A discrete version of Liouville's theorem on conformal maps
Differential Geometry
2025-01-07 v3 Combinatorics
Complex Variables
Geometric Topology
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.
Keywords
Cite
@article{arxiv.1911.00966,
title = {A discrete version of Liouville's theorem on conformal maps},
author = {Ulrich Pinkall and Boris Springborn},
journal= {arXiv preprint arXiv:1911.00966},
year = {2025}
}
Comments
11 pages. v2: typos corrected, references added and updated. v3: final version