Discrete conformal structure on polyhedral surfaces is a discrete analogue of the conformal structure on smooth surfaces, which includes tangential circle packing, Thurstons circle packing, inversive distance circle packing and vertex scaling as special cases and generalizes them to a very general context. Glickenstein conjectured the rigidity of discrete conformal structures on polyhedral surfaces, which includes Luos conjecture on the rigidity of vertex scaling and Bowers-Stephensons conjecture on the rigidity of inversive distance circle packings on polyhedral surfaces as special cases. We prove Glickensteins 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-Luos combinatorial Ricci flow for circle packings and Luos combinatorial Yamabe flow for vertex scaling, 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 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 3-dimensional 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.
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