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 $\mathbb{RP}^2$-surfaces, whereas poles of order $3$ and greater than $3$ correspond to geodesic and piecewise geodesic ends, respectively. Moreover, at each pole of order at least $3$, we construct a natural bordification of the surface such that the $\mathbb{RP}^2$-structure extends to the boundary circle in a metric preserving way.
Citation
Xin Nie. "Poles of cubic differentials and ends of convex $\mathbb{RP}^2$-surfaces." J. Differential Geom. 123 (1) 67 - 140, 1 January 2023. https://doi.org/10.4310/jdg/1679503805
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