多面体曲面的参数化离散一致化定理和曲率流，II
Parameterized discrete uniformization theorems and curvature flows for polyhedral surfaces, II
摘要：
研究了多面体曲面上分片双曲度量顶点尺度的组合$α$ -曲率，它是经典组合曲率的参数化推广.建立了组合$α$ -曲率的离散一致化定理，推广了Gu-Guo-Luo-Sun-Wu关于经典组合曲率的离散一致化定理.我们进一步引入组合$α$ -Yamabe流和组合$α$ -Calabi流的顶点缩放，以找到分段双曲度量与指定的组合$α$ -曲率。为了处理沿组合曲率流的潜在奇异性，我们沿流通过边缘翻转进行手术。利用Gu-Guo-Luo-Sun-Wu建立的离散共形理论，证明了组合$α$ -Yamabe流和组合$α$ -Calabi流的长时间存在性和收敛性，为寻找具有指定组合$α$ -曲率的分段双曲度量提供了有效的算法.
英文摘要：
This paper investigates the combinatorial $α$-curvature for vertex scaling of piecewise hyperbolic metrics on polyhedral surfaces, which is a parameterized generalization of the classical combinatorial curvature. A discrete uniformization theorem for combinatorial $α$-curvature is established, which generalizes Gu-Guo-Luo-Sun-Wu's discrete uniformization theorem for classical combinatorial curvature. We further introduce combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow for vertex scaling to find piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures. To handle the potential singularities along the combinatorial curvature flows, we do surgery along the flows by edge flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu, we prove the longtime existence and convergence of combinatorial $α$-Yamabe flow and combinatorial $α$-Calabi flow with surgery, which provide effective algorithms for finding piecewise hyperbolic metrics with prescribed combinatorial $α$-curvatures.
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