Poles of cubic differentials and ends of convex $\mathbb{RP}^2$-surfaces

Abstract

The affine sphere construction gives, on any oriented surface, a one-to-one correspondence between convex $\mathbb{RP}^2$-structures and holomorphic cubic differentials. Generalizing results of Benoist-Hulin, Loftin and Dumas-Wolf, we show that poles of order less than $3$ of cubic differentials correspond to finite volume ends of convex $\mathbb{RP}^2$-structures, and poles of order $3$ (resp. bigger than $3$) correspond to geodesic (resp. piecewise geodesic) ends. In particular, at a pole of order at least $3$, we bordify the surface by attaching to it a boundary circle in a natural way with respect to the cubic differential, and show that the $\mathbb{RP}^2$-structure extends to the boundary in a metric preserving way.