Discrete conformal structures on surfaces with boundary (III) -- Deformation

TL;DR

This paper develops and analyzes combinatorial Ricci and Calabi flows for discrete conformal structures on surfaces with boundary, providing algorithms for hyperbolic metrics with prescribed boundary lengths.

Contribution

It introduces and proves the convergence of combinatorial curvature flows for surfaces with boundary, advancing the deformation theory of discrete conformal structures.

Findings

Longtime existence of the curvature flows

Global convergence of the flows

Effective algorithms for hyperbolic metrics

Abstract

The present work constitutes the third installment in a series of investigations devoted to discrete conformal structures on surfaces with boundary. In our preceding works \cite{X-Z DCS1, X-Z DCS2}, we established, respectively, a classification of these discrete conformal structures and results on their rigidity and existence. Building on this foundation, the present work focuses on the deformation theory of discrete conformal structures on surfaces with boundary. Specifically, we introduce the combinatorial Ricci flow and the combinatorial Calabi flow, and establish the longtime existence and global convergence of solutions to these combinatorial curvature flows. These results yield effective algorithms for finding discrete hyperbolic metrics on surfaces with totally geodesic boundaries of prescribed lengths.