Combinatorial Calabi flow with surgery on surfaces

TL;DR

This paper introduces a rigorous algorithm using combinatorial Calabi flow with surgery to compute uniformization maps for surfaces, ensuring convergence and handling singularities effectively.

Contribution

It develops a novel combinatorial Calabi flow with surgical flips for vertex scaling, guaranteeing convergence for any initial polyhedral metric on surfaces.

Findings

Flow with surgery exists for all time and converges exponentially

Convergence is independent of initial triangulation

Algorithm effectively handles singularities during flow

Abstract

Computing uniformization maps for surfaces has been a challenging problem and has many practical applications. In this paper, we provide a theoretically rigorous algorithm to compute such maps via combinatorial Calabi flow for vertex scaling of polyhedral metrics on surfaces, which is an analogue of the combinatorial Yamabe flow introduced by Luo. To handle the singularies along the combinatorial Calabi flow, we do surgery on the flow by flipping. Using the discrete conformal theory established by Gu-Guo-Luo-Sun-Wu and Gu-Luo-Sun-Wu, we prove that for any initial Euclidean or hyperbolic polyhedral metric on a closed surface, the combinatorial Calabi flow with surgery exists for all time and converges exponentially fast after finite number of surgeries. The convergence is independent of the combinatorial structure of the initial triangulation on the surface.