Rigidity and deformation of discrete conformal structures on polyhedral surfaces

Abstract

Discrete conformal structure on polyhedral surfaces is a discrete analogue of the conformal structure on smooth surfaces, which includes tangential circle packing, Thurston's circle packing, inversive distance circle packing and vertex scaling as special cases and generalizes them to a very general context. Glickenstein [40] conjectured the rigidity of discrete conformal structures on polyhedral surfaces, which includes Luo's conjecture on the rigidity of vertex scaling [57] and Bowers-Stephenson's conjecture on the rigidity of inversive distance circle packings [6] on polyhedral surfaces as special cases. We prove Glickenstein's conjecture using a variational principle.We further study the deformation of discrete conformal structures on polyhedral surfaces by combinatorial curvature flows. It is proved that the combinatorial Ricci flow for discrete conformal structures, which is a generalization of Chow-Luo's combinatorial Ricci flow for circle packings [7] and Luo's combinatorial Yamabe flow for vertex scaling [57], could be extended to exist for all time and the extended combinatorial Ricci flow converges exponentially fast for any initial data if the discrete conformal structure with prescribed combinatorial curvature exists. This confirms another conjecture of Glickenstein [40] on the convergence of the combinatorial Ricci flow and provides an effective algorithm for finding discrete conformal structures with prescribed combinatorial curvatures.The relationship of discrete conformal structures on polyhedral surfaces and 3dimensional hyperbolic geometry is also discussed. As a result, we obtain some new convexities of the co-volume functions for some generalized 3-dimensional hyperbolic tetrahedra.

Prescribing discrete Gaussian curvature on polyhedral surfaces

“…Luo's development [27] of Bobenko-Pinkall-Spingborn's extension [4] has recently been further developed to handle other cases. See, for instance, [38,39,41,42] and others. In [19], the function F T (u) is extended to be a C 2 -smooth convex function defined on R V by changing the triangulation of the marked surface under the Delaunay condition, based on which Gu-Luo-Sun-Wu proved the Discrete Uniformization Theorem 1.4 for PL metrics on closed surfaces.…”

A new class of discrete conformal structures on surfaces with boundary

“…The formula (3.6) was first obtained by Glickenstein-Thomas [8] (see also [23]) for generic hyperbolic discrete conformal structures on closed surfaces, which has lots of applications. See, for instance, [25,27,28,30,31] and others. This is the first time the formula (3.6) proved for hyperbolic discrete conformal structures on surfaces with boundary.…”

“…To prove that the Jacobian matrix [27,28], we further introduce the following parameterized admissible space…”

“…The structure condition (1.2) is a direct consequence of the cosine law for generalized hyperbolic triangles with all vertices hyper-ideal. Please refer to Section 5.1 in [28] and Remark 8 in Section 6 of this paper for the details of the geometric explanation. The structure condition (1.2) has been previously used in the study of discrete conformal structures on closed surfaces.…”

“…The structure condition (1.2) has been previously used in the study of discrete conformal structures on closed surfaces. See, for instance, [26][27][28]33] and others.…”