Convergence of combinatorial Ricci flows to degenerate circle patterns

TL;DR

This paper studies the behavior of combinatorial Ricci flows on surfaces with nonpositive Euler characteristic, especially when standard convergence conditions are not met, addressing open questions in the field.

Contribution

It provides new insights into the convergence properties of combinatorial Ricci flows under degenerate conditions, extending previous understanding.

Findings

Identifies conditions under which Ricci flows fail to converge

Addresses open questions about degenerate circle patterns

Provides theoretical analysis of flow behavior in non-convergent cases

Abstract

We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.