Combinatorial Yamabe flow on hyperbolic bordered surfaces

TL;DR

This paper investigates the combinatorial Yamabe flow on hyperbolic bordered surfaces, proving its long-term existence and exponential convergence to a conformal factor that matches prescribed boundary lengths, thus offering an effective algorithm.

Contribution

It establishes the global existence and exponential convergence of the flow, providing a new algorithm for prescribing boundary lengths on hyperbolic surfaces.

Findings

Flow exists for all time

Flow converges exponentially fast

Algorithm for boundary length prescription

Abstract

This paper studies the combinatorial Yamabe flow on hyperbolic bordered surfaces. We show that the flow exists for all time and converges exponentially fast to conformal factor which produces a hyperbolic surface whose lengths of boundary components are equal to prescribed positive numbers. This provides an algorithm to such problems.