A discrete version of Liouville's theorem on conformal maps

TL;DR

This paper establishes a discrete analogue of Liouville's theorem for conformal maps, demonstrating that simplicial complexes with certain scale factor relations are discretely conformally equivalent, extending classical conformal geometry to discrete settings.

Contribution

It introduces a novel discrete version of Liouville's theorem applicable to simplicial complexes, bridging continuous conformal maps and discrete combinatorial structures.

Findings

Proves a discrete Liouville's theorem for simplicial complexes.

Defines discrete conformal equivalence via vertex-associated scale factors.

Extends classical conformal geometry principles to discrete combinatorial models.

Abstract

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.