Convex Rp Structures and Cubic Differentials under Neck Separation

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 structures on S and pairs (Σ, U) consisting of a conformal structure Σ on S and a holomorphic cubic differential U over Σ. The pairs (Σ, U), for Σ varying in moduli space, allow us to define natural holomorphic coordinates on the moduli space of convex RP structures. We consider geometric limits of convex RP structures on S in which the RP structure degenerates only along a set of simple, non-intersecting, non-homotopic loops c. We classify the resulting RP structures on S−c and call them regular convex RP structures. We put a natural topology on the moduli space of all regular convex RP structures on S and show that this space is naturally homeomorphic to the total space of the vector bundle over Mg 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 structures along all neck pinches. The proof relies on previous techniques of the author, Benoist-Hulin, and DumasWolf, as well as some details due to Wolpert of the geometry of hyperbolic metrics on conformal surfaces in Mg.