Rigidity of discrete conformal structures on surfaces

TL;DR

This paper proves Glickenstein's conjecture on the rigidity of discrete conformal structures on polyhedral surfaces, unifying previous results and connecting to hyperbolic geometry, using variational principles.

Contribution

It establishes the rigidity conjecture for discrete conformal structures, generalizing prior results and introducing a unified variational approach.

Findings

Proof of Glickenstein's rigidity conjecture.

Unification of Luo's and Bobenko-Pinkall-Springborn's results.

New insights into hyperbolic tetrahedra volume functions.

Abstract

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.