Rigidity of inversive distance circle packings revisited

Abstract

Inversive distance circle packing metric was introduced by P Bowers and K Stephenson \cite{BS} as a generalization of Thurston's circle packing metric \cite{T1}. They conjectured that the inversive distance circle packings are rigid. For nonnegative inversive distance, Guo \cite{Guo} proved the infinitesimal rigidity and then Luo \cite{L3} proved the global rigidity. In this paper, based on an observation of Zhou \cite{Z}, we prove this conjecture for inversive distance in $(-1,+\infty )$ by variational principles. We also study the global rigidity of a combinatorial curvature introduced in \cite{GJ4,GX4,GX6} with respect to the inversive distance circle packing metrics where the inversive distance is in $(-1,+\infty )$.

arXiv admin note: text overlap with arXiv:1701.01795

References in corpus (5)

• Rigidity of Polyhedral Surfaces
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Cited by in corpus (2)

• On a combinatorial curvature for surfaces with inversive distance circle packing metrics
• Tame the flexibility of circle patterns