标题： 双曲圆排列与带边曲面的总测地曲率 显示英文标题

标题： Hyperbolic Circle Packings and Total Geodesic Curvatures on Surfaces with Boundary

摘要： 本文研究了边界曲面上总测地曲率的广义双曲圆盘镶嵌（包括圆、极限圆或双曲圆）。我们主要关注接触图为有限多边形细胞分解的$1$-骨架的圆盘镶嵌的存在性和刚性，这类似于Bobenko和Springborn [4]的构造。受Colin de Verdière方法[6]的启发，我们引入了多边形上广义双曲圆盘镶嵌的变分原理。通过分析多边形上广义圆盘镶嵌的极限行为，我们给出了关于接触图每个顶点上总测地曲率的具有锥形奇点的广义双曲圆盘镶嵌的存在性和刚性。作为结果，我们引入了组合Ricci流，以找到具有预定总测地曲率的接触图每个顶点上的期望圆盘镶嵌。 显示英文摘要

摘要： This paper investigates a generalized hyperbolic circle packing (including circles, horocycles or hypercycles) with respect to the total geodesic curvatures on the surface with boundary. We mainly focus on the existence and rigidity of circle packing whose contact graph is the $1$-skeleton of a finite polygonal cellular decomposition, which is analogous to the construction of Bobenko and Springborn [4]. Motivated by Colin de Verdi\`ere's method [6], we introduce the variational principle for generalized hyperbolic circle packings on polygons. By analyzing limit behaviours of generalized circle packings on polygons, we give an existence and rigidity for the generalized hyperbolic circle packing with conical singularities regarding the total geodesic curvature on each vertex of the contact graph. As a consequence, we introduce the combinatoral Ricci flow to find a desired circle packing with a prescribed total geodesic curvature on each vertex of the contact graph.