Sadayoshi Kojima

Circle Packings on Surfaces with projective Structures I

Abstract: This is the first part of two consecutive talks on circle packings on surfaces with projective structures based on our joint work with Shigeru Mizushima.

A projective structure on a surface is a geometric structure locally modeled on the pair of the Riemann sphere and the group of projective transformations. It is not a Riemanniann structure but it still makes sense to talk about circles since projective transformations map a circle to a circle.

Fixing a graph $\tau$ on a surface $\Sigma_g$ of genus $g \geq 1$ which lifts to a triangulation on the universal cover, we set up the problem of which surfaces with projective structures admit a circle packing on it with nerve isotopic to $\tau$, and if some surface does, then whether the packing is rigid, and the relation with the uniformization map.

In this first part, we formulate the problem in terms of what we call a cross ratio parameter space $\mathcal{C}_{\tau}$, which turns out to be identified with the space of all pairs of a projective structure on $\Sigma_g$ and a packing with nerve $\tau$ on it, and propose the following conjecture: The composition of the forgetting map $f : \mathcal{C}_{\tau} \to \mathcal{P}_g$ to the space $\mathcal{P}_g$ of all projective structures on $\Sigma_g$ with the uniformization map@ $u : \mathcal{P}_g \to \mathcal{T}_g$ to the Teichm\"uller space $\mathcal{T}_g$ is a homeomorphism.

Also in this part, we discuss local results which support the conjecture near the unique hyperbolic surface with a required packing established by Koebe-Andreev-Thurston.