Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I

Abstract

In this paper, we introduce a parameterized discrete curvature ($\alpha$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is established for the parameterized discrete curvature, which generalizes the discrete uniformization theorem obtained by Gu-Luo-Sun-Wu. We also prove the global rigidity of parameterized discrete curvature with respect to the discrete conformal factors, which confirms a generalized Luo conjecture on rigidity of discrete curvatures. We further introduce a parameterized discrete Yamabe flow for piecewise linear metrics on surfaces. To handle the possible singularities along the flow, we do surgery on the flow by flipping. Then we prove that the flow with surgery converges to a piecewise linear metric with constant discrete $\alpha$-curvature, which confirms another generalized Luo conjecture on convergence of discrete Yamabe flow with surgery. We also introduce a parameterized discrete Calabi flow and prove the convergence of the flow with surgery, which generalizes the convergence proved by Zhu and the author.