多面体曲面的参数化离散一致化定理和曲率流，I
Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, I
摘要：
本文对多面体曲面上的分段线性度量引入了参数化离散曲率（$α$ -curvature），它是经典离散曲率的推广.对参数化的离散曲率建立了离散一致化定理，推广了Gu-Luo-Sun-Wu的离散一致化定理。我们还证明了参数化离散曲率关于离散共形因子的整体刚性，从而证实了关于离散曲率刚性的一个推广的Luo猜想。我们进一步介绍了一个参数化离散Yamabe流的分片线性度量的表面。为了处理沿着流动的可能的奇点，我们通过翻转对流动进行手术。证明了带手术的Yamabe流收敛到一个具有常数离散$α$ -曲率的分段线性度量，从而证实了关于带手术的Yamabe流收敛的另一个广义Luo猜想.我们还引入了一个参数化离散Calabi流，并证明了该流的收敛性，推广了Zhu和作者的收敛性证明。
英文摘要：
In this paper, we introduce a parameterized discrete curvature ($α$-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 $α$-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.
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