Rigidity of Polyhedral Surfaces

Abstract

We study rigidity of polyhedral surfaces and the moduli space of polyhedral surfaces using variational principles. Curvature like quantities for polyhedral surfaces are introduced. Many of them are shown to determine the polyhedral metric up to isometry. The action functionals in the variational approaches are derived from the cosine law and the Lengendre transformation of them. These include energies used by Colin de Verdiere, Braegger, Rivin, Cohen-Kenyon-Propp, Leibon and Bobenko-Springborn for variational principles on triangulated surfaces. Our study is based on a set of identities satisfied by the derivative of the cosine law. These identities which exhibit similarity in all spaces of constant curvature are probably a discrete analogous of the Bianchi identity.

73 pages, 9 figures

References in corpus (4)

• A variational principle for domino tilings
• An analog of a modular functor from quantized Teichm"uller theory
• Geometric angle structures on triangulated surfaces
• On parameterizations of Teichmüller spaces of surfaces with boundary

Cited by in corpus (32)

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• A discrete uniformization theorem for polyhedral surfaces II
• Combinatorial Ricci flows and the hyperbolization of a class of compact 3-manifolds
• Circle patterns with obtuse exterior intersection angles
• On the deformation of ball packings
• Variational Principles on Triangulated Surfaces
• A new proof of Bowers-Stephenson conjecture
• Rigidity of discrete conformal structures on surfaces
• Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I
• On the global rigidity of sphere packings on 3-dimensional manifolds
• Second-order integrable Lagrangians and WDVV equations
• On Teichmuller Space of Surface with Boundary
• Rigidity of inversive distance circle packings revisited
• Fractional combinatorial Calabi flow on surfaces
• Local rigidity of inversive distance circle packing
• On a combinatorial curvature for surfaces with inversive distance circle packing metrics
• On parameterizations of Teichmüller spaces of surfaces with boundary
• 3-Dimensional Schlaefli Formula and Its Generalization
• The skein algebra of arcs and links and the decorated Teichmüller space
• Cell decompositions of Teichmüller spaces of surfaces with boundary
• Existence and uniqueness theorem for convex polyhedral metrics on compact surfaces
• A variational proof of Alexandrov's convex cap theorem
• Combinatorial Yamabe flow on hyperbolic surfaces with boundary
• The Unified Surface Ricci Flow
• Combinatorial Ricci flow on cusped 3-manifolds
• Calculus of generalized hyperbolic tetrahedron
• A note on circle patterns on surfaces
• A deformation of Penner's simplicial coordinate
• On the deformation of discrete conformal factors on surfaces
• Discrete uniformization of finite branched covers over the Riemann sphere via hyper-ideal circle patterns
• Non-rigidity of spherical inversive distance circle packings
• Combinatorial Calabi flow with surgery on surfaces