A Discrete Ricci Flow on Surfaces in Hyperbolic Background Geometry

TL;DR

This paper extends discrete Ricci flow to hyperbolic surfaces, introducing a new curvature measure and proving convergence conditions without relying on traditional existence assumptions.

Contribution

It generalizes discrete Ricci flow to hyperbolic geometry and introduces a new curvature definition, establishing convergence criteria independent of classical conditions.

Findings

Flow converges iff a zero curvature metric exists

Flow converges with negative initial curvatures

Generalized curvature definition for arbitrary area elements

Abstract

In this paper, we generalize our results in \cite{GX3} to triangulated surfaces in hyperbolic background geometry, which means that all triangles can be embedded in the standard hyperbolic space. We introduce a new discrete Gaussian 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 converges if and only if there exists a circle packing metric with zero curvature. We also prove that the flow converges if the initial curvatures are all negative. 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 existence of zero curvature metric and the convergence of the corresponding flow.