Combinatorial Calabi flows on surfaces with boundary

TL;DR

This paper introduces and analyzes combinatorial Calabi flows on surfaces with boundary, proving their long-term existence and convergence, and presents algorithms for constructing hyperbolic surfaces with specified boundary lengths.

Contribution

It develops the combinatorial Calabi flow and fractional Calabi flow on surfaces with boundary, extending previous flows and establishing their convergence properties.

Findings

Proved long-term existence of combinatorial Calabi flow.

Established global convergence of the flows.

Provided algorithms for hyperbolic surface construction.

Abstract

Motivated by Luo's combinatorial Yamabe flow on closed surfaces \cite{L1} and Guo's combinatorial Yamabe flow on surfaces with boundary \cite{Guo}, we introduce combinatorial Calabi flow on ideally triangulated surfaces with boundary, aiming at finding hyperbolic metrics on surfaces with totally geodesic boundaries of given lengths. Then we prove the long time existence and global convergence of combinatorial Calabi flow on surfaces with boundary. We further introduce fractional combinatorial Calabi flow on surfaces with boundary, which unifies and generalizes the combinatorial Yamabe flow and the combinatorial Calabi flow on surfaces with boundary. The long time existence and global convergence of fractional combinatorial Calabi flow are also proved. These combinatorial curvature flows provide effective algorithms to construct hyperbolic surfaces with totally geodesic boundaries with prescribed lengths.