A discrete version of Liouville's theorem on conformal maps

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.

11 pages. v2: typos corrected, references added and updated. v3: final version

References in corpus (5)

• Discrete conformal maps and ideal hyperbolic polyhedra
• Ideal hyperbolic polyhedra and discrete uniformization
• An extended correction to ``Combinatorial Scalar Curvature and Rigidity of Ball Packings,'' (by D. Cooper and I. Rivin)
• $C^{\infty }$-convergence of conformal mappings on triangular lattices
• The Cotton tensor and Chern-Simons invariants in dimension $3$: an introduction