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Ser Peow Tan

Circle Packings on Surfaces with Projective Stuructures II

Abstract: This is the second part of two consecutive talks on circle packings on surfaces with projective structures.

The main aim of this talk is to discuss global results under the assumption that a circle packing is formed by {\it one} circle, that is, the associated graph $\tau$ has exactly {\it one} vertex. Using the notation in the abstract of the first part, we show that:

Theorem 1. Whan $\tau$ has one vertex, then the cross rario parameter space $\mathcal{C}_{\tau}$ is homeomorphic to $\mathcal{R}^{6g-6}$.

Theorem 2. When $\tau$ has one vertex, the forgetting map $f : \mathcal{C}_{\tau} \to \mathcal{P}_g$ is injective. In particular, the image $\mathcal{I}_{\tau} = f(\mathcal{C}_{\tau})$ is homeomorphic to $\mathbb{R}^{6g-6}$, and for any surface in $\mathcal{I}_{\tau} $, the circle packing on the surface with nerve $\tau$ is rigid.

Theorem 3. When $\tau$ has one vertex, the compsition $u \circ f : \mathcal{C} \to \mathcal{T}_g$ of the forgetting map with the uniformization map is proper.

These results provide strong evidence towards the validity of our conjecture that the composition of the forgetting map with the uniformization map is a homeomorphism for a general graph $\tau$.