Cubic differentials and hyperbolic convex sets. (English) Zbl 1301.53040

Benoist, Yves; Hulin, Dominique

J. Differ. Geom. 98, No. 1, 1-19 (2014).

The authors show that a properly convex domain \(\Omega \subset \mathbb R\text{P}^2\) is Gromov hyperbolic if and only if the volumes of the metric balls in \((\Omega, d_\Omega)\) have a uniform exponential growth rate and if they present a natural bijection between the moduli space of marked Gromov hyperbolic properly convex domains of \(\mathbb R\text{P}^2\) and the space of bounded holomorphic cubic differentials on the disk.

Reviewer: Witold Mozgawa (Lublin)

MSC:

53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

moduli space; properly convex domain; Gromov hyperbolicity; cubic differential; Pick form; curvature estimate; growth profile; marked hyperbolic domain