Projective Rigidity of Circle Packings

TL;DR

This paper proves the projective rigidity of circle packings on surfaces of genus at least two, showing they form a submanifold within the space of complex projective structures, thus not deformable within that structure.

Contribution

It establishes the projective rigidity of circle packings and characterizes their space as a submanifold in the complex projective structure space.

Findings

Circle packings are projectively rigid on high-genus surfaces.

The space of circle packings forms a submanifold within complex projective structures.

Circle packings cannot be deformed within the complex projective structure.

Abstract

We prove that the space of circle packings consistent with a given triangulation on a surface of genus at least two is projectively rigid, so that a packing on a complex projective surface is not deformable within that complex projective structure. More broadly, we show that the space of circle packings is a submanifold within the space of complex projective structures on that surface.