Volume 13, issue 3 (2009)

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Rigidity of polyhedral surfaces, II

Ren Guo and Feng Luo

Geometry & Topology 13 (2009) 1265–1312

We study the rigidity of polyhedral surfaces using variational principles. The action functionals are derived from the cosine laws. The main focus of this paper is on the cosine law for a nontriangular region bounded by three possibly disjoint geodesics. Several of these cosine laws were first discovered and used by Fenchel and Nielsen. By studying the derivative of the cosine laws, we discover a uniform approach to several variational principles on polyhedral surfaces with or without boundary. As a consequence, the work of Penner, Bobenko and Springborn and Thurston on rigidity of polyhedral surfaces and circle patterns are extended to a very general context.

derivative cosine law, energy function, variational principle, edge invariant, circle packing metric, circle pattern metric, polyhedral surface, rigidity, metric, curvature
Mathematical Subject Classification 2000
Primary: 52C26, 52B70, 58E30
Secondary: 51M10, 57Q15
Received: 5 November 2007
Accepted: 17 January 2009
Published: 13 February 2009
Proposed: David Gabai
Seconded: Peter Teichner, Walter Neumann
Ren Guo
School of Mathematics
University of Minnesota
Minneapolis, MN 55455
Feng Luo
The Center of Mathematical Science
Zhejiang University
Hangzhou, Zhejiang 310027, China
Department of Mathematics
Rutgers University
Piscataway, NJ 08854