On circle patterns and spherical conical metrics

TL;DR

This paper establishes a unique existence result for circle patterns with spherical conical metrics by prescribing geodesic total curvature, overcoming non-uniqueness issues caused by M"obius transformations.

Contribution

It introduces a new approach to spherical circle patterns by fixing geodesic total curvature instead of cone angles, ensuring uniqueness.

Findings

Unique existence of spherical circle patterns with prescribed geodesic total curvature

Overcomes non-uniqueness caused by M"obius transformations in spherical metrics

Extends circle pattern theory to spherical conical metrics

Abstract

The Koebe-Andreev-Thurston circle packing theorem, as well as its generalization to circle patterns due to Bobenko and Springborn, holds for Euclidean and hyperbolic metrics possibly with conical singularities, but fails for spherical metrics because of the non-uniqueness coming from M\"obius transformations. In this paper, we show that a unique existence result for circle pattern with spherical conical metric holds if one prescribes the geodesic total curvature of each circle instead of the cone angles.