A discrete version of Liouville's theorem on conformal maps
Ulrich Pinkall, Boris Springborn
TL;DR
This work provides a discrete Liouville-type theorem for conformal maps on simplicial complexes in dimensions $n\ge 3$, showing that two locally Delaunay discrete domains are discretely conformally equivalent if and only if they are Möbius equivalent under the same combinatorial isomorphism. The proof separates into an easy direction (Möbius $\Rightarrow$ discrete conformal) valid for all complexes and a hard direction (discrete conformal $\Rightarrow$ Möbius) that relies on local Delaunay geometry, inversions, and Cauchy-type rigidity (with Pak’s higher-dimensional extension) to glue local Möbius maps into a global one. The authors illuminate the relationship between discrete conformal equivalence and hyperbolic geometry, propose a notion of discrete conformal flatness, and extend the theory to triangulated piecewise Euclidean manifolds, highlighting how induced hyperbolic metrics encode discrete conformal data. This work thereby provides a rigorous higher-dimensional analogue of Liouville theory and lays groundwork for discrete uniformization and rigidity phenomena in geometric topology and Regge calculus.
Abstract
Liouville's theorem says that in dimension greater than two, all conformal maps are Möbius 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.