Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type

TL;DR

This paper establishes a precise link between the geometric property of Gromov hyperbolicity of the Kobayashi metric and the finite type condition of convex domains with smooth boundaries, using tools from Hilbert metric theory.

Contribution

It provides necessary and sufficient conditions for Gromov hyperbolicity of the Kobayashi metric on convex domains of finite type, extending understanding of complex geometric structures.

Findings

Gromov hyperbolicity characterized by finite type condition

Equivalence for convex domains with smooth boundary

Gromov hyperbolicity for locally convexifiable domains

Abstract

In this paper we prove necessary and sufficient conditions for the Kobayashi metric on a convex domain to be Gromov hyperbolic. In particular we show that for convex domains with $C^\infty$ boundary being of finite type in the sense of D'Angelo is equivalent to the Gromov hyperbolicity of the Kobayashi metric. We also show that bounded domains which are locally convexifiable and have finite type in the sense of D'Angelo have Gromov hyperbolic Kobayashi metric. The proofs use ideas from the theory of the Hilbert metric.