Convex RP^2 Structures and Cubic Differentials under Neck Separation

Abstract

Let S be a closed oriented surface of genus at least two. Labourie and the author have independently used the theory of hyperbolic affine spheres to find a natural correspondence between convex RP^2 structures on S and pairs (\Sigma,U) consisting of a conformal structure \Sigma on S and a holomorphic cubic differential U over \Sigma. The pairs (\Sigma,U$, for \Sigma varying in moduli space, allow us to define natural holomorphic coordinates on the moduli space of convex RP^2 structures. We consider geometric limits of convex RP^2 structures on S in which the RP^2 structure degenerates only along a set of simple, non-intersecting, non-homotopic loops c. We classify the resulting RP^2 structures on S-c and call them regular convex RP^2 structures. We put a natural topology on the moduli space of all regular convex RP^2 structures on S and show that this space is naturally homeomorphic to the total space of the vector bundle over the Deligne-Mumford compactification of the moduli space of curves each of whose fibers over a noded Riemann surface is the space of regular cubic differentials. In other words, we can extend our holomorphic coordinates to bordify the moduli space of convex RP^2 structures along all neck pinches. The proof relies on previous techniques of the author, Benoist-Hulin, and Dumas-Wolf, as well as some details due to Wolpert of the geometry of hyperbolic metrics on conformal surfaces in the Deligne-Mumford compactification.