A Discrete Ricci Flow on Surfaces with Hyperbolic Background Geometry

作者：Ge, Huabin; Xu, Xu^*

来源：International Mathematics Research Notices, 2017, 2017(11): 3510-3527.

DOI：10.1093/imrn/rnw142

In this article, we generalize our results in [6] to triangulated surfaces with hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gauss curvature by dividing the classical discrete Gauss curvature by an area element, which could be taken as the area of the hyperbolic disk packed at each vertex. We prove that the corresponding discrete Ricci flow with a nonpositive prescribing curvature function converges if and only if the prescribing curvature function is admissible. In the special case that the prescribing curvature function is zero, this provides another characterization of Thurston's circle packing theorem. In the case of zero prescribing curvature function, we further prove that the flow converges to a zero curvature metric if the initial curvatures are all nonpositive. Note that, this result does not require the existence of zero curvature metric or Thurston's combinatorial-topological condition. We further generalize the definition of combinatorial curvature to any given area element and prove the equivalence between the admissibility of a nonpositive prescribing curvature function and the convergence of the corresponding flow.

• 出版日期2017-6
• 单位北京交通大学; 武汉大学