标题： 曲面上离散共形映射的收敛性 显示英文标题

标题： Convergence of discrete conformal mappings on surfaces

摘要： 基于圆盘排列、顶点缩放和相关结构的离散共形映射自20世纪80年代Thurston提出圆盘排列作为近似共形映射的一种方法以来，一直受到广泛关注。 Rodin-Sullivan（1987）的第一个收敛结果证明了圆盘排列映射确实收敛到到圆盘的共形映射。 最近的结果表明，其他离散共形结构的映射也收敛到共形映射。 我们给出了一个关于曲面之间离散共形映射收敛的一般定理，该定理允许各种离散共形结构以及有界或无界的流形。 这些映射是分段线性离散共形映射和黎曼重心坐标组成的，称为重心离散共形映射。 对重心离散共形映射的估计允许提取收敛的子序列和黎曼度量的拉回估计，从而证明共形性。 该定理需要对单形的饱满性做出假设，以防止退化三角形，并需要一种局部离散共形刚性，该刚性推广了圆盘排列的六边形刚性。 显示英文摘要

摘要： Discrete conformal mappings based on circle packing, vertex scaling, and related structures has had significant activity since Thurston proposed circle packing as a way to approximate conformal maps in the 1980s. The first convergence result of Rodin-Sullivan (1987) proved that circle packing maps do indeed converge to conformal maps to the disk. Recent results have shown convergence of maps of other discrete conformal structures to conformal maps as well. We give a general theorem of convergence of discrete conformal mappings between surfaces that allows for a variety of discrete conformal structures and manifolds with or without boundary. The mappings are a composition of piecewise linear discrete conformal mappings and Riemannian barycentric coordinates, called barycentric discrete conformal maps. Estimates of the barycentric discrete conformal maps allow extraction of convergent subsequences and estimates for the pullback of the Riemannian metric, proving conformality. The theorem requires assumptions on fullness of simplices to prevent degenerate triangles and a local discrete conformal rigidity generalizing hexagonal rigidity of circle packings.