Projective rigidity of circle patterns and polyhedral surfaces in hyperbolic ends

TL;DR

This paper establishes a detailed geometric correspondence between Delaunay circle patterns and ideal polyhedral surfaces in hyperbolic ends, extending previous results to more general polyhedral types and structures.

Contribution

It proves that the space of complex projective structures with prescribed Delaunay circle patterns forms a manifold and that the associated forgetful map is a Lagrangian immersion, extending prior circle packing results.

Findings

The space of structures with prescribed circle patterns is a manifold of dimension 6g-6.

The forgetful map to CP^1-structures is a Lagrangian immersion.

Extension of results to ideal polyhedral surfaces with prescribed edge lengths and other types.

Abstract

Let $S$ be a closed, orientable surface of genus $g\geq 2$. We consider Delaunay circle patterns on $S$ equipped with a complex projective structure. We prove that the space of complex projective structures on $S$ equipped with a Delaunay circle pattern of prescribed combinatorics and intersection angles is a manifold of dimension $6g-6$, and that the forgetful map to the space $\cC_S$ of $\CP^1$-structures on $S$ is a Lagrangian immersion. This extends a recent result of Bonsante and Wolf for circle packings. This statement, and its proof, are more conveniently stated in terms of ideal polyhedral surfaces (surfaces with vertices at infinity) in hyperbolic ends, with the angles between the circles corresponding to the dihedral angles. Seen from this angle, we extend the statement to ideal polyhedral surfaces with prescribed edge lengths (or induced metrics), and to other types of polyhedral surfaces, either compact or hyperideal.