Convex $\mathbb{RP}^2$ 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 $\mathbb{RP}^2$ structures on $S$ and pairs $(\Sigma, U)$ consisting of a conformal structure $\Sigma$ on $S$ and a holomorphic cubic differential $U$ over $\Sigma$. We consider geometric limits of convex $\mathbb{RP}^2$ structures on $S$ in which the $\mathbb{RP}^2$ structure degenerates only along a set of simple, non-intersecting, nontrivial, non-homotopic loops $c$. We classify the resulting $\mathbb{RP}^2$ structures on $S - c$ and call them regular convex $\mathbb{RP}^2$ structures. Under a natural topology on the moduli space of all regular convex $\mathbb{RP}^2$ structures on $S$, this space is homeomorphic to the total space of the vector bundle over $\overline{M}_g$ each of whose fibers over a noded Riemann surface is the space of regular cubic differentials. 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 $\overline{M}_g$.