A discrete version of Liouville’s theorem on conformal maps

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 lengt...

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Bibliographic Details
Published inGeometriae dedicata Vol. 214; no. 1; pp. 389 - 398
Main Authors Pinkall, Ulrich, Springborn, Boris
Format Journal Article
LanguageEnglish
Published Dordrecht Springer Netherlands 01.10.2021
Springer Nature B.V
Subjects
Algebraic Geometry
Apexes
Conformal mapping
Convex and Discrete Geometry
Differential Geometry
Equivalence
Geometric transformation
Graph theory
Hyperbolic Geometry
Mathematics
Mathematics and Statistics
Original Paper
Projective Geometry
Theorems
Topology
52C26
Conformal flatness
53A30
Möbius transformation
Discrete conformal map
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Summary: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.
ISSN:0046-5755
1572-9168
DOI:10.1007/s10711-021-00621-2