The existence and uniqueness of infinite combinatorial Yamabe flows

TL;DR

This paper investigates the combinatorial Yamabe flow on infinite triangulated surfaces, establishing existence, uniqueness, and convergence results, thereby advancing the understanding of discrete geometric flows on noncompact surfaces.

Contribution

It proves short-time existence and uniqueness, introduces an extended flow with long-time existence, and demonstrates convergence on hexagonal plane triangulations.

Findings

Short-time existence and uniqueness of the flow

Extended flow has long-time existence

Flow converges on hexagonal plane triangulations

Abstract

In this paper, we study the combinatorial Yamabe flow on infinite triangulated surfaces in Euclidean background geometry, aiming for solving discrete Yamabe problem on noncompact surfaces. Under suitable conditions, we establish the short-time existence and uniqueness of the flow. We further introduce an extended version of the flow and prove its long-time existence. As an application, we prove the convergence result of the Yamabe flow in the case of hexagonal triangulations of the plane.