Rigidity of discrete conformal structures on surfaces

In \cite{G3}, Glickenstein introduced the discrete conformal structures on polyhedral surfaces in an axiomatic approach from Riemannian geometry perspective. Glickenstein's discrete conformal structures include Thurston's circle packings, Bowers-Stephenson's inversive distance circle packings and Luo's vertex scalings as special cases. Glickenstein \cite{G5} further conjectured the rigidity of the discrete conformal structures on polyhedral surfaces. Glickenstein's conjecture includes Luo's conjecture on the rigidity of vertex scalings \cite{L1} and Bowers-Stephenson's conjecture on the rigidity of inversive distance circle packings \cite{BSt} as special cases. In this paper, we prove Glickenstein's conjecture using variational principles. This unifies and generalizes the well-known results of Luo \cite{L4} and Bobenko-Pinkall-Springborn \cite{BPS}. Our method provides a unified approach to similar problems. We further discuss the relationships of Glickenstein's discrete conformal structures on polyhedral surfaces and $3$-dimensional hyperbolic geometry. As a result, we obtain some new results on the convexities of the co-volume functions of some generalized $3$-dimensional hyperbolic tetrahedra.Comment: 35 pages, 4 figures. We divide the previous version of this paper into two parts. This paper is on the rigidity of Glickenstein's discrete conformal structures. In another paper, we deal with the deformation of Glickenstein's discrete conformal structure