Combinatorial Yamabe flow on hyperbolic surfaces with boundary

TL;DR

This paper investigates the combinatorial Yamabe flow on hyperbolic surfaces with boundary, demonstrating that boundary lengths are uniquely determined by the conformal factor and analyzing the flow's long-term behavior.

Contribution

It establishes the uniqueness of boundary lengths via a variational principle and explores the flow's properties as a gradient flow of a concave function.

Findings

Boundary lengths are uniquely determined by the conformal factor.

The flow is a gradient flow of a concave function.

Long-term behavior and geometric implications are analyzed.

Abstract

This paper studies the combinatorial Yamabe flow on hyperbolic surfaces with boundary. It is proved by applying a variational principle that the length of boundary components is uniquely determined by the combinatorial conformal factor. The combinatorial Yamabe flow is a gradient flow of a concave function. The long time behavior of the flow and the geometric meaning is investigated.