Combinatorial curvature flows with surgery for inversive distance circle packings on surfaces

TL;DR

This paper introduces new combinatorial curvature flows with surgery techniques to find Euclidean metrics with prescribed curvatures on surfaces, overcoming singularity issues and ensuring long-term convergence.

Contribution

It develops and proves the convergence of combinatorial curvature flows with surgery for inversive distance circle packings, enabling effective metric computation.

Findings

Proves long-term existence of curvature flows with surgery.

Establishes convergence of flows to prescribed curvatures.

Provides algorithms for surface metric determination.

Abstract

Inversive distance circle packings introduced by Bowers-Stephenson are natural generalizations of Thurston's circle packings on surfaces. To find piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures, we introduce the combinatorial Calabi flow, the fractional combinatorial Calabi flow and the combinatorial $p$-th Calabi flow for the Euclidean inversive distance circle packings. Due to the singularities possibly developed by these combinatorial curvature flows, the longtime existence and convergence of these combinatorial curvature flows have been a difficult problem for a long time. To handle the potential singularities along these combinatorial curvature flows, we do surgery along these flows by edge flipping under the weighted Delaunay condition. Using the discrete conformal theory recently established by Bobenko-Lutz for decorated piecewise Euclidean metrics on surfaces, we prove the longtime existence and global convergence for the solutions of these combinatorial curvature flows with surgery. This provides effective algorithms for finding piecewise Euclidean metrics on surfaces with prescribed combinatorial curvatures.