Goldilocks domains, a weak notion of visibility, and applications

TL;DR

This paper introduces Goldilocks domains in complex Euclidean space, characterized by specific growth bounds of the Kobayashi metric, and explores their geometric properties and applications in boundary behavior and Wolff-Denjoy theorems.

Contribution

It defines Goldilocks domains with growth bounds on the Kobayashi metric and demonstrates their geometric properties, including visibility and boundary extension results.

Findings

Goldilocks domains satisfy a visibility condition similar to negatively curved spaces.

The Kobayashi metric on these domains exhibits behavior akin to negatively curved Riemannian metrics.

Applications include boundary extension of maps and Wolff-Denjoy theorems.

Abstract

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary. Strongly pseudoconvex domains and weakly pseudoconvex domains of finite type always satisfy this Goldilocks condition, but we also present families of Goldilocks domains that have low boundary regularity or have boundary points of infinite type. We will show that the Kobayashi metric on these domains behaves, in some sense, like a negatively curved Riemannian metric. In particular, it satisfies a visibility condition in the sense of Eberlein and O'Neill. This behavior allows us to prove a variety of results concerning boundary extension of maps and to establish Wolff-Denjoy theorems for a wide collection of domains.