Gromov hyperbolicity, the Kobayashi metric, and $\mathbb{C}$-convex sets

TL;DR

This paper investigates the conditions under which the Kobayashi metric on complex domains is Gromov hyperbolic, focusing on the role of boundary geometry and extending known results to $ ext{C}$-convex sets with smooth boundaries.

Contribution

It extends the characterization of Gromov hyperbolicity of the Kobayashi metric to $ ext{C}$-convex sets with smooth boundaries and demonstrates the necessity of boundary regularity through counterexamples.

Findings

Gromov hyperbolicity is obstructed by boundary complex affine disks in convex domains.

Extension of the boundary condition results to $ ext{C}$-convex sets with $C^1$-smooth boundary.

Existence of $ ext{C}$-convex domains with Gromov hyperbolic Kobayashi metric despite boundary containing complex affine balls.

Abstract

In this paper we study the global geometry of the Kobayashi metric on domains in complex Euclidean space. We are particularly interested in developing necessary and sufficient conditions for the Kobayashi metric to be Gromov hyperbolic. For general domains, it has been suggested that a non-trivial complex affine disk in the boundary is an obstruction to Gromov hyperbolicity. This is known to be the case when the set in question is convex. In this paper we first extend this result to $\mathbb{C}$-convex sets with $C^1$-smooth boundary. We will then show that some boundary regularity is necessary by producing in any dimension examples of open bounded $\mathbb{C}$-convex sets where the Kobayashi metric is Gromov hyperbolic but whose boundary contains a complex affine ball of complex codimension one.