Combinatorial curvature flows for generalized circle packings on surfaces with boundary

TL;DR

This paper develops combinatorial curvature flows to deform generalized circle packings on surfaces with boundary, enabling the computation of hyperbolic metrics with prescribed boundary lengths through proven long-term convergence.

Contribution

It introduces combinatorial Ricci and Calabi flows for generalized circle packings on bordered surfaces and proves their long-term existence and convergence.

Findings

Proven long-term existence of the flows.

Established global convergence to desired metrics.

Provided algorithms for hyperbolic surface metrics with boundary.

Abstract

In this paper, we investigate the deformation of generalized circle packings on ideally triangulated surfaces with boundary, which is the $(-1,-1,-1)$ type generalized circle packing metric introduced by Guo-Luo \cite{GL2}. To find hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths, we introduce combinatorial Ricci flow and combinatorial Calabi flow for generalized circle packings on ideally triangulated surfaces with boundary. Then we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows, which provide effective algorithms for finding hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.