Geometry and Topology Seminar

Labourie and I independently proved that n a closed oriented surface S of genus g at least 2, a convex real projective structure is equivalent to a pair (Σ, U) , where Σ is a conformal structure and U is a holomorphic cubic differential. It is then natural to allow Σ to go to the boundary of the moduli space of Riemann surfaces. The bundle of cubic differentials then extends over the boundary to form the bundle of regular cubic differentials, which is an orbifold vector bundle over the Deligne-Mumford compactification M_g of moduli space. We define regular convex real projective structures corresponding to the regular cubic differentials over nodal Riemann surfaces and define a topology on these structures. Our topology is an extension of Harvey's use of the Chabauty topology to analyze M_g via limits of Fuchsian groups. The main theorem is that the total space of the bundle of regular cubic differentials over M_g is homeomorphic to the space of regular real projective structures. The proof involves recent techniques of Benoist-Hulin and Dumas-Wolf.