Infinite combinatorial Ricci flow in spherical background geometry

TL;DR

This paper studies an infinite combinatorial Ricci flow in spherical geometry, proving long-term existence and convergence under certain conditions, marking a first in the field for such flows.

Contribution

It introduces the first analysis of an infinite combinatorial Ricci flow in spherical background geometry, establishing existence and convergence results.

Findings

Existence of solutions for all time

Convergence of solutions under additional conditions

First study of infinite combinatorial Ricci flow in spherical geometry

Abstract

Since the fundamental work of Chow-Luo \cite{CL03}, Ge \cite{Ge12,Ge17} et al., the combinatorial curvature flow methods became a basic technique in the study of circle pattern theory. In this paper, we investigate the combinatorial Ricci flow with prescribed total geodesic curvatures in spherical background geometry. For infinite cellular decompositions, we establish the existence of a solution to the flow equation for all time. Furthermore, under an additional condition, we prove that the solution converges as time tends to infinity. To the best of our knowledge, this is the first study of an infinite combinatorial curvature flow in spherical background geometry.