Circle packings on surfaces with projective structures
by
Ser Peow Tan
National University of Singapore, Singapore
Coauthors: Sadayoshi Kojima (Tokyo Institute of Technology, Japan), Shigeru Mizushima (Tokyo Institute of Technology, Japan)

The Andreev-Thurston theorem states that for any triangulation of a closed orientable surface \Sigma_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 greater than or equal to 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 \Sigma_g where circle packings are also defined. We show that the space of projective structures on a surface of genus g greater than or equal to 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.

Date received: February 6, 2002