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A discrete version of Liouville’s theorem on conformal maps

Pinkall, Ulrich; Springborn, Boris

FG Differentialgeometrie

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.