Circle packings on surfaces with projective structures and uniformization

Sadayoshi Kojima, Shigeru Mizushima and Ser Peow Tan

Vol. 225 (2006), No. 2, 287–300

DOI: 10.2140/pjm.2006.225.287

Abstract

Let ${\Sigma }_g$ be a closed orientable surface of genus $g\ge 2$ and $\tau$ a graph on ${\Sigma }_g$ with one vertex that lifts to a triangulation of the universal cover. We have shown before that the cross ratio parameter space ${\mathcal{C}}_{\tau }$ associated with $\tau$, which can be identified with the set of all pairs of a projective structure and a circle packing on it with nerve isotopic to $\tau$, is homeomorphic to ${ℝ}^{6g-6}$, and moreover that the forgetting map of ${\mathcal{C}}_{\tau }$ to the space of projective structures is injective. Here we show that the composition of the forgetting map with the uniformization from ${\mathcal{C}}_{\tau }$ to the Teichmüller space ${\mathcal{T}}_g$ is proper.

Dedicated to Professor Yukio Matsumoto on his sixtieth birthday

Keywords

circle packing, projective structure, uniformization, Teichmüller space

Mathematical Subject Classification 2000

Primary: 57M27

Milestones

Received: 18 August 2004

Revised: 14 December 2004

Accepted: 19 January 2005

Published: 1 June 2006