On a combinatorial curvature for surfaces with inversive distance circle packing metrics

TL;DR

This paper introduces a new combinatorial curvature for triangulated surfaces with inversive distance circle packings, proves its global rigidity, and explores its associated Ricci flow and Yamabe problem, extending to a generalized curvature.

Contribution

It presents a novel combinatorial curvature, establishes its global rigidity, and develops a Ricci flow approach to analyze the associated Yamabe problem, including a generalization to alpha-curvature.

Findings

The new combinatorial curvature is globally rigid.

The combinatorial Ricci flow converges if and only if a constant curvature metric exists.

The alpha-curvature generalization also exhibits global rigidity.

Abstract

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evolves almost in the same way as that of scalar curvature along the surface Ricci flow obtained by Hamilton \cite{Ham1}. Then we study the long time behavior of the combinatorial Ricci flow and obtain that the existence of a constant curvature metric is equivalent to the convergence of the flow on triangulated surfaces with nonpositive Euler number. We further generalize the combinatorial curvature to $\alpha$-curvature and prove that it is also globally rigid, which is in fact a generalized Bower-Stephenson conjecture \cite{BS}. We also use the combinatorial Ricci flow to study the corresponding $\alpha$-Yamabe problem.