Properness for circle packings and Delaunay circle patterns on complex projective structures

TL;DR

This paper proves that the map from complex projective structures with specified circle packings or Delaunay circle patterns to the underlying complex structures is proper, advancing understanding of geometric structures on surfaces.

Contribution

It establishes the properness of the forgetful map for circle packings and Delaunay circle patterns on complex projective surfaces, confirming a conjecture of Kojima, Mizushima, and Tan.

Findings

Properness of the forgetful map is proven.

Supports conjecture by Kojima, Mizushima, and Tan.

Enhances understanding of geometric structures on surfaces.

Abstract

We consider circle packings and, more generally, Delaunay circle patterns - arrangements of circles arising from a Delaunay decomposition of a finite set of points - on surfaces equipped with a complex projective structure. Motivated by a conjecture of Kojima, Mizushima and Tan, we prove that the forgetful map sending a complex projective structure admitting a circle packing with given nerve (resp. a Delaunay circle pattern with given nerve and intersection angles) to the underlying complex structure is proper.