A discrete version of Liouville's theorem on conformal maps

Liouville's theorem says that in dimension greater than two, all conformal maps are M\"obius transformations. We prove an analogous statement about simplicial complexes, where two simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices.


Introduction
Liouville's theorem says that in dimension three and higher, conformal maps are Möbius transformations. More precisely: Theorem (Liouville). If U ⊂ R n is a domain and n ≥ 3, then any sufficiently regular conformal map f : U → R n is the restriction of a Möbius transformation.
The purpose of this article is to extend the theorem in a different direction. We establish the following discrete version of Liouville's theorem for simplicial complexes (see Section 2 for precise definitions): Theorem 1. If n ≥ 3, then two locally Delaunay discrete domains in R n are discretely conformally equivalent if and only if they are Möbius equivalent.
Roughly, simplicial complexes are considered discretely conformally equivalent if they are combinatorially equivalent and the lengths of corresponding edges are related by scale factors associated with the vertices. They are considered Möbius equivalent if they are combinatorially equivalent and the vertex positions are related by a Möbius transformation.
One implication of the equivalence statement, "Möbius equivalence implies discrete conformal equivalence", holds for arbitrary simplicial complexes and for any dimension n (see Section 3.1). The other implication, "discrete conformal equivalence implies Möbius equivalence", is only true for n ≥ 3 and for a more restrictive class of simplicial complexes (see Section 3.3). In Definitions 2.3 and 2.4 we therefore define a locally Delaunay discrete domain to be a locally finite, full-dimensional simplicial complex that satisfies some additional conditions that are sufficient and necessary to deduce Möbius equivalence from discrete conformal equivalence in dimension three or greater.
The basic concepts are explained in Section 2. Section 3 is devoted to a proof of Theorem 1, which may in hindsight appear rather obvious. Theorem 1 and its proof suggest a necessary and sufficient condition for discrete conformal flatness. This, the connection between discrete conformal equivalence and hyperbolic geometry, and some open questions are discussed in Section 4.

Basic definitions
In this article, a combinatorial isomorphism of simplicial complexes K and K in R n is understood to be a bijection between the vertex sets V and V of K and K , respectively, such that for any subset {v 0 , . . . , v k } ⊆ V the simplex is an element of K if and only if the simplex is an element of K . Thus, a combinatorial isomorphism φ induces a bijection between the complexes K and K , as well as a piecewise linear simplicial map between their carriers |K| and |K |. Simplicial complexes are combinatorially equivalent if there exists a combinatorial isomorphism between their vertex sets.
where | · | denotes the euclidean norm on R n . In other words, each edge length is scaled by the geometric mean of the scale factors e u attached to its vertices. We say that K and K are discretely conformally equivalent if they are discretely conformally equivalent with respect to some combinatorial isomorphism.
This notion of discrete conformal equivalence appeared first in the four dimensional Lorentz-geometric context of the Regge calculus [19]. In the two-dimensional setting of surfaces, it has lead to a rich theory which is intimately connected with hyperbolic geometry [2,4,7,8,9,13,17,20] and useful in diverse applications; see, e.g., [11,12,21].
To fix ideas and introduce some notation, let us collect some basic facts about Möbius transformations, beginning with the definition: A Möbius transformation of R n is a composition of inversions in hyperspheres and reflections in hyperplanes, where A Möbius transformation preserves or reverses orientation, depending on whether it is a composition of an even or an odd number of inversions and reflections. The Möbius transformations of R n form a Lie group Möb(n) of dimension 1 2 (n + 1)(n + 2) which is isomorphic to the projectivized orthogonal group PO(n + 1, 1). Indeed, we may identify R n with the n-dimensional unit sphere S n ⊂ R n+1 via stereographic projection and consider R n+1 as the real projective space RP n+1 minus a projective hyperplane "at infinity". This identifies the Möbius group Möb(n) with the group PO(n + 1, 1) of projective transformations of RP n+1 that map the sphere S n to itself.
The group Sim(n) of similarity transformations of R n , i.e., of transformations of the form is the subgroup of Möbius transformations that fix ∞ ∈ R n : Conversely, the Möbius group Möb(n) is generated by the similarity group Sim(n) together with one sphere inversion.
In Möbius geometry, a hypersphere in R n is either a euclidean hypersphere in R n or the union of a hyperplane in R n with {∞}. Möbius transformations map hyperspheres to hyperspheres. A Möbius transformation that is not a similarity transformation does not map simplices in R n to simplices, except for zero-dimensional simplices, i.e., vertices. Two simplicial complexes are considered Möbius equivalent if their vertices are related by a Möbius transformation. More precisely: Definition 2.2 (Möbius equivalence). Simplicial complexes K and K in R n are called Möbius equivalent with respect to a combinatorial isomorphism Simplicial complexes K and K are called Möbius equivalent if they are Möbius equivalent with respect to some combinatorial isomorphism.  Definition 2.4 (local Delaunay condition). A discrete domain K in R n is called locally Delaunay if, for every n-simplex σ ∈ K, the open ball bounded by the circumsphere of σ contains no vertices of n-simplices sharing a common (n − 1)-face with σ.
Remark 2.5. Let σ and σ be two n-simplices in R n that share a common (n − 1)-face, say Then the following statements are equivalent: • v 0 is contained in the open ball bounded by the circumsphere of σ .
• v n+1 is contained in the open ball bounded by the circumsphere of σ. Thus, the local Delaunay condition imposes one condition for every (n − 1)simplex in K that is incident with two n-simplices.
To state the obvious, Theorem 1 refers to discrete conformal equivalence and Möbius equivalence with respect to the same combinatorial isomorphism, i.e., we will prove the following statement: Theorem 1 (pedantic version). Suppose n ≥ 3, K and K are locally Delaunay discrete domains in R n , and φ is a combinatorial isomorphism between K and K . Then K and K are discretely conformally equivalent with respect φ if and only if they are Möbius equivalent with respect to φ.
3 Proof of Theorem 1 3.1 The easy implication: "Möbius equivalence implies discrete conformal equivalence" This implication holds for arbitrary simplicial complexes in R n and for arbitrary dimension n. So let K and K be simplicial complexes in R n and assume they are Möbius equivalent. We will show that they are discretely conformally equivalent. If the Möbius transformation T ∈ Möb(n) relating K and K is a similarity transformation of R n , then K and K are obviously discretely conformally equivalent, because the relation (2) holds with a constant scale factor e u .
If T is the inversion in the unit sphere, then the identity implies that K and K are discretely conformally equivalent. Indeed, in this case the relation (2) holds with e u(v) = |v| −2 .
Since the similarity transformations and the inversion in the unit sphere generate the Möbius group, the implication holds for all T ∈ Möb(n).

The equivalence of simplices
In this section, we consider conformal equivalence and Möbius equivalence for pairs of simplices. In the next section, we will use the results to prove the harder implication, "discrete conformal equivalence implies Möbius equivalence." for all two-element subsets {i, j} of {0, . . . , n}.
(ii) There is a Möbius transformation T of R n such that Section 3.1 proves the implication "(ii) ⇒ (i)", so it remains to show the converse statement, "(i) ⇒ (ii)". This is based on the following observations, which will also be useful by themselves: Lemma 3.2. Assume condition (i) of Lemma 3.1 holds. Let S v 0 and S v 0 be the inversions in the spheres with radius 1 centered at v 0 and v 0 , respectively, and let To show the implication "(i) ⇒ (ii)" of Lemma 3.1 using Lemma 3.2, let F be a similarity transformation of R n mapping w i to w i for i ∈ {1, . . . , n}, then for i ∈ {1, . . . , n}, and a similar equation holds for w i . Using the identity (3), one obtains for i, j ∈ {1, . . . , n}, and a similar equation for | w i − w j |. Now (4) implies and hence the simplices are similar with scale factor e −u 0 . This completes the proof of Lemma 3.2. We will also use the following fact, the proof of which we leave to the reader: Lemma 3.3. If there exists any Möbius transformation T satisfying (5), then there exist exactly two of them, say, T 1 and T 2 , of which one preserves orientation while the other reverses orientation, and which are related by where C and C are the inversions in the circumspheres of the simplices [v 0 , . . . , v n ] and [v 0 , . . . , v n ], respectively.

The harder implication: "Discrete conformal equivalence implies Möbius equivalence"
Let K and K be two locally Delaunay discrete domains in R n , where n ≥ 3, and assume K and K are discretely conformally equivalent with respect to the combinatorial isomorphism φ.
Note that φ may be orientation preserving or orientation reversing. We may assume without loss of generality that φ is orientation preserving. (If φ is orientation reversing, consider orientation preserving isomorphism between K and a mirror image of K .) Lemmas 3.1 and 3.3 say that for each n-simplex σ ∈ K, there is a unique orientation preserving Möbius transformation T σ such that T σ (v) = φ(v) for every vertex v ∈ σ. Note that the assumptions about orientation ensure that T σ maps the inside of the circumsphere of σ ∈ K to the inside of the circumsphere of φ(σ) ∈ K .
It remains to show that these Möbius transformations T σ are in fact all equal. To this end, it is enough show the equality for n-simplices in the star of an interior interior vertex, i.e., to show the following lemma: If v is an interior vertex of K and if σ andσ are two n-simplices contained in star(v), then T σ = Tσ.
Indeed, suppose Lemma 3.4 holds and σ andσ are any n-simplices of K. By assumption, both contain interior vertices of K, say v andṽ. Furthermore, by assumption, there is a path from v toṽ in the 1-skeleton of K traversing only interior vertices. By induction on the length of the path, Lemma 3.4 implies that T σ = Tσ.
It remains to show Lemma 3.4. To this end, let v = φ(v), and let Q and Q be the links of v and v , respectively, i.e., Let S v and S v be the inversions in the spheres with radius 1 centered at v and v , respectively. Apply the inversions S v and S v to the vertices of Q and Q , respectively, to obtain the Möbius equivalent polyhedra P and P . The local Delaunay condition for K and K implies that P and P are convex polyhedra in R n . Indeed the Möbius inversions S v and S v map the empty circumspheres of the stars to empty half-spaces whose boundary planes contain the faces of P and P . As in Section 3.2, one sees that the facets of P and P are similar. By Cauchy's rigidity theorem for convex polyhedra and its higher dimensional generalization [16], there is a similarity transformation F of R n that maps P to P . By the orientation assumption, F is orientation preserving. Hence is an orientation preserving Möbius transformation that maps star(v) to star(v ). Therefore, T = T σ for all σ ∈ star(v). This proves Lemma 3.4 and hence the implication "discrete conformal equivalence implies Möbius equivalence", and this completes the proof of Theorem 1.

Discrete conformal flatness and induced hyperbolic metric: Concluding remarks, outlook and open questions
The definition of discrete conformal equivalence (Definition 2.1) extends in an obvious way from simplicial complexes in R n to triangulated piecewise euclidean manifolds, possibly with boundary, i.e., to manifolds that consist of euclidean simplices glued together along their facets. We propose the following notion of discrete conformal flatness: Definition 4.1. A triangulated piecewise euclidean manifold is discretely conformally flat if the vertex star of every interior vertex is discretely conformally equivalent to a locally Delaunay discrete domain.
Applying the same idea as in the proof of the discrete Liouville theorem, and in particular equation (6), one obtains the following result: Theorem 2. A n-dimensional triangulated piecewise euclidean manifold is discretely conformally flat if and only if every interior vertex v 0 satisfies the following condition: There exists a convex polyhedron in R n that is combinatorially equivalent to the link of v 0 and whose edge lengths arẽ Here, ij denotes the length of the edge between two adjacent vertices v i , v j of the triangulated piecewise euclidean manifold, and for vertices v i , v j in the link of v 0 ,˜ ij denotes length of the corresponding edge of the convex polyhedron.
Note that for n = 2, the condition on the vertex link is equivalent to the polyhedral inequalities for the˜ ij , i.e., each˜ ij is larger than the sum of the others.
Note also that a connection between discrete conformal equivalence and hyperbolic geometry, which plays an important role in the theory in dimension two [2,20], extends to higher dimensions: If you interpret the circumsphere of a euclidean n-simplex as the boundary of n-dimensional hyperbolic space in the Beltrami-Klein model, this induces a hyperbolic metric on the simplex minus its vertices, turning the simplex into an ideal hyperbolic simplex. If you perform this construction on all simplices of a triangulated piecewise euclidean manifold, this induces a hyperbolic metric on the manifold with cusps at the vertices and cone-like singularities in the faces of codimension two. Just as in the two-dimensional setting, one can prove the following theorem: Theorem 3. Two triangulated piecewise euclidean manifolds are discretely conformally equivalent if and only if they are isometric with respect to the induced hyperbolic metrics.
As in the two-dimensional setting [2,20], this observation suggests extending the definition of discrete conformal equivalence to triangulations that are not combinatorially equivalent: Definition 4.2 (discrete conformal equivalence, extended). Two triangulated piecewise euclidean manifolds (which need not be combinatorially equivalent) are discretely conformally equivalent if they are isometric with respect to the induced hyperbolic metrics.
In the 2-dimensional setting, the known uniformization results [9,20] show that any triangulated surface is discretely conformally flat, provided the notion of discrete conformal flatness is based on the extended notion of discrete conformal equivalence.
In dimensions 3 and higher, the situation more complicated. The induced hyperbolic metric will in general have cone-like singularities along faces of codimension 2, even if the piecewise euclidean manifold is flat. The total dihedral angles at the faces of codimension 2 are discrete conformal invariants. This, any codimension-2-face whose hyperbolic cone angle is not equal to 2π occurs in any discretely conformally equivalent triangulated piecewise euclidean manifold.
Maybe the most intriguing question opened by this line of research is how far the analogy between smooth and discrete conformal flatness extends. Consider for example the case of closed 3-dimensional manifolds M . The Chern-Simons functional CS M (g) is an R/Z-valued conformally invariant function on the space of Riemannian metrics g on M , and the critical points of CS M are precisely the conformally flat metrics on M ; see, e.g., [15]. Is there an analogous invariant on the set of discrete conformal classes whose critical points are exactly the discretely conformally flat classes?