Tame the flexibility of circle patterns

TL;DR

This paper establishes a comprehensive theorem characterizing deformation of circle patterns with interstices, linking combinatorics, intersection angles, and conformal structures, and derives related surface and approximation results.

Contribution

It introduces a deformation circle pattern theorem that fully describes circle patterns with interstices based on combinatorial and geometric data, extending Rivin's theorem and approximation theory.

Findings

Complete description of circle pattern deformations

Surface version of Rivin's theorem proved

Approximation property of packable surfaces established

Abstract

This paper proves a deformation circle pattern theorem, which gives a complete description of those circle patterns with interstices in terms of the combinatorial type, the exterior intersections angles and the conformal structures of interstices. As results, the surface version of Rivin's theorem and the approximation property of packable surfaces are obtained.