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

Abstract

We consider circle packings and, more generally, Delaunay circle patternsarrangements 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.

Quadratic differentials and circle patterns on complex projective tori

“…(ii) Surfaces with genus g > 1: Similarly, Kojima, Mizushima and Tan [21] showed that in the space of circle packings on any given triangulation, the circle packing on a hyperbolic surface is contained in a neighborhood homeomorphic to R 6g−6 . Recently, Schlenker and Yarmola [25] showed that in the setting of circle patterns, the projection to the Teichmüller space is a proper map. However it remains unknown whether the space of circle patterns with fixed intersection angles is a manifold and whether its projection to the Teichüller space is an immersion.…”