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

Abstract

On any oriented surface, the affine sphere construction gives 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, while poles of order at least $3$ correspond to geodesic or piecewise geodesic boundary components. More specifically, in the latter situation, we prove asymptotic results for the convex $\mathbb{RP}^2$-structure around the pole in terms of the cubic differential.