带边界且具有离散高斯曲率的广义双曲圆填充的存在性、刚性与离散Schwarz–Pick引理
Existence, Rigidity, and Discrete Schwarz--Pick Lemma for Generalized Hyperbolic Circle Packings with Boundary and Discrete Gaussian Curvatures
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中文总结 AI 辅助
本文针对带有限多边形胞腔分解的紧带边曲面，研究具有指定边界顶点测地曲率、内部顶点总测地曲率及对偶圆圆心离散高斯曲率的广义双曲圆填充，给出其存在的充要条件与唯一性，建立含比较结果及刚性的离散Schwarz–Pick引理。
中文摘要 AI 辅助
本文研究赋予有限多边形胞腔分解的紧带边曲面上的广义双曲圆填充，探讨实现此类圆填充的问题，要求边界顶点处具有指定测地曲率、内部顶点处具有指定总测地曲率、对偶圆圆心处具有指定离散高斯曲率。本文给出此类广义双曲圆填充存在的充要条件并证明其唯一性，还在该框架下建立离散Schwarz–Pick引理，包含顶点曲率、广义圆弧长度、距离与面积的比较结果及对应的刚性结论。
英文摘要
This paper is concerned with generalized hyperbolic circle packings on compact bordered surfaces, endowed with finite polygonal cellular decompositions. We investigate the problem of realizing generalized hyperbolic circle packings with prescribed geodesic curvatures at boundary vertices, prescribed total geodesic curvatures at interior vertices, and prescribed discrete Gaussian curvatures at the centers of dual circles. We give a necessary and sufficient condition for the existence of such generalized hyperbolic circle packings and show their uniqueness. We also establish a discrete Schwarz--Pick lemma in this setting, including comparison results for vertex curvatures, generalized circle arc lengths, distances, and areas, together with the corresponding rigidity statements.
