Circle Packings on Surfaces with Projective Structures

Abstract

The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface Σ g of genus g which is covered by a simple graph in the universal cover, there exists a unique metric of curvature 1, 0 or −1 on the surface depending on whether g = 0, 1 or ≥ 2 such that the surface with this metric admits a circle packing with combinatorics given by the triangulation. Furthermore, the circle packing is essentially rigid, that is, unique up to conformal automorphisms of the surface isotopic to the identity. In this paper, we consider projective structures on the surface Σ g where circle packings are also defined. We show that the space of projective structures on a surface of genus g ≥ 2 which admits a circle packing by one circle is homeomorphic to R 6g−6 and furthermore that the circle packing is rigid on such surfaces. 1991 Mathematics Subject Classification. Primary 52C15; Secondary 30F99, 57M50.

Circle packings on surfaces with projective structures and uniformization

“…Suppose that S is a projective Riemann surface lying in f (Ꮿ τ ) with a reference homeomorphism h : g → S, and let P be a circle packing on S with nerve isotopic to h(τ ). In [Kojima et al 2003], we defined the cross ratio parameter for the pair (S, P), which is a function…”

“…In [Kojima et al 2003], we conjectured that this holds in general, regardless of the (positive) genus g and the graph τ . In this paper we take the first step towards solving that conjecture, by proving the following properness theorem for graphs τ having a single vertex: Theorem 1.1.…”

“…In [Kojima et al 2003], we initiated the study of circle packings on Riemann surfaces with complex projective structures. Since the topic is rather new, we recall the setting briefly.…”

“…We are interested in the moduli space of all pairs (S, P) consisting of a projective Riemann surface S with a reference homeomorphism h : g → S and a circle packing P on S whose nerve is isotopic to h(τ ). In [Kojima et al 2003], this moduli space is shown to be identifiable with what we call the cross ratio parameter space Ꮿ τ .…”

“…One of main results in [Kojima et al 2003] is that when τ has exactly one vertex, Ꮿ τ is homeomorphic to euclidean space of dimension 6g − 6 and f is injective. Injectivity implies that each projective Riemann surface admits at most one circle packing dominated by τ .…”