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

TL;DR

This paper establishes a correspondence between poles of cubic differentials and types of ends in convex real projective structures, extending the geometric understanding of these structures via affine sphere theory.

Contribution

It generalizes previous results by classifying ends of convex $ ext{RP}^2$-surfaces based on the order of poles of associated cubic differentials, and introduces a natural bordification at poles of order at least 3.

Findings

Poles of order less than 3 correspond to finite volume ends.

Poles of order 3 correspond to geodesic ends.

Poles greater than 3 correspond to piecewise geodesic ends.

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.