Gromov hyperbolicity of intrinsic metrics from isoperimetric inequalities

Abstract

In this paper we investigate the Gromov hyperbolicity of the classical Kobayashi and Hilbert metrics, and the recently introduced minimal metric. Using the linear isoperimetric inequality characterization of Gromov hyperbolicity, we show if these metrics have an "expanding property" near the boundary, then they are Gromov hyperbolic. This provides a new characterization of the convex domains whose Hilbert metric is Gromov hyperbolic, a new proof of Balogh-Bonk's result that the Kobayashi metric is Gromov hyperbolic on a strongly pseudoconvex domain, a new proof of the second author's result that the Kobayashi metric is Gromov hyperbolic on a convex domain with finite type, and a new proof of Fiacchi's result that the minimal metric is Gromov hyperbolic on a strongly minimally convex domain. We also characterize the smoothly bounded convex domains where the minimal metric is Gromov hyperbolic.