Observing that the notion of disk in 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.
