by cells $\mathfrak{D}_{0}(\mathcal{T})$ is
piecewise-smooth and locally finite, we can make $d_{t}$ intersecting the
boundaries of this decompostion only finitely many times. If we multiply a
Euclidean cone-metric by $\lambda>0$, we stay in the same discrete conformal
class and in the same cells. Thus, we can multiply all $d_{t}$ by sufficiently
small $\lambda$ so that all the circumscribed circles of the Delaunay
decompositions of $\lambda\cdot d_{t}$ become smaller than 1. By linearly
connecting the corresponding end points of $d_{t}$ and $\lambda\cdot d_{t}$,
we obtain a path in $\Phi(\mathfrak{C}_{+}(d))$ connecting $\Phi(d^{\prime})$
and $\Phi(d^{\prime\prime})$.

∎

###  2.3 Cusp-metrics

In the study of the discrete conformality, it was necessary to introduce
hyperbolic cusp-metrics. See also [5, 16, 15, 40] for employing this
connection to study the discrete conformality of Euclidean and hyperbolic
cone-metrics.

######  Definition 2.14.

A _hyperbolic cusp-metric_ $g$ on $(S,V)$ is a complete hyperbolic metric of
finite area on $S\backslash V$.

Points of $V$ are called _cusps_ in a hyperbolic cusp-metric.

######  Definition 2.15.

A _decoration_ $h$ of $(S,V)$ with a hyperbolic cusp-metric $g$ is a choice of
a horocycle at each cusp.

After we chose a decoration, we can speak about the distance between two
cusps: we consider the signed distance between the corresponding horocycles
taken with the minus sign if they intersect. Although $g$ is actually a metric
on the surface with punctures $S\backslash V$, we can still speak about
realizations of triangulations with vertices at $V$. The edges are infinite
geodesics approaching the cusps like ideal points of
$\overline{\mathbb{H}}^{2}$. It is well-known that

######  Lemma 2.16 ([27], Proposition 7.4.6).

For each hyperbolic cusp-metric $g$ on $(S,V)$ each triangulation
$\mathcal{T}$ of $(S,V)$ is realized by $g$.

The reader can find more details about cusp-metrics in [32]. The basic aspects
are also treated well in [27, Chapter 7.4].

In the work [16] a discrete conformal class in the Euclidean case
$\mathfrak{C}_{0}(d)$ was associated with a hyperbolic cusp-metric $g$. It was
shown that a Delaunay triangulation $\mathcal{T}$ of a metric
$d^{\prime}\in\mathfrak{C}_{0}(d)$ is a Delaunay triangulation of $g$ for some
choice of decoration $h$, also called an _Epstein–Penner triangulation_. Here
we do not give the definition of a Delaunay triangulation of a decorated
hyperbolic cusp-metric, but refer to [40, 8]. The work of Akiyoshi [1] implies
then

######  Lemma 2.17.

Let $d\in\mathfrak{M}_{0}(S,V)$. There are finitely many triangulations
$\mathcal{T}$ of $(S,V)$ such that
$\mathfrak{D}_{0}(\mathcal{T})\cap\mathfrak{C}_{0}(d)$ is non-empty.

Together with Lemmas 2.11 and 2.12 this implies

######  Corollary 2.18.

Let $d\in\mathfrak{M}_{+,T}(S,V)$. There are finitely many triangulations
$\mathcal{T}$ of $(S,V)$ such that
$\mathfrak{D}_{+}(\mathcal{T})\cap\mathfrak{C}_{+}(d)$ is non-empty.

###  2.4 Semi-ideal triangles and tetrahedra

We will use some basic trigonometry of hyperbolic triangles having
simultaneously ideal and non-ideal vertices. We consider ideal points in
$\mathbb{H}^{2}$ or $\mathbb{H}^{3}$ always equipped with horocycles or
horospheres respectively, called _canonical_. By a distance between two ideal
points we mean the signed distance between their canonical
horocycles/horospheres. Similarly, the distance from an ideal point to a non-
ideal one means the signed distance from the canonical horocycle/horosphere to
the non-ideal point.

######  Definition 2.19.

A triangle $BA_{1}A_{2}$ in $\mathbb{H}^{2}$ is called _semi-ideal_ if the
vertices $A_{1}$ and $A_{2}$ are ideal and the vertex $B$ is not.

We denote the distances $BA_{1}$, $BA_{2}$, $A_{1}A_{2}$ by $u_{1}$, $u_{2}$,
$a_{12}$ respectively. By $l_{12}$ we denote the angle at $B$. By $\rho_{12}$
and $\rho_{21}$ we denote the lengths of the parts of the canonical horocycles
at $A_{1}$ and $A_{2}$ respectively cut out by the angles of the triangle.

######  Lemma 2.20 (Cosine laws).

We have

$\rho^{2}_{12}=\exp(u_{2}-u_{1}-a_{12})-\exp(-2u_{1}),~{}~{}~{}~{}~{}\rho^{2}_{21}=\exp(u_{1}-u_{2}-a_{12})-\exp(-2u_{2}),$
$\cos(l_{12})=\frac{\exp(u_{1}+u_{2})-2\exp(a_{12})}{\exp(u_{1}+u_{2})}.$

The first two formulas are shown in [34, Lemma 2.3]. The last formula is
obtained from the cosine law for hyperbolic triangles with all non-ideal
vertices by passing to the limit. We also have an analogue of the sine law:

######  Lemma 2.21 (Sine law).

We have

$\frac{1}{2}\sin(l_{12})\exp(-a_{12})=\rho_{12}\exp(-u_{2})=\rho_{21}\exp(-u_{1}).$

Indeed, it is not hard to see from the cosine laws that the square of each
expression is equal to

$\exp(-u_{1}-u_{2}-a_{12})-\exp(-2u_{1}-2u_{2}).$

Let $BA_{12}$ be the perpendicular from $B$ to $A_{1}A_{2}$. By $b_{12}$ and
$b_{21}$ we denote the distances $A_{12}A_{1}$ and $A_{12}A_{2}$ respectively,
by $u_{12}$ denote the distance $A_{1}A_{2}$.

######  Lemma 2.22.

We have

$\exp(u_{1})=\exp(b_{12})\cosh(u_{12}),~{}~{}~{}~{}~{}\exp(u_{2})=\exp(b_{21})\cosh(u_{12}),$
$\rho^{2}_{12}=\exp(-2b_{12})-\exp(-2u_{1}),~{}~{}~{}~{}~{}\rho^{2}_{21}=\exp(-2b_{21})-\exp(-2u_{2}).$

Here the first two formulas are just the limits of the Pythagoras theorem for
hyperbolic triangles. The last two formulas are obtained from substituting
them in the cosine laws.

######  Definition 2.23.

A tetrahedron $BA_{1}A_{2}A_{3}$ in $\mathbb{H}^{3}$ is called _semi-ideal_ if
the vertices $A_{1}$, $A_{2}$ and $A_{3}$ are ideal and the vertex $B$ is not.

The notation for the parameters of a semi-ideal tetrahedron $BA_{1}A_{2}A_{3}$
is inherited from semi-ideal triangles.

###  2.5 Ideal cone-polyhedra

Let $d\in\mathfrak{M}_{+}(\mathcal{T})$ for some triangulation $\mathcal{T}$
of $(S,V)$. For each triangle $T$ of $\mathcal{T}$ consider a semi-ideal
tetrahedron $\bar{T}$ in $\mathbb{H}^{3}$ that has $T$ as its spherical link
at the non-ideal vertex $B$. Here by the spherical link we mean the part of
the unit sphere in $T_{B}\mathbb{H}^{3}$ that is cut off by our tetrahedron.
Note that $\bar{T}$ exists and is determined up to isometry by $T$. We
decorate $\bar{T}$ by considering at each ideal vertex the horosphere that
passes through the non-ideal vertex. Glue all such tetrahedra together with
respect to $\mathcal{T}$ so that at the ideal boundary the canonical
horospheres match together. We obtain a complex $P=P(d,\mathcal{T})$, which is
a complete cone-manifold with ideal boundary, called _ideal cone-polyhedron_.
We consider them up to marked isometry: isometry such that the induced map on
$(S,V)$ fixes $V$ and is isotopic to identity with respect to $V$. We say that
the point of $P$ obtained from gluing all the non-ideal vertices of the
tetrahedra is _the marked point_ of $P$.

We note that the total dihedral angle at the interior edge corresponding to
$v\in V$ is $\omega_{v}(d)$. We can say that $\kappa_{v}(d)$ is the curvature
of this interior edge. For $e\in E(\mathcal{T})$ we denote the dihedral angle
of an edge $e\in E(\mathcal{T})$ at the boundary of $P$ by $\alpha_{e}$. We
say that $P$ is _convex_ if all $\alpha_{e}\leq\pi$. With the help of the
central projection from the marked point we identify the boundary of $P$ with
$(S,V)$, equipping the latter with a decorated cusp-metric.

We can construct ideal cone-polyhedra in another way. Let $(g,h)$ be a
decorated hyperbolic cusp-metric on $(S,V)$, $\mathcal{T}$ be a triangulation
of $(S,V)$ and $u\in\mathbb{R}^{V}$. Assume that for every triangle of
$\mathcal{T}$ there exists a (non-degenerate) semi-ideal tetrahedron with the
ideal face being a decorated ideal triangle coming from the realization of
$\mathcal{T}$ in $(g,h)$ and edge lengths to the non-ideal point being
determined by $u$. Then we can glue these tetrahedra together so that the
decorations given by $h$ match together. We obtain an ideal cone-polyhedron,
which in this case we denote by $P=\bar{P}(g,h,\mathcal{T},u)$.

For an ideal cone-polyhedron $P=P(d,\mathcal{T})$ or
$P=\bar{P}(g,h,\mathcal{T},u)$ we say that an edge $e$ of $\mathcal{T}$ is an
edge of $P$ if $\alpha_{e}\neq\pi$. A face of $P$ is a connected component of
the boundary of $P$ minus all edges. This determines _the face decomposition
of $P$_. A priori there could be non-simply connected faces in the face
decomposition, then it would not be a polygonal decomposition in the sense of
Definition 2.2. We will soon show that this is not the case.

Let $o\in\mathbb{H}^{3}$, $\Pi\subset\mathbb{H}^{3}$ be a geodesic plane not
containing $o$ and $q$ be the closest point from $\Pi$ to $o$. By
$\mu:\Pi\rightarrow\mathbb{R}$ we denote the distance function from $\Pi$ to
$o$. Then for each $p\in\Pi$ we have

$\cosh\mu(p)=\cosh\mu(q)\cosh d_{\mathbb{H}^{3}}(p,q).$ (2.3)

Now consider an ideal cone-polyhedron $P$. Let $\mu:S\backslash
V\rightarrow\mathbb{R}$ be the _distance function_ from the boundary of $P$,
identified with $S\backslash V$, to the marked point of $P$. Let
$\psi:[0,\tau]\rightarrow S$ be a unit-speed geodesic in the cusp-metric of
the boundary. Then it may have kink points coming from the edges of $P$.
Denote their coordinates by $t_{1},\ldots,t_{k}\in[0,\tau]$, also set
$t_{0}:=0$, $t_{k+1}:=\tau$. Due to (2.3), the restriction of $\mu\circ\psi$
to $(t_{i},t_{i+1})$ has the form

$\mu\circ\psi(t)=\operatorname{arcosh}(b_{i}\cosh(t-a_{i}))$ (2.4)

for some real numbers $a_{i}$, $b_{i}$ where $i=0,\ldots,k$. If $P$ is convex,
then at the kink points $t_{i}$ the left derivative of $\mu\circ\psi$ is
greater than the right derivative.

######  Lemma 2.24.

All faces of an ideal cone-polyhedron $P$ are simply connected.

###### Proof.

In the same way as in [34, Lemma 4.12] one can show that if we have a non-
simply connected face, then it contains a closed geodesic $\psi$ (in the
extrinsic sense, so intersecting no edges of $P$). Then the restriction of
$\mu$ to $\psi$ must be periodic. On the other hand, it must have the form
(2.4), which is not periodic, hence the contradiction. ∎

It follows that the face decomposition of $P$ is a polygonal decomposition.

######  Lemma 2.25.

The ideal cone-polyhedron $P(d,\mathcal{T})$ is convex if and only if
$\mathcal{T}$ is Delaunay for $d$. Moreover, if $e\in E(\mathcal{T})$ is an
interior edge of a face in the Delaunay decomposition containing
$\mathcal{T}$, then $\alpha_{e}=\pi$.

###### Proof.

Indeed, the dihedral angle of the intersection of two planes in
$\mathbb{H}^{3}$ is equal to the intersection angle of the two circles in
$\partial_{\infty}\mathbb{H}^{3}$ that are their boundaries at infinity.
Clearly, the conformal structure at $\partial_{\infty}\mathbb{H}^{3}$
coincides with the conformal structure of the unit sphere. This implies the
desired statement. ∎

This shows that the face decomposition of $P(d,\mathcal{T})$ is exactly the
Delaunay decomposition containing $\mathcal{T}$. Now we are ready to show that
it exists and is uniquely determined by $d$.

###### Proof of Lemma 2.7..

Let $\mathcal{T}$ and $\mathcal{T}^{\prime}$ be two different Delaunay
triangulations of $d$, $e\in E(\mathcal{T})$ and $p\in S$ be its intersection
point with an edge $e^{\prime}\in E(\mathcal{T}^{\prime})$ in their
realizations in $d$. Due to Lemma 2.25, both ideal cone-polyhedra
$P=P(d,\mathcal{T})$ and $P^{\prime}=P(d,\mathcal{T}^{\prime})$ are convex.

By $\mu$, $\mu^{\prime}$ we denote the distance functions from the boundaries
of $P$, $P^{\prime}$ respectively to the marked point where the boundaries are
identified with $S\backslash V$. From the convexity of $P^{\prime}$, by
considering a semi-ideal triangle subtended by $e$ one sees that
$\mu^{\prime}(p)\geq\mu(p)$. Similarly, by considering the semi-ideal triangle
subtended by $e^{\prime}$ we see that $\mu(p)\geq\mu(p^{\prime})$. Thus,
$\mu(p)=\mu^{\prime}(p)$.

The union of $\mathcal{T}$ and $\mathcal{T}^{\prime}$ cuts $(S,d)$ into convex
polygons. We can cut it further without adding new vertices to obtain a
triangulation. This decomposes $P$, $P^{\prime}$ into tetrahedra with some
ideal vertices and some non-ideal ones, all with a marked non-ideal point. It
is easy to see that if two hyperbolic tetrahedra, both with a marked non-ideal
point, have isometric spherical links at the marked points and equal distances
from these points to the respective non-ideal points, then the tetrahedra are
isometric. This allows us to construct a marked isometry between $P$ and
$P^{\prime}$. Thus, both $\mathcal{T}$ and $\mathcal{T}^{\prime}$ refine the
face decomposition of $P=P^{\prime}$. Thus, the Delaunay decomposition of $d$
is unique and is the face decomposition of $P$.

Now we show that the Delaunay decomposition exists. Take any triangulation
$\mathcal{T}$ realized by $d$. If it is not Delaunay, then there exists an
edge $e\in E(\mathcal{T})$ such that its dihedral angle in $P(d,\mathcal{T})$
is greater than $\pi$. It is easy to see that $e$ is adjacent to two distinct
triangles and that the union of their realizations in $d$ is a convex
quadrilateral. Thus, we can flip $e$. The volume of $P(d,\mathcal{T})$
increases under this operation. Due to Lemma 2.4, there are finitely many
triangulations realized by $d$, thus, the algorithm finishes in finitely many
steps. ∎

Let $\mathcal{T}$ be a Delaunay triangulation for
$d\in\mathfrak{M}_{+}(\mathcal{T})$. Due to Lemma 2.25, the ideal cone-
polyhedron $P(d,\mathcal{T})$ is independent on a particular choice of
$\mathcal{T}$ in case if there are several Delaunay triangulations. We denote
it just by $P(d)$ and denote its boundary cusp-metric by $g_{d}$. It also
comes with a canonical decoration $h_{d}$ of $g_{d}$.

Now we need to return to the second construction of ideal cone-polyhedra. So
take a decorated hyperbolic cusp-metric $(g,h)$ on $(S,V)$. We can show

######  Lemma 2.26.

Let $P=\bar{P}(g,h,\mathcal{T},u)$ and
$P^{\prime}=\bar{P}(g,h,\mathcal{T}^{\prime},u)$ be two convex ideal cone-
polyhedra. Then they are marked isometric.

###### Proof.

The proof is similar to the proof of Lemma 2.7 just above, but is more
involved as we need to use that the distance functions of $P$ and $P^{\prime}$
coincide asymptotically at the cusps. This is almost identical to the proof of
a similar fact in [34, Section 4.1], but we sketch it here for self-
completeness. The proof was inspired by [4, Section 2.3] or [11, Section 3.1],
but the situation there was dealing with compact boundaries, so without the
need to investigate the asymptotic behaviour.

Consider an edge $e$ of $\mathcal{T}$. We parametrize it by the signed
distance along $e$ to the horocycle of $h$ at one of its endpoints $v$. Let
$\mu$, $\mu^{\prime}$ be the distance functions of $P$ and $P^{\prime}$
respectively and $f$, $F$ be the restrictions of $\cosh\mu$,
$\cosh\mu^{\prime}$ to $e$.

The function $f$ has the form

$f(t)=b\cosh(t-a).$

It is important to note that two functions of this form either coincide or
have at most one point in common. There exist real numbers
$t_{1}<\ldots<t_{k}\in\mathbb{R}$ such that at $(t_{i},t_{i+1})$ the function
$F(t)$ has the form

$F(t)=b_{i}\cosh(t-a_{i}).$

Here we also assume $t_{0}:=-\infty$, $t_{k+1}:=+\infty$. It might happen that
$k=0$. By $F_{i}(t)$ we denote the function $b_{i}\cosh(t-a_{i})$ extended to
all $\mathbb{R}$. It is clear that for all $i=0,\ldots,k-1$ we have
$F_{i}(t)>F_{i+1}(t)$ for $t>t_{i+1}$ and $F_{i}(t)<F_{i+1}(t)$ for
$t<t_{i+1}$. By induction we get that $F_{i}(t)>F(t)$ for all
$t\notin(t_{i},t_{i+1})$.

We prove that $F(t)\geq f(t)$ for all $t\in\mathbb{R}$. We need two simple
statements.

######  Claim 1.

Let

$f_{1}(x)=b_{1}\cosh(x-a_{1}),$ $f_{2}(x)=b_{2}\cosh(x-a_{2})$

and for $t\in\mathbb{R}$ we have $f_{1}(t)=f_{2}(t)$ and
$\dot{f}_{1}(t)>\dot{f}_{2}(t)$. Then $a_{2}>a_{1}$.

The proof is an easy computation, see [34, Section 4.1]. It then follows by
induction that $a_{0}<\ldots<a_{k}$.

######  Claim 2.

Let $\psi_{1}$ and $\psi_{2}$ be two distinct geodesic lines in
$\mathbb{H}^{2}$ meeting at a point $A\in\partial_{\infty}\mathbb{H}^{2}$ and
$o\in\mathbb{H}^{2}$ be outside of the angle formed by $\psi_{1}$, $\psi_{2}$.
Let $A$ be decorated by an horocycle and $\psi_{1}$, $\psi_{2}$ be
parametrized by the signed distance to this horocycle. Denote the hyperbolic
cosines of the distance functions from $\psi_{1}$ and $\psi_{2}$ to $o$ by

$f_{1}(t)=b_{1}\cosh(t-a_{1}),$ $f_{2}(t)=b_{2}\cosh(t-a_{2})$

respectively. Then ${f_{1}(t)-f_{2}(t)}$ has a constant nonzero sign. Besides,
if $f_{1}(t)>f_{2}(t)$, then $a_{2}>a_{1}$.

The proof is straightforward. Now consider $F_{0}(t)$ and $f(t)$. Their
difference has a constant sign. Suppose that $F_{0}(t)<f(t)$. Then Claim 2
implies that $a<a_{0}$. Thus, we have $a<a_{k}$. Also we see that $f(t)>F(t)$
for all $t\in\mathbb{R}$. Consider now the other parametrization of $e$ by the
signed distance along $e$ to the horocycle at its second endpoint. We get the
reparametrized functions $f(a_{e}-t)$, $F(a_{e}-t)$, where $a_{e}$ is the
length of $e$ in the metric $g$ decorated by $h$. We similarly obtain
$a_{e}-a<a_{e}-a_{k+1}$, or $a_{k+1}<a$. This is a contradiction, so
$F_{0}(t)\geq f(t)$ for all $t\in\mathbb{R}$. Similarly, $f(t)\leq F_{k+1}(t)$
for all $t\in\mathbb{R}$.

Now assume that for some $t^{\prime}\in\mathbb{R}$ we have
$f(t^{\prime})>F(t^{\prime})$. Then either for each $t>t^{\prime}$ or for each
$t<t^{\prime}$ we have $f(t)>F(t)$. But this contradicts with $F_{0}(t)\geq
f(t)$, $F_{k+1}(t)\geq f(t)$ for all $t\in\mathbb{R}$.

Thus, for each point $p\in S$ that is an intersection point of two distinct
edges $e\in E(\mathcal{T})$ and $e^{\prime}\in E(\mathcal{T})$ realized in
$(S,g)$ we have $\mu(p)\leq\mu^{\prime}(p)$. Similarly we can obtain
$\mu(p)\geq\mu^{\prime}(p)$. Hence, $\mu(p)=\mu^{\prime}(p)$.

The union of $\mathcal{T}$ and $\mathcal{T}^{\prime}$ cuts $(S,g)$ into convex
hyperbolic polygons with some vertices possibly at cusps. We can cut it
further to a triangulation without adding new vertices. At each vertex, which
is not a cusp, the distances in $P$, $P^{\prime}$ to the marked point
coincide. At each cusp the signed distances from the horospheres determined by
$h$ to the marked point coincide. One can see that each tetrahedron is
determined up to isometry by this data. It follows that $P$ is marked
isometric to $P^{\prime}$. ∎

Lemma 2.26 implies that if an ideal cone-polyhedron
$\bar{P}(g,h,\mathcal{T},u)$ is convex, we can denote it just by
$\bar{P}(g,h,u)$. By $U^{\prime}(g,h)$ we denote the set of those
$u\in\mathbb{R}^{V}$ that there exists a convex ideal cone-polyhedron
$\bar{P}(g,h,u)$. In the case $g=g_{d}$, $h=h_{d}$ for a spherical cone-metric
$d$ we also denote $U^{\prime}(g_{d},h_{d})$ by $U^{\prime}(d)$, and denote
$\bar{P}(g_{d},h_{d},u)$ by $P(d,u)$.

######  Lemma 2.27.

The set $U^{\prime}(g,h)$ is open.

###### Proof.

Let $u\in U^{\prime}(g,h)$ and $\mathcal{T}$ be a face triangulation of
$\bar{P}(g,h,u)$. As all semi-ideal tetrahedra determined by $g$, $h$,
$\mathcal{T}$ and $u$ are non-degenerate, they exist also for all $u^{\prime}$
sufficiently close to $u$. Then the ideal cone-polyhedron
$\bar{P}(g,h,\mathcal{T},u^{\prime})$ is well-defined. However, it can be non-
convex.

Due to Lemma 2.24, there are finitely many face triangulations of
$\bar{P}(g,h,u)$. Choose a neighbourhood $N\ni u$ so that all edges of
$\bar{P}(g,h,u)$ remain to have $\alpha_{e}<\pi$ also in
$\bar{P}(g,h,\mathcal{T}^{\prime},u^{\prime})$ for all $u^{\prime}\in N$ and
all face triangulations $\mathcal{T}^{\prime}$ of $\bar{P}(g,h,u)$. Then we
perform the flip-algorithm for $\bar{P}(g,h,\mathcal{T},u^{\prime})$ from the
proof of Lemma 2.7: at each step take a non-convex edge and flip it. The flip
is always possible and the volume increases under a flip. Due to our choice of
$N$, an edge of $\bar{P}(g,h,u)$ can never be flipped. Thus, all appearing
triangulations are face triangulations of $\bar{P}(g,h,u)$. Since there are
finitely many of them, the algorithm finishes in finitely many steps,
providing a triangulation $\mathcal{T}^{\prime}$ such that
$\bar{P}(g,h,\mathcal{T}^{\prime},u^{\prime})$ is convex. Hence, $N\subset
U^{\prime}(g,h)$. ∎

By $U(d)$ we denote the connected component of $U^{\prime}(d)$ containing $o$.
We actually think that $U(g,h)$ is always connected, so $U(d)=U^{\prime}(d)$.
However, we do not need this for the needs of the present paper, so we do not
dwell upon this.

For a triangulation $\mathcal{T}$ of $(S,V)$ let $U^{\prime}(g,h,\mathcal{T})$
be the set of $u\in U^{\prime}(g,h)$ such that $\mathcal{T}$ is isotopic with
respect to $V$ to a face triangulation of $\bar{P}(g,h,u)$. By
$U(d,h,\mathcal{T})$ we denote the intersection
$U^{\prime}(g,h,\mathcal{T})\cap U(g,h)$. So $U(g,h,\mathcal{T})$ provide a
locally finite decomposition of $U(g,h)$ into cells with piecewise-smooth
boundaries. In case of $g=g_{d}$, $h=h_{d}$ we denote
$U(g_{d},h_{d},\mathcal{T})$ just by $U(d,\mathcal{T})$.

Using geometric arguments similar to the proof of Lemma 4.14 in [34] one can
show that if $u\in U(g,h,\mathcal{T})$, then $\mathcal{T}$ is an
Epstein–Penner triangulation for $g$ and a decoration $h^{\prime}$ having the
horocycle at $v$ at the signed distance $u_{v}$ from the respective horocycle
of $h$. Combined with the Akiyoshi result, this would give another proof of
Corollary 2.18. We will not rely on this, so we do not clarify this more.

Consider two metrics $d,d^{\prime}\in\mathfrak{M}_{+,T}(S,V)$. Assume that the
respective cusp-metrics $g_{d},g_{d^{\prime}}$ are isometric by an isometry
isotopic to identity with respect to $V$. Let $h_{d}$ be the decoration of
$g_{d}$ coming from the ideal cone-polyhedron $P(d)$. Identify $g_{d}$ and
$g_{d^{\prime}}$ with the help of the isometry we have. However, the
decoration $h_{d^{\prime}}$ coming from the ideal cone-polyhedron
$P(d^{\prime})$ is different from $h_{d}$. By
$u_{d}(d^{\prime})\in\mathbb{R}^{V}$ we denote the signed distances of the
horocycles of $h_{d^{\prime}}$ from the horocycles of $h_{d}$. Clearly, the
ideal cone-polyhedron $P(d^{\prime})$ coincides with the ideal cone-polyhedron
$P(d,u_{d}(d^{\prime})).$ We can show

######  Lemma 2.28.

Two metrics $d,d^{\prime}\in\mathfrak{M}_{T}(S,V)$ are discretely conformally
equivalent if and only if the respective cusp-metrics $g_{d},g_{d^{\prime}}$
are isometric by an isometry isotopic to the identity with respect to $V$ and
$u_{d}(d^{\prime})\in U(d)$.

###### Proof.

Let $u^{\prime}$ be in $U(d)$ and $d^{\prime}$ be the spherical cone-metric
induced on the spherical link of the marked point in $P(d,u^{\prime})$. We
show that $d^{\prime}$ is discretely conformally equivalent to $d$. Consider a
path $u_{t}$ connecting $u^{\prime}$ with the origin $o$. As the decomposition
of $U(d)$ into cells $U(d,\mathcal{T})$ is locally finite and piecewise-
smooth, we can choose $u_{t}$ so that it intersects the boundaries finitely
many times. Let $u_{1},\ldots,u_{k}$ be the intermediate points of the path at
the boundaries of $U(d,\mathcal{T})$ and $d_{1},\ldots,d_{k}$ be the induced
spherical cone metrics. We also set $u_{0}:=o$, $u_{k+1}:=u^{\prime}$,
$d_{0}:=d$, $d_{k+1}:=d^{\prime}$. By construction, each $u_{i}$ and $u_{i+1}$
belongs to the same $U(d,\mathcal{T})$, we denote this $\mathcal{T}$ by
$\mathcal{T}_{i}$. Due to Lemma 2.25,
$d_{i}\in\mathfrak{D}_{+}(\mathcal{T}_{i})$. So we need to check that $d_{i}$
and $d_{i+1}$ are discretely conformally equivalent in the sense of (b) from
Definition 2.8.

Let $e$ be an edge of $\mathcal{T}_{i}$ with endpoints $v$ and $w$. Then, due
to Lemma 2.20, we have

$\cos(l_{e}(d_{i}))=1-2\exp(a_{e}-u_{i,v}-u_{i,w}),$
$\cos(l_{e}(d_{i+1}))=1-2\exp(a_{e}-u_{i+1,v}-u_{i+1,w}).$

Here $a_{e}$ is the length of the edge $e$ in the metric $g_{d}$ with the
decoration $h_{d}$. We can rewrite it as

$\sin\left(\frac{l_{e}(d_{i})}{2}\right)=\sqrt{\frac{1-\cos
l_{e}(d_{i})}{2}}=\exp\left(\frac{a_{e}-u_{i,v}-u_{i,w}}{2}\right),$
$\sin\left(\frac{l_{e}(d_{i+1})}{2}\right)=\sqrt{\frac{1-\cos
l_{e}(d_{i+1})}{2}}=\exp\left(\frac{a_{e}-u_{i+1,v}-u_{i+1,w}}{2}\right),$
$\sin\left(\frac{l_{e}(d_{i+1})}{2}\right)=\exp\left(\frac{u_{i,v}-u_{i+1,v}}{2}+\frac{u_{i,w}-u_{i+1,w}}{2}\right)\sin\left(\frac{l_{e}(d_{i})}{2}\right).$

This shows that each $d_{i}$ and $d_{i+1}$ are discretely conformally
equivalent and so are $d$ and $d^{\prime}$.

Now suppose that we have two discretely conformal metrics $d$ and
$d^{\prime}$. We want to show that $g_{d}$ is marked isometric to
$g_{d^{\prime}}$. Due to Definition 2.8, the uniqueness of Delaunay
decomposition and Lemma 2.25, it is enough to show this under the assumption
that both $d,d^{\prime}\in\mathfrak{D}_{+}(\mathcal{T})$ for some
triangulation $\mathcal{T}$. Then there exists $u\in\mathbb{R}^{V}$ such that
for each edge $e$ of $\mathcal{T}$ with endpoints $v$ and $w$ we have

$\sin\left(\frac{l_{e}(d^{\prime})}{2}\right)=\exp\left(\frac{u_{v}+u_{w}}{2}\right)\sin\left(\frac{l_{e}(d)}{2}\right).$
(2.5)

Note that the lengths $a_{e}$, $a^{\prime}_{e}$ of the edge $e$ in the cusp-
metric $g_{d}$ with the decoration $h_{d}$ and in $g_{d^{\prime}}$ with
$h_{d^{\prime}}$ respectively satisfy

$\sin\left(\frac{l_{e}(d)}{2}\right)=\exp\left(\frac{a_{e}}{2}\right),$ (2.6)
$\sin\left(\frac{l_{e}(d^{\prime})}{2}\right)=\exp\left(\frac{a^{\prime}_{e}}{2}\right).$
(2.7)

For each $v\in V$ choose at the respective cusp of $P(d^{\prime})$ a horoball
so that the marked point is at the signed distance $-u_{v}$ from it. This
determines a decoration $h^{\prime}$ of $g_{d^{\prime}}$. It follows from
(2.5), (2.6) and (2.7) that the length of $e$ in $g_{d^{\prime}}$ with the
decoration $h^{\prime}$ is $a_{e}$. Thus, the pairs $(g_{d},h_{d})$ and
$(g_{d^{\prime}},h^{\prime})$ have the same Penner coordinates. It follows
that $g_{d^{\prime}}$ is marked isometric to $g_{d}$.

The operation above allows us to define a map from $\mathfrak{C}_{+}(d)$ to
$U^{\prime}(d)$, which is continuous. By Corollary 2.13, $\mathfrak{C}_{+}(d)$
is connected. It follows that for every discretely conformal metrics $d$ and
$d^{\prime}$ we have $u_{d}(d^{\prime})\in U(d)$. ∎

The construction above produces a continuous map
$\Psi_{d}:U(d)\rightarrow\mathfrak{C}_{+}(d)$, which is injective due to
Lemmas 2.25 and 2.7. It was also shown that it has a continuous injective
inverse. Hence, $\Psi_{d}$ is a homeomorphism. One can show that it is
$C^{1}$, but we will only use that the curvature function
$\kappa\circ\Psi_{d}$ is $C^{1}$. We will obtain this in next subsection. For
$u\in U(d)$ we denote the metric $\Psi_{d}(u)$ by $u*d$.

######  Remark 2.29.

It follows from the considerations above that if $u,u^{\prime}\in
U(g,h,\mathcal{T})$ for some triangulation $\mathcal{T}$, then the respective
spherical cone-metrics $d$, $d^{\prime}$ satisfy
$d,d^{\prime}\in\mathfrak{D}_{+}(\mathcal{T})$ and for each $e\in
E(\mathcal{T})$ with endpoints $v$, $w$ we have

$\sin\left(\frac{l_{e}(d^{\prime})}{2}\right)=\exp\left(\frac{u_{v}+u_{w}-u^{\prime}_{v}-u^{\prime}_{w}}{2}\right)\sin\left(\frac{l_{e}(d)}{2}\right).$

Also Lemma 2.28 and Corollary 2.18 in turn imply

######  Corollary 2.30.

Let $d\in\mathfrak{M}_{+,T}(S,V)$. There are finitely many triangulations
$\mathcal{T}$ of $(S,V)$ such that $U(d,\mathcal{T})$ is non-empty.

###  2.6 Discrete curvature

Let $(g,h)$ be a decorated cusp metric. For a convex ideal cone polyhedron
$P=\bar{P}(g,h,u)$ define its _total discrete curvature_ $H(P)\in\mathbb{R}$
as

$H(P)=-2{\rm vol}(P)+\sum_{v\in V}u_{v}\kappa_{v}+\sum_{e\in
E(\mathcal{T})}a_{e}(\pi-\alpha_{e}).$

Here $\kappa_{v}$ are the curvatures of the interior edges of $P$,
$\mathcal{T}$ is a face triangulation of $P$, $\alpha_{e}$ are the dihedral
angles of edges of $P$ and $a_{e}$ are their lengths in $g$ with $h$.

This defines the _discrete curvature functional_ $H$ over $U^{\prime}(g,h)$.
It is also sometimes called _the discrete Hilbert–Einstein functional_. Here
the sum $-2{\rm vol}(P)+\sum_{v\in V}u_{v}\kappa_{v}$ can be regarded as the
discrete version of the integral of the scalar curvature, and the sum
$\sum_{e\in E(\mathcal{T})}a_{e}(\pi-\alpha_{e})$ is the discrete version of
the integral of the mean curvature of the boundary. This functional was first
introduced by Volkov in his PhD thesis from 1955 to give a variational proof
of the Alexandrov realization theorem, see [45, 44]. It was notably used in
various similar problems since then, see [19] for a survey and [4, 11, 12, 34,
35] for examples of its use.

######  Lemma 2.31.

The functional $H$ is $C^{1}$ and we have

$\frac{\partial H}{\partial u_{v}}=\kappa_{v}.$

###### Proof.

The proof basically is an application of the celebrated Schläffli formula,
which was adapted to partially ideal hyperbolic polyhedra in [36, Theorem
14.5].

Consider a decorated semi-ideal tetrahedron $\bar{T}$ with the edge lengths of
the ideal face $a_{1}$, $a_{2}$, $a_{3}$, their dihedral angles $\alpha_{1}$,
$\alpha_{2}$, $\alpha_{3}$ respectively, the lengths of semi-ideal edges
$u_{1}$, $u_{2}$, $u_{3}$ and their dihedral angles $\omega_{1}$,
$\omega_{2}$, $\omega_{3}$ respectively. Then the Schläffli formula says that

$-2d{\rm vol}(\bar{T})=\sum u_{i}d\omega_{i}+\sum a_{i}d\alpha_{i}.$

Suppose that $u$ is in the interior of a cell $U(g,h,\mathcal{T})$. Then the
combinatorics does not change locally around $P$ and, as the dihedral angles
are smooth functions of the edge lengths of semi-ideal polyhedra, we see that
$H$ is smooth. Summing all the equalities for particular tetrahedra we get

$-2d{\rm vol}(P)=-\sum_{v\in
V}u_{v}d\kappa_{v}+\sum_{e\in\mathcal{T}}a_{e}d\alpha_{e}.$

This implies

$dH(P)=\sum_{v\in V}\kappa_{v}du_{v},$

which is the desired statement.

When $u$ is not in the interior of $U(g,h,\mathcal{T})$, we recall that the
decomposition of $U(g,h)$ into cells $U(g,h,\mathcal{T})$ is finite and the
cells have piecewise-smooth boundaries. Thus, one can compute directional
derivatives at $u$ with the help of the method above. One sees then that all
the partial derivatives

$\frac{\partial H}{\partial u_{v}}=\kappa_{v}$

exist and are continuous around $u$. This implies that $H$ is $C^{1}$. ∎

Now we are going to show that $H$ is actually $C^{2}$ and to compute its
second derivatives. By $\vec{E}_{v}(\mathcal{T})$ we denote the set of
oriented edges of $\mathcal{T}$ emanating from $v\in V$. By
$\vec{E}_{vw}(\mathcal{T})$ we denote the set of oriented edges emanating from
$v\in V$ and ending at $w\in V$. For a convex ideal cone-polyhedron
$P=\bar{P}(g,h,u)$ and an oriented edge $\vec{e}$ of its face triangulation we
denote by $\alpha^{+}_{\vec{e}}$ and by $\alpha^{-}_{\vec{e}}$ the dihedral
angles of the two tetrahedra adjacent to $\vec{e}$. By $\rho_{\vec{e}}$ we
denote the length of the part of the canonical horocycle between $\vec{e}$ and
the interior edge of $P$ at the starting endpoint of $\vec{e}$.

######  Lemma 2.32.

The functional $H$ is $C^{2}$ and we have

$v\neq w:\frac{\partial^{2}H}{\partial u_{v}\partial
u_{w}}=\frac{\partial\kappa_{v}}{\partial
u_{w}}=\frac{\partial\kappa_{w}}{\partial
u_{v}}=\sum_{\vec{e}\in\vec{E}_{vw}(\mathcal{T})}\frac{\cot(\alpha^{+}_{\vec{e}})+\cot(\alpha^{-}_{\vec{e}})}{\rho_{\vec{e}}\exp(u_{v})\sin(l_{\vec{e}})},$
(2.8)
$v=w:\frac{\partial^{2}H}{\partial^{2}u_{v}}=\frac{\partial\kappa_{v}}{\partial
u_{v}}=-\sum_{\vec{e}\in\vec{E}_{v}(\mathcal{T})}\frac{\cot(\alpha^{+}_{\vec{e}})+\cot(\alpha^{-}_{\vec{e}})}{\rho_{\vec{e}}\exp(u_{v})\sin(l_{\vec{e}})}\cos(l_{\vec{e}})+\sum_{\vec{e}\in\vec{E}_{vv}(\mathcal{T})}\frac{\cot(\alpha^{+}_{\vec{e}})+\cot(\alpha^{-}_{\vec{e}})}{\rho_{\vec{e}}\exp(u_{v})\sin(l_{\vec{e}})}.$
(2.9)

###### Proof.

Similarly to the proof of Lemma 2.31 it is enough to compute the derivatives
for the case when $u$ belongs to the interior of a cell $U(g,h,\mathcal{T})$.
In the general case one checks that the partial derivatives exist and are
continuous, so the functional is $C^{2}$.

The computation follows from the computation of the derivatives of the
dihedral angles in a single semi-ideal tetrahedron $\bar{T}=BA_{1}A_{2}A_{3}$.
We inherit the notation from the previous proof and from Section 2.4. It
remains to denote the lengths of the parts of the canonical horocycles in the
ideal triangle $A_{1}A_{2}A_{3}$ by $\lambda_{1}$, $\lambda_{2}$,
$\lambda_{3}$ respectively.

The solid angle at the vertex $A_{1}$ of $\bar{T}$ cuts off a Euclidean
triangle from the canonical horosphere at $A_{1}$. It has edge lengths
$\lambda_{1}$, $\rho_{12}$ and $\rho_{13}$, and angles $\omega_{1}$,
$\alpha_{3}$ and $\alpha_{2}$ respectively. We have

$\cos\omega_{1}=\frac{\rho_{12}^{2}+\rho_{13}^{2}-\lambda_{1}^{2}}{2\rho_{12}\rho_{13}}.$

Differentiating it we get

$\frac{\partial\omega_{1}}{\partial\rho_{12}}=-\frac{\cot(\alpha_{3})}{\rho_{12}},~{}~{}~{}~{}~{}\frac{\partial\omega_{1}}{\partial\rho_{13}}=-\frac{\cot(\alpha_{2})}{\rho_{13}}.$

Now we differentiate the cosine law for the semi-ideal triangle $BA_{1}A_{2}$
given in Lemma 2.20, use the sine law from Lemma 2.21 and get the derivatives

$\frac{\partial\rho_{12}}{\partial
u_{1}}=-\frac{\cot(l_{12})}{\exp(u_{1})},~{}~{}~{}~{}~{}\frac{\partial\rho_{12}}{\partial
u_{2}}=\frac{1}{\exp(u_{1})\sin(l_{12})}.$

This gives us

$\frac{\partial\omega_{1}}{\partial
u_{1}}=\frac{\partial\omega_{1}}{\partial\rho_{12}}\frac{\partial\rho_{12}}{\partial
u_{1}}+\frac{\partial\omega_{1}}{\partial\rho_{13}}\frac{\partial\rho_{13}}{\partial
u_{1}}=\frac{1}{\exp(u_{1})}\left(\frac{\cot(\alpha_{3})\cot(l_{12})}{\rho_{12}}+\frac{\cot(\alpha_{2})\cot(l_{13})}{\rho_{13}}\right).$

One can see that summing this up for all tetrahedra constituting $P$ gives us
the first part of (2.9).

We also have

$\frac{\partial\omega_{1}}{\partial
u_{2}}=\frac{\partial\omega_{1}}{\partial\rho_{12}}\frac{\partial\rho_{12}}{\partial
u_{2}}=-\frac{\cot(\alpha_{3})}{\rho_{12}\exp(u_{1})\sin(l_{12})}.$

Summing this up for all tetrahedra gives us (2.8) and the second part of
(2.9). The proof is finished. ∎

In particular, Lemma 2.32 implies that $\kappa\circ\Psi_{d}$ is a $C^{1}$-map.

##  3 Proof of Theorem 1.5

###  3.1 Outline of the proof

We fix $d\in\mathfrak{M}_{+,T}(S,V)$. Abusing the notation, in the rest of the
paper we denote $\kappa\circ\Psi_{d}$ just by $\kappa$, so now

$\kappa:U(d)\rightarrow(-\infty,2\pi)^{V}$

and it is a $C^{1}$-map. Denote

$U_{C}(d):=\\{u\in U(d):\text{for each $v\in V$,
$\kappa_{v}(u)>0$}\\}=\kappa^{-1}((0,2\pi)^{V}),$

which is an open domain in $\mathbb{R}^{V}$. Denote $K$ as the set of
$\kappa\in(0,2\pi)^{V}$ satisfying (1.2) and (1.3). Then $K$ is a convex open
domain in $\mathbb{R}^{V}$. Then Theorem 1.5 is equivalent to the fact that
the discrete curvature map $\kappa$ is a bijection from $U_{C}(d)$ to $K$.

It follows from Theorem 1.2 that the image of $\kappa$ is in $K$. Then Theorem
1.5 is obtained from the following three lemmas.

######  Lemma 3.1.

The differential $d\kappa$ is non-degenerate on $U_{C}(d)$.

######  Lemma 3.2.

The map $\kappa:U_{C}(d)\rightarrow K$ is proper.

######  Lemma 3.3.

$U_{C}(d)$ is connected.

###### Proof of Theorem 1.5.

By Lemmas 3.1 and 3.2, $\kappa$ is a proper local homeomorphism. Thus it is a
covering map. Since $K$ is simply connected and by Lemma 3.3 we get that
$U_{C}(d)$ is connected, $\kappa$ is a global homeomorphism. ∎

###  3.2 Infinitesimal rigidity

####  3.2.1 Euclidean cone-polygons

We show the non-degeneracy of $d\kappa$ by a “dimensional reduction” argument
appearing in [4, 20]. To this purpose we need to introduce new objects.

######  Definition 3.4.

A _Euclidean cone-polygon_ $Q$ is a complete metric space homeomorphic to the
2-disk with a locally Euclidean metric in the interior except one point $v$,
where it is locally isometric to a Euclidean cone of total angle $\omega$, and
with piecewise geodesic boundary.

We denote the kink points of the boundary of $Q$ by $w_{1},\ldots,w_{n}$ in
the cyclic order. The cone-polygon $Q$ can be naturally triangulated by
geodesics from $v$ to the kink-points. By $\rho_{i}$ we denote the length of
the geodesic from $v$ to $w_{i}$, by $\alpha_{i}$ we denote the angle at
$w_{i}$, by $\lambda_{i}$ we denote the length of the geodesic segment between
$w_{i}$ and $w_{i+1}$ and by $\omega_{i}$ we denote the angle between the
geodesics from $v$ to $w_{i}$ and $w_{i+1}$.

The cone-polygon $Q$ is uniquely determined by the lengths $\rho_{i}$ and
$\lambda_{i}$. Fix $\lambda_{i}$ and consider the point
$\rho\in\mathbb{R}^{n}$. For small deformations of $\rho$ the cone-polygon
remains to exist. Let $x\in T_{\rho}\mathbb{R}^{n}$ be a tangent vector. Every
angle $\alpha_{j}$ is a smooth function around $\rho$. By
$\dot{\alpha}_{j}(x)$ we denote its derivative in the direction $x$, which is
a linear form in $x_{i}$. Similarly, $\dot{\omega}(x)$ is the directional
derivative of $\omega$.

Consider the bilinear form $I$ on $T_{\rho}\mathbb{R}^{n}$ defined by

$I(x,y)=\sum_{i}\rho_{i}x_{i}\dot{\alpha}_{i}(y)=\sum_{i,j}\rho_{i}\frac{\partial\alpha_{i}}{\partial\rho_{j}}x_{i}y_{j}.$

######  Remark 3.5.

One can see that $I$ is the Hessian of the functional

$G(Q):=-2{\rm area}(Q)+\sum_{i}\rho_{i}^{2}\alpha_{i}.$

######  Lemma 3.6.

Let $\omega<2\pi$. Then the signature of $I$ is $(+,-,\ldots,-)$.

###### Proof.

First we show the non-degeneracy. Suppose the converse, then there exists
$x\in T_{\rho}\mathbb{R}^{n}$ such that for every other vector $y\in
T_{\rho}\mathbb{R}^{n}$ we have $I(y,x)=0$. As the coefficients of $I$ are
equal to $\rho_{i}\frac{\partial\alpha_{i}}{\partial\rho_{j}},$ we get
$\dot{\alpha}_{i}(x)=0$ for each $i$.

Now cut $Q$ along the geodesic $vw_{1}$ and develop it on the Euclidean plane
$\mathbb{E}^{2}$ as a polygon $Q^{\prime}$. Slightly abusing the notation we
denote the vertices of $Q^{\prime}$ by
$v,w^{\prime}_{1},w_{2},\ldots,w_{n},w^{\prime\prime}_{1}$. The infinitesimal
deformation $x$ gives rise to an infinitesimal deformation of $Q^{\prime}$. As
$\dot{\lambda}_{i}(x)$ and $\dot{\alpha}_{i}(x)$ are zero, it is clear that
the infinitesimal deformation of all vertices of $Q^{\prime}$ except $v$ is
induced by a global Killing field on $\mathbb{E}^{2}$. We can assume that it
is zero. Then, as the infinitesimal change of the distance from $v$ to
$w^{\prime}_{1}$ is the same as of the distance from $v$ to
$w^{\prime\prime}_{1}$, we see that the infinitesimal deformation of $v$ is
orthogonal to the segment through $w^{\prime}_{1}$ and $w^{\prime\prime}_{1}$.
But every such non-zero deformation induces a non-trivial change on the sum of
angles $\angle vw^{\prime}_{1}w_{2}+\angle
vw^{\prime\prime}_{1}w_{n}=\alpha_{1}$. Thus, it is zero and $x=0$.

Now we deform $\rho_{i}$ and $\omega_{i}$ continuously so that $\rho_{i}$
remain positive and the sum of $\omega_{i}$ remains constant until all
$\rho_{i}$ become equal and all $\omega_{i}$ become equal. As $I$ stays non-
degenerate during this transformation, its signature does not change. It is
easy to compute that in the end the matrix of $I$ becomes

$\frac{1}{\rho^{2}\sin\frac{\omega}{n}}\begin{pmatrix}-2\cos\frac{\omega}{n}&1&0&\ldots&1\\\
1&-2\cos\frac{\omega}{n}&1&\ldots&0\\\ 0&1&-2\cos\frac{\omega}{n}&\ldots&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots\\\
1&0&0&\ldots&-2\cos\frac{\omega}{n}\end{pmatrix}.$

It has the spectrum

$\left\\{\frac{2(\cos\frac{2\pi
k}{n}-\cos\frac{\omega}{n})}{\rho^{2}\sin\frac{\omega}{n}},k=0,\ldots,n-1\right\\}.$

This proves the claim on the signature. ∎

######  Lemma 3.7.

Assume that $\omega<2\pi$ and all $\alpha_{i}<\pi$. If for $x\in
T_{\rho}\mathbb{R}^{n}$ we have $\dot{\omega}(x)=0$, then $I(x,x)\leq 0$.
Moreover, $I(x,x)=0$ implies that $x=0$.

###### Proof.

Note that by the Gauss–Bonnet theorem we have

$2\pi-\omega+\sum_{i}(\pi-\alpha_{i})=2\pi.$ (3.1)

Consider the vector $x^{0}\in T_{\rho}\mathbb{R}^{n}$ defined by
$x^{0}_{i}:=1/\rho_{i}$. From (3.1) we get

$I(x^{0},x^{0})=\sum_{i}\dot{\alpha}_{i}(x^{0})=-\dot{\omega}(x^{0})=-\sum_{i}\dot{\omega}_{i}(x^{0})=\sum_{i}\frac{\cot\alpha_{i}^{+}+\cot\alpha_{i}^{-}}{\rho_{i}^{2}},$

where $\alpha_{i}^{+}:=\angle vw_{i}w_{i+1}$ and $\alpha^{-}_{i}:=\angle
vw_{i}w_{i-1}.$ As $\alpha_{i}^{+}+\alpha_{i}^{-}=\alpha_{i}<\pi$, we see that
$I(x^{0},x^{0})>0$. By Lemma 3.6, the positive subspace of the quadratic form
associated to $I$ is 1-dimensional, so $x^{0}$ spans it.

Now from $\dot{\omega}(x)=0$ and (3.1) we have

$0=\dot{\omega}(x)=-\sum_{i}\dot{\alpha}_{i}(x)=\sum_{i,j}\frac{\partial\alpha_{i}}{\partial\rho_{j}}x_{j}=I(x^{0},x).$

This means that $x$ either belongs to the negative subspace of the associated
quadratic form to $I$ or is zero. This finishes the proof. ∎

####  3.2.2 Infinitesimal rigidity of cone-polyhedra

Now we return to the setting of our problem. Similarly to the previous
subsection, if $x\in T_{u}\mathbb{R}^{V}$, then by $\dot{\kappa}_{v}(x)$,
$\dot{\rho}_{\vec{e}}(x)$ or $\dot{\alpha}_{\vec{e}}(x)$ we denote the
directional derivatives of various geometric quantities of $P(d,u)$, which are
linear forms. By $\ddot{H}$ we denote the Hessian of $H$ considered as a
bilinear form on $T_{u}\mathbb{R}^{V}$.

######  Lemma 3.8.

We have two expressions

$\ddot{H}(x,y)=\sum_{v\in V}x_{v}\dot{\kappa}_{v}(y),$ (3.2)
$\ddot{H}(x,y)=-\sum_{v\in
V}e^{2u_{v}}\sum_{\vec{e}\in\vec{E}_{v}(\mathcal{T})}\rho_{\vec{e}}\dot{\rho}_{\vec{e}}(x)\dot{\alpha}_{\vec{e}}(y).$
(3.3)

###### Proof.

The formula (3.2) is just Lemma 2.31 in a rewritten form. For (3.3) consider
an oriented edge $\vec{e}$ emanating from $v$ and ending at $w$.

The horospherical link of $P(d,u)$ at a vertex $v$ is a Euclidean cone-polygon
of total angle $\omega_{v}$ with angles $\alpha_{\vec{e}}$ for
$\vec{e}\in\vec{E}_{v}(\mathcal{T})$. Thus, we have

$\dot{\kappa}_{v}(y)=-\dot{\omega}_{v}(y)=\sum_{\vec{e}\in\vec{E}_{v}(\mathcal{T})}\dot{\alpha}_{\vec{e}}(y)$

and we can rewrite (3.2) as

$\ddot{H}(x,y)=\sum_{e=\\{vw\\}\in
E(\mathcal{T})}\dot{\alpha}_{e}(y)(x_{v}+x_{w}).$ (3.4)

Consider the decorated semi-ideal triangle containing in $P(d,u)$ an edge $e$
with vertices $v$ and $w$. By $b_{vw}$, $b_{wv}$ we denote the lengths of the
parts of $e$ in the triangle obtained by drawing a perpendicular from the
marked point and by $\rho_{vw}$, $\rho_{wv}$ denote the respective parts of
the canonical horocircles. Due to Lemma 2.22, we have

$\rho_{vw}^{2}=\exp(-2b_{vw})-\exp(-2u_{v}),~{}~{}~{}~{}~{}\rho_{wv}^{2}=\exp(-2b_{wv})-\exp(-2u_{w}).$

Also $b_{vw}+b_{wv}=a_{e}$. Differentiating this we get

$\exp(2b_{vw}-2u_{v})\dot{u}_{v}+\exp(2b_{wv}-2u_{w})\dot{u}_{w}-\exp(2b_{vw})\rho_{vw}\dot{\rho}_{vw}-\exp(2b_{wv})\rho_{wv}\dot{\rho}_{wv}=0.$
(3.5)

From Lemma 2.22 we also see that

$\exp(2b_{vw}-2u_{v})=\exp(2b_{wv}-2u_{w}).$

So we can rewrite (3.5) as

$\dot{u}_{v}+\dot{u}_{w}-\exp(2u_{v})\rho_{vw}\dot{\rho}_{vw}-\exp(2u_{w})\rho_{wv}\dot{\rho}_{wv}=0.$

Considering the derivatives in the direction $x$ and substituting this to
(3.4) we get exactly (3.3). ∎

Now we are ready to establish Lemma 3.1.

###### Proof of Lemma 3.1..

Due to Lemma 2.31, the Jacobian matrix
$D\kappa=\left(\frac{\partial\kappa_{v}}{\partial u_{w}}\right)$ is the
Hessian matrix $\ddot{H}$. We need to show its non-degeneracy.

Suppose the converse that for some $x$ and every $y$ we have
$\ddot{H}(y,x)=0$. Due to (3.2), we obtain that $\dot{\kappa}_{v}(x)=0$ for
all $v$. In particular,

$\ddot{H}(x,x)=\sum_{v\in V}x_{v}\dot{\kappa}_{v}(x)=0.$

On the other hand, from Lemma 3.7 we get

$\ddot{H}(x,x)=-\sum_{v\in
V}e^{2u_{v}}\sum_{\vec{e}\in\vec{E}_{v}(\mathcal{T})}\rho_{\vec{e}}\dot{\rho}_{\vec{e}}(x)\dot{\alpha}_{\vec{e}}(x)\geq
0$

and its equality to zero means that all $\dot{\rho}_{\vec{e}}(x)$ are zero.
One computes from the cosine laws of Lemma 2.20 and the sine laws of Lemma
2.21 (see also the computations in the proof of Lemma 2.32)

$\dot{\rho}_{\vec{e}}(x)=\frac{x_{v}-\cos(l_{\vec{e}})x_{w}}{\exp(u_{v})\sin(l_{\vec{e}})}.$

Clearly, from every point $v\in V$ there is at least one edge in a Delaunay
triangulation $\mathcal{T}$. Also $\cos(l_{\vec{e}})<1$. Thus, all
$\dot{\rho}_{\vec{e}}(x)=0$ implies that $x=0$. ∎

###  3.3 Properness

####  3.3.1 Comparison lemma

We will make use of the following lemma:

######  Lemma 3.9.

There is no convex spherical cone-metric on the 2-disk $D$ with perimeter
$2\pi$ and at least three kink-points at the boundary.

Here by a convex spherical cone-metric on the 2-disk we mean that the
conditions of Definition 2.1 hold in the interior, the boundary is piecewise
geodesic and the angle of each kink point is less than $\pi$.

For the proof of Lemma 3.9 we need two classical facts.

######  Theorem 3.10 (Alexandrov’s realization theorem).

Let $d$ be a convex spherical cone-metric on the 2-sphere $S$. Then there
exists a convex polyhedron in the standard 3-sphere $\mathbb{S}^{3}$ with
boundary isometric to $(S,d)$.

The proof of this theorem in the Euclidean case is given in [2, Section 4.3].
The discussion of its adaptation to the spherical case is in [2, Section 5.3].

######  Theorem 3.11 (Alexandrov’s rigidity theorem).

Let $\Omega_{1}$, $\Omega_{2}$ be two convex polyhedra in $\mathbb{S}^{3}$ and
$f:\partial\Omega_{1}\rightarrow\partial\Omega_{2}$ be an isometry. Then there
exists an isometry $F:\mathbb{S}^{3}\rightarrow\mathbb{S}^{3}$ inducing $f$.

Similarly, this theorem is commonly proven in the Euclidean case, see [2,
Section 3.3.2], but the proof works the same in the spherical case as noted in
[2, Section 3.6.4].

###### Proof of Lemma 3.9.

Suppose that such metric $d$ exists. Double $(D,d)$ along the boundary and
obtain the 2-sphere with a convex spherical cone-metric, which we denote by
$\Omega=(S,d^{S})$. By Theorem 3.10, it admits a realization as a convex
polyhedron $P$ in the standard 3-sphere $\mathbb{S}^{3}$.

Consider an orientation-inverting isometry $f:\Omega\rightarrow\Omega$ that
changes the halves of $\Omega$. From Theorem 3.11 there exists an isometry
$F:\mathbb{S}^{3}\rightarrow\mathbb{S}^{3}$ inducing $f$ at $\partial P$. We
see that $F$ is orientation-reversing and the set of its fixed points includes
a closed curve $\tau\subset\partial P$, which is the boundary of two glued
disks from $\Omega$, and is the set of the fixed points of $f$. We get that
$F$ is a symmetry with respect to a geodesic 2-sphere, which we denote by
$\mathbb{S}^{2}$. Clearly, $\partial P\cap\mathbb{S}^{2}=\tau$. The broken
line $\tau$ bounds a convex spherical polygon $\psi$, which has perimeter
$2\pi$ and at least three kink points.

As $\psi$ is convex, it belongs to the spherical lune $\sigma$ determined by
any its kink point. The perimeter of $\sigma$ is $2\pi$, which is the same as
the perimeter of $\psi$. It is easy to see that the only option is that $\psi$
is a spherical lune itself. As $\psi$ has at least three kink points, this is
a contradiction. ∎

####  3.3.2 Proof of Lemma 3.2

###### Proof of Lemma 3.2.

If not, there exists a sequence $\\{u^{k}\\}$, $k=1,2,\ldots$ in $U_{C}(d)$
such that $\\{u^{k}\\}$ is leaving any compact subset of $U_{C}(d)$ and
$\kappa^{k}:=\kappa(u^{k})$ converge to some point $\bar{\kappa}\in K$. Denote
$d^{k}:=u^{k}*d$ and up to taking a subsequence assume that $u^{k}$ converge
to some $\bar{u}\in[-\infty,+\infty]^{V}$.

As there are finitely many triangulations $\mathcal{T}$ of $(S,V)$ for which
$\mathfrak{D}_{+}(\mathcal{T})\cap\mathfrak{C}_{+}(d)$ is non-empty, after
picking a subsequence we may assume that there exists a triangulation
$\mathcal{T}$ of $(S,V)$ such that $d^{k}\in\mathfrak{D}_{+}(\mathcal{T})$ for
all $k$. Denote its edge lengths by $l^{k}\in(0,\pi)^{E(\mathcal{T})}$. It
follows that for every edge $e\in E(\mathcal{T})$ with endpoints $v$ and $w$
we have

$\sin\left(\frac{l_{e}^{k}}{2}\right)=\exp\left(\frac{u^{1}_{v}+u^{1}_{w}-u^{k}_{v}-u^{k}_{w}}{2}\right)\sin\left(\frac{l_{e}^{1}}{2}\right),$

see Remark 2.29.

By picking a subsequence we may also assume that $l^{k}$ converge to some
$\bar{l}\in[0,\pi]^{E(\mathcal{T})}$, and all the inner angles of triangles of
$\mathcal{T}$ converge. We have the following three possible cases:

(a) for some $v\in V$ we have $\bar{u}_{v}=-\infty$, or

(b) for all $w\in V$ we have $\bar{u}_{w}\neq-\infty$, but for some $v\in V$
we have $\bar{u}_{v}=+\infty$, or

(c) all $\bar{u}_{v}\in(-\infty,+\infty)$.

Case (a). Clearly, for every $w$ incident to $v$ we have
$\bar{u}_{w}=+\infty$. Thus, in every triangle adjacent to $v$ the length of
the opposite edge is zero in the limit. We need to consider further two
subcases.

Case (a.1): For every edge $e$ incident to $v$ we have $\bar{l}_{e}\neq\pi$.

Let $T$ be a triangle adjacent to $v$ and $\omega^{k}_{v,T}$ be the angle of
$T$ at $v$ in $d^{k}$. We claim that $\omega^{k}_{v,T}\rightarrow 0$. Indeed,
if the limiting length $\bar{l}_{e}$ of an edge $e$ of $T$ incident to $v$ is
not zero, then the claim is clear. If not, denote the other endpoint of $e$ by
$w$, the other edge of $T$ incident to $w$ by $e^{\prime}$, the other end of
$e^{\prime}$ by $w^{\prime}$ and the angle of $T$ at $w$ in $d^{k}$ by
$\omega^{k}_{w,T}$. Then

$\lim_{k}\sin\omega^{k}_{v,T}=\lim_{k}\frac{\sin l^{k}_{e^{\prime}}}{\sin
l^{k}_{e}}\sin\omega^{k}_{w,T}=\lim_{k}\frac{\sin\left(l^{k}_{e^{\prime}}/2\right)}{\sin\left(l^{k}_{e}/2\right)}\sin\omega^{k}_{w,T}=$
$=\lim_{k}\exp\left(\frac{u^{1}_{w^{\prime}}-u^{k}_{w^{\prime}}-u^{1}_{v}+u^{k}_{v}}{2}\right)\frac{\sin\left(l^{1}_{e^{\prime}}/2\right)}{\sin\left(l^{1}_{e}/2\right)}\sin\omega^{k}_{w,T}=0.$

Also one can see that for sufficiently large $k$ the edge $e^{\prime}$ is the
smallest in $T$. Thus, $\omega^{k}_{v,T}\rightarrow 0$. We get
$\bar{\kappa}_{v}=2\pi$, which contradicts to $\bar{\kappa}\in K$.

Case (a.2): For some edge $e$ incident to $v$ we have $\bar{l}_{e}=\pi$. In
this case, if $e^{\prime}$ is another edge adjacent to $v$ in the same
triangle with $e$, then by the triangle inequality we get
$\bar{l}_{e^{\prime}}=\pi$. By induction we see that this holds for every edge
incident to $v$. Let $T$ be a triangle adjacent to $v$ and
$\bar{\omega}^{k}_{v,T}$ be the angle of $T$ at $v$ in the limit. Then

${\rm area}(T,d^{k})\rightarrow 2\bar{\omega}^{k}_{v,T}$

and

$\sum_{T:v\in T}{\rm area}(T,d^{k})\rightarrow 2\cdot(2\pi-\bar{\kappa}_{v}).$

Then by the Gauss–Bonnet theorem

$\left(\sum_{w\neq
v}\bar{\kappa}_{w}\right)-\bar{\kappa}_{v}=4\pi-2\bar{\kappa}_{v}-\lim_{k}{\rm
area}(S,d^{k})\leq 0.$

This contradicts to $\bar{\kappa}\in K$.

Case (b). If all $\bar{u}_{w}=+\infty$, then ${\rm area}(S,d^{k})\rightarrow
0$ and by the Gauss–Bonnet theorem we get $\sum_{w\in
V}\bar{\kappa}_{w}=4\pi$. This contradicts to $\bar{\kappa}\in K$.

Suppose that there exist adjacent vertices $w$ and $v$ such that
$\bar{u}_{v}=+\infty$ and $\bar{u}_{w}\in(-\infty,\infty)$. Then by the
triangle inequality and induction we get $\bar{u}_{v^{\prime}}=+\infty$ for
any $v^{\prime}$ adjacent to $w$. It follows that for every $w^{\prime}$ such
that $\bar{u}_{w^{\prime}}\in(-\infty,\infty)$ and for every $v^{\prime}$
adjacent to $w^{\prime}$ we have $\bar{u}_{v^{\prime}}=+\infty$. Thus, ${\rm
area}(S,d^{k})\rightarrow 0$, which, again, contradicts to $\bar{\kappa}\in
K$.

Case (c). As all $\bar{u}_{v}\neq+\infty$, for each $e\in E(\mathcal{T})$ we
have $\bar{l}_{e}\neq 0$. Note also that if for a triangle $T$ the limit shape
degenerates to an arc, then the Delaunay condition implies that the triangle
$T^{\prime}$ adjacent to $T$ by the largest side of $T$ degenerates to the
half-sphere in the limit. In any case we see that the metrics $d^{k}$ converge
to a convex spherical cone-metric $\bar{d}$ on $(S,V)$ with strictly positive
curvature at every point of $V$.

We need to examine what happens if a triangle $T$ of $\mathcal{T}$ converges
in $\bar{d}$ to a non-convex one, i.e., to a spherical lune. If this does not
happen, then the argument above shows that also all triangle inequalities stay
strict, thus $\bar{d}\in\mathfrak{D}_{+}(\mathcal{T})$ and Lemma 3.2 is
proved. Consider two subcases.

Case (c.1): no edge of $T$ equals $\pi$ in $d$. Then $T$ converges to a half-
sphere in $\bar{d}$. Its vertices can not coincide, as otherwise we get a
vertex of non-positive curvature. Then $\overline{S-T}$ with the limit metric
is a disk with convex spherical cone-metric, perimeter $2\pi$ and three kink
points at the boundary. This contradicts to Lemma 3.9.

Case (c.2): some edge $e$ of $T$ equals $\pi$. Let $T^{\prime}$ be the
triangle adjacent to $T$ by $e$. As the other edges of $T$ are strictly
smaller than $\pi$, such $T^{\prime}$ exists. Then $T^{\prime}$ is also a
spherical lune in $\bar{d}$. Moreover, due to the Delaunay condition, the
union $T\cup T^{\prime}$ is a spherical lune that is at least a half-sphere.
The vertices of $T\cup T^{\prime}$ can not coincide, as otherwise we get a
vertex of non-positive curvature. Then $\overline{S-T-T^{\prime}}$ with the
limit metric is a disk with convex spherical cone-metric, perimeter $2\pi$ and
four kink points at the boundary. This contradicts to Lemma 3.9.

∎

###  3.4 Connectivity

Let $\mathbf{1}=(1,...,1)\in\mathbb{R}^{V}$. It is easy to see that if $u\in
U(d)$, then for all $\lambda>0$ we have $u+\lambda\mathbf{1}\in U(d)$.
Moreover, if $u*d\in\mathfrak{D}_{+}(\mathcal{T})$, then
$(u+\lambda\mathbf{1})*d\in\mathfrak{D}_{+}(\mathcal{T})$.

The Euclidean cone-metric $\Phi((u+\lambda\mathbf{1})*d)$ is obtained from
$\Phi(u*d)$ by multiplication of all edges in any geodesic triangulation by
$e^{-\lambda}$. Thus,

$\delta(u+\lambda\mathbf{1})=\delta(u)$

where by $\delta(u)\in\mathbb{R}^{V}$ we denote the discrete curvature of the
Euclidean cone-metric $\Phi(u*d)$.

It is also not hard to see

######  Lemma 3.12.

For every $u\in U(d)$ and $v\in V$

$\lim_{\lambda\rightarrow+\infty}\kappa_{v}(u+\lambda\mathbf{1})=\delta_{v}(u).$

###### Proof.

Let $T^{+}_{0}$ be a convex spherical triangle, $v$ be its vertex and
$\omega^{+}_{0}$ be its angle at $v$. Consider $T^{+}_{0}$ embedded in the
unit 2-sphere in the Euclidean 3-space and let $T^{0}_{0}$ be the subtending
Euclidean triangle. By $\omega$ denote its angle at $v$. By $T^{0}_{\lambda}$
denote the Euclidean triangle obtained from $T^{0}_{0}$ by multiplication of
all its side length by $e^{-\lambda}$, $\lambda\geq 0$. We consider the
vertices of $T^{0}_{\lambda}$ still at the unit sphere, $v$ still being its
vertex. By $T^{+}_{\lambda}$ we denote the respective spherical triangle
subtended by $T^{0}_{\lambda}$, by $\omega^{+}_{\lambda}$ denote its angle at
$v$. It is enough to show that

$\lim_{\lambda\rightarrow+\infty}\omega^{+}_{\lambda}=\omega.$

Let $\beta_{1,\lambda}$, $\beta_{2,\lambda}$ be the angles between sides of
$T^{0}_{\lambda}$ at $v$ and the ray from $v$ to the origin. In the spherical
link at $v$ we have the spherical triangle with sides $\beta_{1,\lambda}$,
$\beta_{2,\lambda}$ and $\omega$. The angle of this triangle opposite to the
side of length $\omega$ is equal to $\omega^{+}_{\lambda}$. Clearly,
$\beta_{1,\lambda},\beta_{2,\lambda}$ tend to $\pi/2$ as
$\lambda\rightarrow+\infty$. This shows the desired claim. ∎

We need another simple statement

######  Lemma 3.13.

For every $u\in U(d)$ and $v\in V$

$\dot{\kappa}_{v}(\mathbf{1})=\sum_{w\in V}\frac{\partial\kappa_{v}}{\partial
u_{w}}>0.$

Here we consider $\mathbf{1}$ as an element of $T_{u}\mathbb{R}^{V}$ and
$\dot{\kappa}_{v}(\mathbf{1})$ is the directional derivative in this
direction.

###### Proof.

Indeed, due to Lemma 2.32 we have

$\sum_{w\in V}\frac{\partial\kappa_{v}}{\partial u_{w}}=\sum_{\vec{e}\in
E_{v}(\mathcal{T})}\frac{\cot(\alpha^{+}_{\vec{e}})+\cot(\alpha^{-}_{\vec{e}})}{\rho_{\vec{e}}\exp(u_{v})\sin(l_{\vec{e}})}\left(1-\cos(l_{\vec{e}})\right)>0.$

∎

###### Proof of Lemma 3.3.

Let $u$, $u^{\prime}\in U_{C}(d)$. Due to Theorem 2.10, there exists a path of
metrics $d_{t}\subset\mathfrak{C}_{0}(\Psi(d))$, $t\in[0,1]$, connecting
$d_{0}:=\Psi(u*d)$ with $d_{1}:=\Psi(u^{\prime}*d)$ such that

$\delta(d_{t})=t\delta(u^{\prime})+(1-t)\delta(u).$

Moreover, as $\delta$ is constant on the homothetic changes of the metric, one
can choose $d_{t}$ so that the radii of the circumscribed circles of all
triangles in the Delaunay triangulations of $d_{t}$ are less than one. Thus,
there exists the preimage $\Psi^{-1}(d_{t})\subset\mathfrak{C}_{+}(d)$
descending to a path $u_{t}\subset U(d)$ connecting $u$ and $u^{\prime}$.

From Lemma 3.12 for a sufficiently large $\lambda$ the path
$u_{\lambda,t}:=u_{t}+\lambda\mathbf{1}$ belongs to $U_{C}(d)$. Lemma 3.13
implies that we can connect in $U_{C}(d)$ the endpoints of this path
$u_{\lambda,t}$ with $u$ and $u^{\prime}$ respectively along the direction
$\mathbf{1}$. This finishes the proof. ∎

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