Variational principles for circle patterns and Koebe's theorem

2002/03/31 by Alexander I. Bobenko, Boris A. Springborn · 5 citations
Mathematics · #math.GT #math.CV #math.MG #msc:52C26 #msc:53A30

paper · pdf

published as Trans. Amer. Math. Soc. 356 (2004), 659-689. · 33 pages, 12 figures, Appendix. Revised version. Appendix on cellular surfaces removed, references added and removed, typos corrected, a few minor changes

arxiv created 2002/06/03 · arxiv updated 2009/11/30

Abstract

We prove existence and uniqueness results for patterns of circles with prescribed intersection angles in constant curvature surfaces. Our method is based on two new functionals--one for the Euclidean and one for the hyperbolic case. We show how Colin de Verdi`ere's, Br"agger's and Rivin's functionals can be derived from ours.

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• Infinite Ideal Polyhedra in Hyperbolic 3-Space: Existence and Rigidity
• MicroRicci: A Greedy and Local Ricci Flow Solver for Self-Tuning Mesh Smoothing
• Combinatorial p-th Calabi flow for finite and infinite ideal circle patterns
• Characterization of Infinite Ideal Polyhedra in Hyperbolic 3-Space via Combinatorial Ricci Flow
• A prescribed curvature flow on hyperbolic surfaces with infinite topological type

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