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Rigidity of polyhedral
surfaces, II
Ren Guo and Feng Luo |
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Geometry & Topology 13 (2009) 1265–1312
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Abstract |
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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. |
Keywords
derivative cosine law, energy function, variational
principle, edge invariant, circle packing metric, circle
pattern metric, polyhedral surface, rigidity, metric,
curvature
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Mathematical Subject Classification 2000
Primary: 52C26, 52B70, 58E30
Secondary: 51M10, 57Q15
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References |
Publication
Received: 5 November 2007
Accepted: 17 January 2009
Published: 13 February 2009
Proposed: David Gabai
Seconded: Peter Teichner, Walter Neumann
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Authors |
| Ren Guo | |
| School of Mathematics University of Minnesota Minneapolis, MN 55455 USA |
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| Feng Luo | |
| The Center of Mathematical
Science Zhejiang University Hangzhou, Zhejiang 310027, China and Department of Mathematics Rutgers University Piscataway, NJ 08854 USA |
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