Discrete Yamabe Problem for Polyhedral Surfaces

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi cell corresponding to the singularity. We divide polyhedral surfaces into discrete conformal classes using a generalization of discrete conformal equivalence pioneered by Feng Luo. We subsequently show that, in every discrete conformal class, there exists a polyhedral surface with constant discrete Gaussian curvature. We also provide explicit examples to demonstrate that this surface is in general not unique.


Introduction
The Yamabe problem asks if every closed Riemannian manifold is conformally equivalent to one with constant scalar curvature.More precisely: Yamabe Problem Let g be a Riemannian metric on a closed smooth manifold M. Does there exist a smooth function u on M such that the Riemannian metric e 2u g has constant scalar curvature?
For two-dimensional manifolds the scalar and the Gaussian curvature are equivalent, and thus the Yamabe problem is answered by the celebrated Poincaré-Koebe uniformization theorem, which states that any closed oriented Riemannian surface is conformally equivalent to one with constant Gaussian curvature.
The purpose of this article is to translate the Yamabe problem for two-dimensional manifolds into the setting of polyhedral surfaces.The essential ingredient of the translation is the introduction of a new discretization of Gaussian curvature.
Defining the discrete Gaussian curvature requires a little preparation.We characterize a polyhedral or a piecewise flat surface by a triple (S,V, d), where S is the underlying topological surface, d denotes the PL-metric (PL stands for piecewise linear), and V ⊆ S is a finite set containing the conical singularities of d.
Let α i denote the cone angle at a point i ∈ V .The angle defect, evaluates for each i ∈ V how far the piecewise flat surface is from being flat at a neighborhood of i.This notion, introduced by Tullio Regge [10], is best understood as the discretization of the Gaussian curvature two-form.
The Voronoi cell of a point i ∈ V consists of all points on the piecewise flat surface (S,V, d) that are as close or closer to i than to any other point in V .It arises as a natural neighborhood of the point i.
Definition 1.1 The discrete Gaussian curvature at a point i ∈ V is the quotient of the angle defect W i and the area A i of the Voronoi cell of i: The discrete Gaussian curvature shares the following characteristic properties with the smooth Gaussian curvature: it is defined intrinsically, it satisfies the Gauss-Bonnet formula, and it scales by a factor of 1 r 2 upon a global rescaling of the metric by factor r. The latter characteristic is perhaps of the biggest contribution, since the formula most commonly used for discrete Gaussian curvature -the angle defect -is scaling invariant.
Discrete Yamabe problem asks if for every PL-metric there exists a discrete conformally equivalent one with constant discrete Gaussian curvature.It can be answered affirmatively by the following theorem.
Theorem 1.2 (Discrete uniformization theorem) For every PL-metric d on a marked surface (S,V ), there exists a discrete conformally equivalent PL-metric d such that the piecewise flat surface (S,V, d) has constant discrete Gaussian curvature.
The proof of Theorem 1.2 presented here is variational in nature.We translate the problem into a non-convex optimization problem with inequality constraints, which we solve using a classical theorem from calculus.
The PL-metric d of constant curvature from Theorem 1.2 is, in general, not unique.
Discrete conformal equivalence for piecewise flat surfaces with a fixed triangulation was introduced by Martin Roček and Ruth Williams [11], and Feng Luo [7], and is a straightforward discretization of the conformal equivalence on smooth surfaces.Recall that two Riemannian metrics g and g on a surface S are conformally equivalent if there exists a smooth function u on S such that g = e 2u g.
To discretize conformal equivalence, triangulate the piecewise flat surface (S,V, d) such that V is the set of vertices and every edge e ∈ E is a geodesic.The metric d is then uniquely determined by the edge lengths Two PL-metrics on a surface with a fixed triangulation are discrete conformally equivalent if their edge lengths , ˜ : E → R >0 are related by a factor u : V → R: We work with a generalization of discrete conformal equivalence to piecewise flat surfaces (Definition 2.11) introduced by Alexander Bobenko, Ulrich Pinkall and Boris Springborn in [2,Definition 5.1.4].This generalization reveals that hyperbolic geometry is the right setting for problems involving discrete conformal equivalence.The essential relation between piecewise flat surfaces and its hyperbolic equivalentdecorated hyperbolic surfaces with cusps -has been explored and described in detail by Boris Springborn in [12].
Another formulation of the discrete Yamabe problem for polyhedral surfaces due to Feng Luo [7] asks for the existence of PL-metrics with a constant angle defect within a discrete conformal class.It was solved affirmatively by Gu et al. [5], as well as by Boris Springborn [12].For surfaces of genus one, Luo's and our formulation of the discrete Yamabe problem are indeed equivalent.However, we believe that Luo's formulation is not a suitable discretization of the smooth Yamabe problem in general, since the angle defect is not a proper discretization of the smooth Gaussian curvature.This article is organized as follows.In Section 2 we revise the basic concepts and provide a dictionary between piecewise flat surfaces and decorated hyperbolic surfaces with cusps.Section 3 is devoted to the discussion of (non)-uniqueness of PL-metrics with constant discrete Gaussian curvature.In Section 4 we translate the statement of Theorem 1.2 into a non-convex optimization problem with inequality constraints.In Section 5 we prove Theorem 1.2.

Fundamental definitions and results
In this section we explain the correspondence between piecewise flat surfaces and decorated hyperbolic surfaces with cusps.Since the results in this section are wellknown, we only refer to the proofs.
Throughout the article we work with a closed oriented topological surface S and a non-empty finite set V ⊆ S of marked points.A triangulation of the marked surface (S,V ) is a triangulation of S with the vertex set equal to V .We denote a triangulation by ∆ and the set of edges and faces of ∆ by E ∆ and F ∆ , respectively.
A metric d on (S,V ) is called piecewise linear or a PL-metric if is flat everywhere but on a finite set of points contained in V , where it develops conical singularities.
A geodesic triangulation of the piecewise flat surface (S,V, d) is any triangulation of (S,V ) where the edges are geodesics with respect to the metric d.Let ∆ be a geodesic triangulation of a piecewise flat surface (S,V, d).The edge i j ∈ E ∆ is called a Delaunay edge if the vertex l of the adjacent triangle i jl ∈ F ∆ is not contained in the interior of the circumcircle of the other adjacent triangle i jk ∈ F ∆ .
Proposition 2.1 A geodesic triangulation of a piecewise flat surface is Delaunay if and only if each of its edges is Delaunay.
Discrete metric Let ∆ be a triangulation of the marked surface (S,V ).
such that for every triangle i jk ∈ F ∆ , the (sharp) triangle inequalities are satisfied.That is, The logarithm of the discrete metric , is called the logarithmic lengths.
Fact 2.3 Let ∆ be a geodesic triangulation of the piecewise flat surface (S,V, d).
Then the PL-metric d induces a discrete metric on (S,V, ∆ ) by measuring the lengths of the edges in E ∆ .
Vice versa, each discrete metric on a marked triangulated surface (S,V, ∆ ) induces a PL-metric on (S,V ).
Indeed, imposes a Euclidean metric on each triangle i jk ∈ F ∆ by transforming it into a Euclidean triangle with edge lengths i j , jk , ki .The metrics on two neighboring triangles fit isometrically along the common edge.Thus, by gluing each pair of neighboring triangles in ∆ along their common edge we equip the marked surface with a PL-metric.

Hyperbolic metrics, ideal tessellations, and Penner coordinates
Consider a marked surface (S,V ) equipped with a complete finite area hyperbolic metric d hyp with cusps at the marked points.We decorate the surface (S,V, d hyp ) with a horocycle H i at each cusp i ∈ V .Each horocycle is small enough such that, altogether, the horocycles bound disjoint cusp neighborhoods.The set of all horocycles decorating (S,V, d hyp ) is denoted by H .

Ideal Delaunay tessellations and triangulations
Definition 2.4 An ideal Delaunay tessellation of a decorated hyperbolic surface (S,V, d hyp , H ) is an ideal geodesic cell decomposition of (S,V, d hyp ), such that for each face f of the lift of (S,V, d hyp ) to the hyperbolic plane H 2 via an isometry of the universal cover, the following condition is satisfied.There exists a circle that touches all lifted horocycles anchored at the vertices of f externally and does not meet any other lifted horocycles.
An ideal Delaunay triangulation is any refinement of an ideal Delaunay tessellation by decomposing the non-triangular faces into ideal triangles by adding geodesic edges.
Theorem 2.5 ([12, Theorem 4.3]) For each decorated hyperbolic surface with at least one cusp, there exists a unique ideal Delaunay tessellation.
Let ∆ be a geodesic triangulation of a decorated hyperbolic surface (S,V, d hyp , H ).
An edge i j ∈ E ∆ is called Delaunay if the circle touching the horocycles at vertices i, j, k of one adjacent triangle i jk ∈ F ∆ and the horocycle at vertex l of the other adjacent triangle i jl ∈ F ∆ are externally disjoint or externally tangent.We illustrate the difference between a Delaunay and a non-Delaunay edge in Figure 1.Penner coordinates Penner coordinates, introduced by Robert Penner in [9], are the analogue of the discrete metric (see Definition 2.2) for decorated hyperbolic surfaces.Definition 2.7 Let i and j be two ideal points of the hyperbolic plane.Let H i and H j be two horocycles, anchored at ideal points i and j, respectively.The signed horocycle distance between H i and H j is the length of the segment of the geodesic line connecting the cusps i and j, truncated by the horocycles.The length is taken negative if H i and H j intersect.
The signed distances between horocycles of a decorated ideal hyperbolic triangle are illustrated in Figure 2. The distance λ i j is negative, whereas the distances λ jk and λ ki are positive.
Definition 2.8 Penner coordinates is a pair consisting of a triangulation ∆ of (S,V ) and a map λ : Fact 2.9 Penner coordinates (∆ , λ ) on a marked surface (S,V ) define a decorated hyperbolic surface (S,V, d hyp , H ), such that the signed distance between the horocycles H i and H j , with i j ∈ E ∆ , is λ i j .
Vice versa, let ∆ be a geodesic triangulation of a decorated hyperbolic surface (S,V, d hyp , H ). Then (S,V, d hyp , H ) induces Penner coordinates (∆ , λ ) by measuring the signed horocycle distance between horocycles H i and H j for each i j ∈ E ∆ .Theorem 2.10 ([12, Theorem 4.12]) Let (S,V ) be a marked surface with a triangulation ∆ .
Let : E ∆ → R >0 be a discrete metric on (S,V, ∆ ) such that ∆ is a Delaunay triangulation of the piecewise flat surface (S,V, d ).Let λ be the logarithmic lengths of defined by Equation (1).Then ∆ is an ideal Delaunay triangulation of the decorated hyperbolic surface defined on the marked surface (S,V ) by Penner coordinates (∆ , λ ).
Vice versa, let (∆ , λ ) be Penner coordinates on (S,V ) such that ∆ is an ideal Delaunay triangulation of the decorated hyperbolic surface defined on (S,V ) by (∆ , λ ).
Then the map : E ∆ → R ≥0 , defined by Equation (1), is a discrete metric on (S,V, ∆ ), and ∆ is a Delaunay triangulation of the polyhedral surface (S,V, d ).

Discrete conformal classes
Theorem 2.10 tells us that each piecewise flat surface induces a decorated hyperbolic surface, and vice versa.
Definition 2.11 Two PL-metrics on a marked surface (S,V ) are discrete conformally equivalent if the two induced decorated hyperbolic surfaces are isometric, through a map ϕ, where ϕ is homotopic to the identity in S −V relative to V .
Discrete conformal equivalence is an equivalence relation on the space of PL-metrics of a marked surface (S,V ).The corresponding equivalence classes are called confor-mal classes.In particular, discrete conformally equivalent PL-metrics induce different decorations on the -up to isometry -same hyperbolic surface.
Let d and d be two discrete conformally equivalent PL-metrics on (S,V ), and let H and H denote the two decorations induced on the hyperbolic surface (S,V, d hyp ) by d and d, respectively.
Let u i denote the signed distance from the horocycle H i to the horocycle Hi .The distance is taken positive if Hi is closer to the cusp at i than H i -as illustrated in Figure 3 in the halfplane model -and negative otherwise.The map is called a conformal factor, or a conformal change from d to d.
The position of each horocycle in H is completely determined by the decorated hyperbolic surface (S,V, d hyp , H ) and the conformal factor u. Thus, for a fixed marked surface (S,V ), each PL-metric d in the conformal class of the PL-metric d is uniquely defined by d and the conformal factor u.
To express this relation, we denote PL-metric d and the decoration H by d(u) and H (u), respectively.Further, if ∆ is a Delaunay triangulation of (S,V, d), the Penner coordinates ( ∆ , λ ) are denoted by (∆ (u), λ (u)).
Vice versa, each conformal factor defines a PL-metric in the conformal class of d.In other words: Proposition 2.12 The conformal class of the piecewise flat surface (S,V, d) is parametrized by the vector space As shown by Robert Penner in [9], the vector space R V admits a cell decomposition into Penner cells.
Definition 2.13 Let (S,V, d) be a piecewise flat surface, and let ∆ be a triangulation of the marked surface (S,V ).The Penner cell of ∆ in the conformal class of (S,V, d) is the set The set of all triangulations with non-empty Penner cells in the conformal class of (S,V, d) is denoted by D(S,V, d).
Discrete conformal equivalence also induces a relation on discrete metrics.
Proposition 2.14 Let d and d be two conformally equivalent PL-metrics on a marked surface (S,V ), related by the conformal factor u : V → R, and let ∆ be a geodesic triangulation of the surface (S,V, d), as well as the surface (S,V, d).Then the discrete metrics and ˜ , induced by d and d, respectively, satisfy for every edge i j ∈ E ∆ .
Remark 2.15 Proposition 2.14 is the definition of discrete conformal equivalence for piecewise flat surfaces with fixed triangulation, introduced by Feng Luo [7].
3 Counterexamples to uniqueness of metrics with constant curvature Uniqueness of PL-metrics with constant discrete Gaussian curvature up to global scaling in discrete conformal classes holds in three special cases: -S is of genus zero and |V | = 3.This follows from the positive semi-definiteness of the second derivative of the function F, defined in Fact 4.12.-S is of genus one.
In this case the Yamabe problem is equivalent to the discrete uniformization problem.The uniqueness follows from the positive semi-definiteness of the second derivative of function E (Definition 4.3) and was proved by Xianfeng Gu, Feng Luo, Jian Sun and Tianqi Wu in [5].-S is of genus larger than one and |V | = 1.
This case is trivial, since every discrete conformal class consists of one PL-metric up to a global scaling.
In order to show that uniqueness does not hold in general, we construct several examples of pairs of discrete conformally equivalent PL-metrics with constant discrete Gaussian curvature on the sphere with four marked points -that is, a tetrahedronand the surface of genus two with two marked points.
Tetrahedra with constant curvature We start with a combinatorial tetrahedron, denoting the vertices and edges as in Figure 4 (left).On this tetrahedron we define the PL-metric d 0 by prescribing the following lengths to the edges: Fact 3.1 Let ∆ be a geodesic triangulation of a piecewise flat surface (S,V, d) and let i jk and i jl be two neighboring triangles in F ∆ .Let α k , α l be the angles opposite of the edge i j in the triangles i jk and i jl, respectively.The edge i j is Delaunay if one of the following equivalent Delaunay conditions holds: a) cot α k + cot α l ≥ 0, The values of b 0 and c 0 need to be greater than 1 and chosen so that the edges of the tetrahedron are Delaunay.This is the case if and only if the triangle with edge lengths 1, b 0 , c 0 is acute.Due to condition c) in Fact 3.1, this is further equivalent to the following inequality: Denoting the area of the triangle with edge lengths 1, b 0 , c 0 by A, one sees that the PL-metric d 0 has constant discrete Gaussian curvature, We now apply the following family of discrete conformal changes to d 0 : The PL-metric d(v), defined by applying the discrete conformal change u(v) to the metric d 0 , has Delaunay edges if v ∈ S (b 0 ,c 0 ) .Its discrete Gaussian curvature at two pairs of vertices is equal, Proof For each v ∈ R the tetrahedron with metric d(v) has edge lengths The tetrahedron thus consists of two triangles with edge lengths a, b, c and two triangles with edge lengths ā, b, c.The equality of the curvatures follows immediately from the fact that The minimal and maximal value of the parameter v follow from the properties of Delaunay edges (Fact 3.1) and Equation (2).
Lemma 3.2 implies that the PL-metric d(v) has constant discrete Gaussian curvature if K 1 = K 3 .In order to test if, for a fixed value of b 0 and c 0 , this equality holds, we transform it into an expression more favorable for calculations.
Let A and Ā denote the area of the triangles with side lengths a, b, c and ā, b, c, respectively, and let F a , . . ., F c denote the areas as in Figure 4 (right).

Lemma 3.3
The PL-metric d(v) has constant discrete Gaussian curvature if and only if v is a zero of the map Proof Follows by a straightforward calculation: We plotted the graphs of the function g (b 0 ,c 0 ) for various values of b 0 and c 0 in Figure 6.
Fig. 5 Surfaces of genus two with two marked points and constant curvature The initial metric d 0 is defined on a triangulation with combinatorics as in Figure 5, with edge lengths prescribed as follows: for two values b 0 , c 0 ≥ 1 satisfying Condition (2).As in the previous paragraph, one can easily check that d 0 has constant discrete Gaussian curvature where A is the area of the triangle with edge lengths 1, b 0 , c 0 .We now apply the following family of discrete conformal changes to d 0 : The following lemma is the analogon of Lemmata 3.2 and 3.3.

Lemma 3.4
The PL-metric d(v), given by applying the discrete conformal change u(v) to the metric d 0 , has Delaunay edges if v ∈ S (b 0 ,c 0 ) .It has constant discrete Gaussian curvature if and only if v is a zero of the map Proof Analogous to the proofs of Lemmata 3.2 and 3.3.
-  The number of critical points of the maps g (b 0 ,c 0 ) and h (b 0 ,c 0 ) varies depending on the choice of (b 0 , c 0 ). Figure 6 illustrates the graphs of g (b 0 ,c 0 ) and h (b 0 ,c 0 ) for various values of (b 0 , c 0 ).In each graph the red and green curves correspond to discrete conformal classes with more than one metric with constant discrete Gaussian curvature.

Variational principles
The goal of this article is to prove the existence of piecewise flat surfaces with constant Gaussian curvature, where the discrete Gaussian curvature is the quotient of the angle defect and the area of the corresponding Voronoi cell.In this section we translate this setting into an optimization problem which we describe by three variational principles.To this end, we define two functions -E and A tot -whose partial derivatives correspond to the angle defect and the area of the Voronoi cell, respectively.
The functions E and A tot form the two essential building blocks of the variational principles.

Two essential building blocks
The function E The function E, which we will introduce shortly, was defined by Alexander Bobenko et al. in [2].As we will see, it is locally convex and its partial derivatives correspond to the angle defects at the vertices.Its building block is a peculiar function f .
The function f is defined as follows:   Recall that the discrete conformal class of a piecewise flat surface (S,V, d) is parameterized by the vector space R V (see Proposition 2.12), which can be decomposed into Penner cells (see Definition 2.13).We first define the function E ∆ on each Penner cell A ∆ and then extend its domain to obtain the function E on R V .
Definition 4.3 Let (S,V, d) be a piecewise flat surface, and let ∆ ∈ D(S,V, d).On the Penner cell A ∆ , the function E ∆ is defined as follows: where λi j are the logarithmic lengths of the discrete metric induced by the PLmetric d(u) on ∆ .

Lemma 4.4
The partial derivatives of the function E ∆ satisfy the equation where W i is the angle defect at vertex i of the piecewise flat surface (S,V, d(u)).
The functions f and E ∆ have the following properties: Theorem 4.6 (Extension) For a conformal factor u ∈ R V , let ∆ (u) be a Delaunay triangulation of the surface (S,V, d(u)).The map is well-defined and twice continuously differentiable.Its second derivative is a positive semidefinite quadratic form with one-dimensional kernel, spanned by (1, . . ., 1) ∈ R V .Explicitly, Proof Follows from [2, Proposition 4.1.6]and [12,Section 7 and 8].
The function A tot The function A tot , whose first partial derivatives correspond to the area of the Voronoi cells, denotes the total area of the surface.We first define the function A ∆ tot on each Penner cell A ∆ and then extend its domain to obtain the function A tot on R V .Definition 4.7 Let (S,V, d) be a piecewise flat surface, and let ∆ ∈ D(S,V, d).On the Penner cell A ∆ , the function A ∆ tot is defined as follows: where A i jk (u) is the area of the triangle with vertices i, j, k ∈ V on the piecewise flat surface (S,V, d(u)).
Let us denote the area of the Voronoi cell of a marked point i ∈ V by A i .

Lemma 4.8
The function A ∆ tot is analytic.Its partial derivatives satisfy the equation Its second derivative is where the vertices k, l ∈ V are the opposite vertices in the neighboring triangles i jk, i jl ∈ 8 cot α i j k + cot α i j l , and R i jk denotes the radius of the circumcircle of the triangle i jk.Proof The function A ∆ tot is analytic since the area A i jk (u) of each triangle i jk ∈ F ∆ is an analytic function with respect to the vector of conformal factors u 1 .Consider a triangle with vertices i, j, and k, and let A i jk denote the signed area of the triangle with vertices k, the circumcentre of the triangle i jk, and the midpoint of the edge jk, as depicted in Figure 8.The sign of A i jk is positive if the circumcentre of i jk lies inside the triangle, and negative otherwise.Then and the area of the Voronoi cell V i of a piecewise flat surface (S,V, d) satisfies the equation Thus, Due to the linearity of the area function, In the upcoming calculations we use the following formula from [2,].
Lemma Let a, b, c be edge lengths of a triangle, α, β , γ angles opposite of a, b, c, respectively, and let λ a , λ b , λ c be the logarithmic lengths.Then we obtain the equation Let i, j ∈ V be two vertices.If j is not adjacent to i, If j is adjacent to i, let k, l ∈ V be the two opposite vertices in the neighboring triangles i jk, i jl ∈ F ∆ .Since the mixed partial derivative equals Thus, Theorem 4.9 (Extension) For a conformal factor u ∈ R V , let ∆ (u) be a Delaunay triangulation of the surface (S,V, d(u)).The map is well-defined and once continuously differentiable.
Proof Due to Lemma 4.8 the function A tot is once continuously differentiable in the interior of every Penner cell.At the boundary between two (or more) Penner cells the triangulations induce the same Delaunay tessellation and thus the same Voronoi tessellation.The areas of the Voronoi cells induced by either of the triangulations are therefore equal.

Remark 4.10
The function A tot is, in fact, twice continuously differentiable.This can be proved by a long and unilluminating calculation [6, Chapter 8].

The variational principles
Proof We use the method of Lagrange multipliers.A conformal factor u ∈ R V is a critical point of the function E under the constraint A tot = 1 if and only if there exists a Lagrange multiplier λ ∈ R, such that

This holds if and only if
The Lagrange multiplier λ satisfies by the discrete Gauss-Bonnet theorem.
Theorem (Discrete Gauss-Bonnet theorem) Let (S,V, d) be a piecewise flat surface with constant discrete Gaussian curvature K av at every vertex.Denote the total area of the surface by A tot .Then, Fact 4.12 (Alternative variational principle to Theorem 4.11) Up to global rescaling, the PL-metrics with constant discrete Gaussian curvature in the conformal class of the metric d are in one-to-one correspondence with the critical points of the function Indeed,

This holds if and only if
Theorem 4.13 (Variational principle with inequality constraints) Let (S,V, d) be a piecewise flat surface with χ(S) = 0.The existence of PL-metrics with constant discrete Gaussian curvature in the conformal class of the metric d follows from the existence of minima of the function E under the following inequality constraints: if the Euler characteristic of S satisfies χ(S) = 2, the inequality constraint is if the Euler characteristic of S satisfies χ(S) < 0, the inequality constraint is Proof Proposition 4.15 shows that if u ∈ R V is a minimum of the function E under one of these constraints, then A tot (u) = 1.Since a minimum is a critical point, the claim follows from Theorem 4.11.
Proposition 4.14 The sets have the following properties: a) The sets A + and A − are closed subsets of R V .
b) Let I = (1, . . ., 1) ∈ R V , and let u ∈ R V be a conformal factor.Then the rays are completely contained in the sets A + and A − , respectively.The sets A + and A − are thus unbounded.
Proof a) The proof follows from the fact that the sets A + and A − satisfy the equation The statement follows from the fact that Proof Let χ(S) = 2 and let u ∈ A + be a minimum of the function E in A + .We show that A tot (u) = 1.Let This implies that c = 0, and thus A tot (u) = 1.
For surfaces with χ(S) < 0 the proof is analogous.

Existence of metrics with constant Gaussian curvature
In this section we prove Theorem 1.2.We build the proof on several key observations of the behaviour of a sequence (u n ) n∈N of conformal factors in R V .These observations are central for the application of Theorem 5.1, from which the proof of Theorem 1.2 follows almost immediately.
In the other two cases (χ(S) < 0 and χ(S) = 2) finding metrics with constant Gaussian curvature is equivalent to finding the minima of the function E in the set A − if χ(S) < 0, and in the set A + if χ(S) = 2.This is due to Theorem 4.13.To prove the existence of these minima we apply Theorem 5.1 -a traditional theorem from calculus.
Theorem 5.1 Let A ⊆ R m be a closed set and let f : A → R be a continuous function.
If every unbounded sequence (x n ) n∈N in A has a subsequence (x n k ) k∈N such that then f attains a minimum in A.
We already verified that the majority of the conditions of Theorem 5.1 is satisfied.Proposition 4.14 ensures that the sets A + and A − are closed.Theorem 4.6 tells us that the function E is continuous.To obtain the minima of E in the sets A + and A − , the following two theorems are left to prove.

Behaviour of sequences of conformal factors
Fix a piecewise flat surface (S,V, d) and let (u n ) n∈N be an unbounded sequence in its discrete conformal class R V .We denote its coordinate sequence at vertex j ∈ V by (u j,n ) n∈N .
Convention 5.4 Throughout this section we assume that the sequence (u n ) n∈N possesses the following properties: -It lies in one Penner cell A ∆ of R V .
-There exists a vertex i * ∈ V such that for all j ∈ V and n ∈ N u i * ,n ≤ u j,n .
-Each coordinate sequence (u j,n ) n∈N either converges, diverges properly to +∞, or diverges properly to −∞. -For all j ∈ V the sequences (u j,n − u i * ,n ) n∈N either converge or diverge properly to +∞.
We may adopt Convention 5.4 without loss of generality because every sequence in R V possesses a subsequence that satisfies these properties.The first property follows from a theorem by Hirotaka Akiyoshi.
In addition, we use the following notation: where is the discrete metric induced by the PL-metric d on (S,V, ∆ ) (see Fact 2.3).Since the sequence (u n ) n∈N lies inside the Penner cell A ∆ , ∆ is a Delaunay triangulation of (S,V, d(u n )) for all n ∈ N (see Definition 2.13).Furthermore, the map n defined by Equation ( 5) is the discrete metric induced on (S,V, ∆ ) by the PL-metric d(u n ) (see Proposition 2.14).
Behaviour of (u n ) n∈N in one triangle Consider a triangle in F ∆ with vertices labeled by 1, 2, 3 ∈ V and initial edge lengths 12 , 23 , 31 , uniquely determined by d.Define In other words, there exists no sequence in A 123 where two of the coordinate sequences would diverge properly to +∞ and the third one would be bounded from above.
Proof Without loss of generality we may assume that u 1,n ≤ u 2,n for all n ∈ N. Then 0 This contradicts the triangle inequality We now make a subtle shift of perspective -instead of studying the development of triangles under sequences of conformal factors (u i,n ) n∈N , we consider their development under the sequences (u i,n − u i * ,n ) n∈N .Geometrically, this corresponds to the rescaling of the whole triangulation by a factor exp (−u i * ,n ) at each step n.
Since we are primarily interested in the conditions under which the triangle inequalities break (such as those in Lemma 5.5), this shift is an elegant way to reduce the number of cases.Indeed, a triangle with conformal factors (2n, 2n, n) n∈N will degenerate just as the triangle with conformal factors (n, n, 0) n∈N will, since the triangles are similar.Lemma 5.5 yields the following key observation: Corollary 5.6 At every triangle i jk ∈ F ∆ , at least two of the three sequences Proof The claim holds for any triangle with vertex i * due to Lemma 5.5.It holds for all remaining triangles in F ∆ due to the connectivity of the triangulation.Behaviour of (u n ) n∈N around a vertex star Let i ∈ V be a vertex such that the sequence (u i,n ) n∈N diverges properly to +∞ and the sequences (u j,n ) n∈N at any neighbour j ∈ V converge.We investigate the behaviour of angles in triangles with vertex i.A vertex star around vertex i is the subset F i ∆ ⊆ F ∆ of triangles in F ∆ that contain the vertex i.We denote the degree of the vertex i by s and label the vertices as in Figure 10.We drop the index n that denotes the elements in the sequence when we label angles.Whenever the labeling requires it we use the conventions 1 − 1 = s and s + 1 = 1.

2. 1
Tessellations of piecewise flat surfaces, discrete metric Voronoi tessellation Every piecewise flat surface (S,V, d) possesses a unique Voronoi tessellation.For p ∈ S let d(p,V ) denote the distance of p to the set V , and let Γ V (p) be the set of all geodesics realizing this distance.The open 2-, 1-and 0-cells of the Voronoi tessellation of (S,V, d) are the connected components of {p ∈ S | |Γ V (p)| = 1}, {p ∈ S | |Γ V (p)| = 2}, and {p ∈ S | |Γ V (p)| ≥ 3}, respectively.We denote the closure of the open Voronoi 2-cell containing i ∈ V by V i .Delaunay tessellation and triangulation Delaunay tessellation of a piecewise flat surface is the dual of the Voronoi tessellation.A Delaunay triangulation arises from the Delaunay tessellation by adding edges to triangulate the non-triangular faces.
The orange circle and the horocycle at l are disjoint, the edge i j is Delaunay.The orange circle and the horocycle at l intersect, the edge i j is not Delaunay.

Fig. 2 :
Fig. 2: Penner coordinates of a decorated ideal hyperbolic triangle i jk, in the Poincaré disc model.

Fig. 4 :
Fig. 4: A tetrahedron (left) and a division of areas in the two triangles (right). b

Fig. 6 :
Fig. 6: Graphs of the functions g (left) and h (right) for various values of b 0 and c 0 .

Definition 4 . 1
Consider a Euclidean triangle with edge lengths a, b, c and angles α, β , γ, opposite to edges a, b, c, respectively.Let x = log a, y = log b, z = log c, as illustrated in Figure 7a.Let A be the set of all triples (x, y, z) ∈ R 3 , such that (a, b, c) satisfy the triangle inequalities: where L(α) = − α 0 log |2 sin(t)| dt is Milnor's Lobachevsky function, introduced by John Milnor in [8].Logarithmic edge lengths of a triangle.(b) Graph of Milnor's Lobachevsky function, y = L(x).

Theorem 4 . 11 (
Variational principle with equality constraints) Let (S,V, d) be a piecewise flat surface.Up to global rescaling, the PL-metrics with constant discrete Gaussian curvature in the conformal class of the metric d are in one-to-one correspondence with the critical points of the function

Proposition 4 .
15 Let (S,V, d) be a piecewise flat surface.If the Euler characteristic of S satisfies χ(S) = 2 and the function E attains a minimum in the set A + , or the Euler characteristic of S satisfies χ(S) < 0 and the function E attains a minimum in the set A − , the minimum lies at the boundary of the sets,

2 and Theorem 5 . 3 . 3 . 5 . 1
In Section 5.2 we study the behaviour of sequences of conformal factors.Finally, in Section 5.3 we prove Theorem 5.2 and Theorem 5.Reduction to Theorem 5.2 and Theorem 5.3

Theorem 5 . 2 Theorem 5 . 3
Let χ(S) < 0 and let (u n ) n∈N be an unbounded sequence in A − .Then there exists a subsequence (u n k ) k∈N of (u n ) n∈N , such that lim k→∞ E(u n k ) = +∞.Let χ(S) = 2 and let (u n ) n∈N be an unbounded sequence in A + .Then there exists a subsequence (u n k ) k∈N of (u n ) n∈N , such that lim k→∞ E(u n k ) = +∞.

Lemma 5 . 7
Assume that the sequence (u 1,n ) n∈N diverges properly to +∞ and the sequences (u 2,n ) n∈N and (u 3,n ) n∈N converge.