  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 conjectured the rigidity of discrete conformal structures on
polyhedral surfaces, which includes Luo's conjecture on the rigidity of vertex
scaling and Bowers-Stephenson's conjecture on the rigidity of inversive
distance circle packings 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 and
Luo's 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.
