Rigidity of circle packings on projective surfaces

Observing that the notion of disk in $CP^1$ is invariant under projective transformations, Kojima, Mizushima and Tan proposed the study of circle packings on surfaces equipped with complex projective structures. The way disks are arranged in a circle packing is described by a triangulation of the surface, called the nerve of the packing. By a celebrated theorem of Koebe and Andreev-Thurston, for any triangulation T of a surface of genus g>1, there exists a unique hyperbolic metric which admits a circle packing with nerve T. However the space of projective structures equipped with a circle packing with a given nerve is a real algebraic set of dimension 6g-6. Kojima Mizushima and Tan conjectured that the natural forgetful map associating to a projective structure equipped with a circle packing the underlying conformal structure is a diffeomorpfismo of this space to the Teichmüller space.

In the talk I will explain some recent developments obtained in collaboration with Mike Wolf.