Perron's method and spherical ideal circle patterns with prescribed total geodesic curvatures

TL;DR

This paper proves existence and uniqueness of spherical circle patterns with prescribed geodesic curvatures on closed surfaces using Perron's method, and demonstrates convergence of Thurston's algorithm to these patterns.

Contribution

It introduces a new proof technique for circle pattern existence and uniqueness, and establishes convergence of an algorithm for constructing such patterns.

Findings

Existence and uniqueness of circle patterns with prescribed total geodesic curvatures.

Convergence of Thurston's algorithm to the circle pattern.

Application of Perron's method in spherical geometry.

Abstract

In this paper, we apply the classical Perron method to give a proof of the existence and uniqueness/rigidity result of a circle pattern on a closed surface equipped with conical spherical metric when prescribed measures of the angles of intersecting circles stay in the range (0,{\pi}/2] and total geodesic curvatures are assigned to the circles, which is recently obtained in [3] via Colin de Verdi\`ere's variation method. Then we show the convergence of Thurston's algorithm, which adjusts the geodesic curvatures of circles one by one based on the prescribed values for total geodesic curvatures of the circles, to the desired circle pattern in the setting of the result.