双曲多面体曲面上反演距离圆填充的存在性
The existence of inversive distance circle packing on hyperbolic polyhedral surface
AI总结：
本文研究双曲多面体曲面上反演距离圆填充的存在唯一性，结合带翻边手术的离散 Ricci 流、加权 Delaunay 不等式及凸 Ricci 势变分原理完成证明。
摘要：
本文证明，给定一个带有反演距离圆填充的双曲多面体度量，以及一个满足 Gauss–Bonnet 公式的目标离散曲率，存在唯一的反演距离圆填充与前者离散共形。我们通过离散 Ricci 流使曲面形变，并在某些面的正交圆即将变为非紧时通过翻边进行手术。双曲情形下修正的加权 Delaunay 不等式蕴含正交圆的紧性。我们利用定义在纤维丛上的凸 Ricci 势的变分原理完成证明，该纤维丛具有基于 Teichmüller 空间的胞腔分解和微分结构。
英文摘要：
In this paper, we prove that given a hyperbolic polyhedral metric with an inversive distance circle packing, and a target discrete curvature satisfying Gauss-Bonnet formula, there exist a unique inversive distance circle packing which is discrete conformal to the former one. We deform the surface by discrete Ricci flow, and do surgery by edge flipping when the orthogonal circles of some faces are about to be non-compact. The revised weighted Delaunay inequality of hyperbolic case implies the compactness of the orthogonal circle. We use a variational principle of a convex Ricci potential defined on the fiber bundles with cell-decomposition and differential structure based on Teichmüller space to finish the proof.
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