Horosphere topology

TL;DR

This paper introduces a new boundary and topology concept called horosphere boundary for complete Kobayashi hyperbolic manifolds, providing insights into boundary behavior of univalent maps, especially in non-smooth domains.

Contribution

It develops a novel horosphere topology theory, extending boundary notions and analyzing boundary behavior of univalent maps in complex hyperbolic manifolds.

Findings

Bounded strongly pseudoconvex domains are homeomorphic to their Euclidean closure under the horosphere topology.

The horosphere topology on the polydisc is non-Hausdorff and differs from the Gromov topology.

Univalent maps from the unit ball to bounded convex domains extend as homeomorphisms up to the boundary.

Abstract

We introduce a prime end-type theory on complete Kobayashi hyperbolic manifolds using horosphere sequences. This allows to introduce a new notion of boundary-new even in the unit disc in the complex space-the horosphere boundary, and a topology on the manifold together with its horosphere boundary, the horosphere topology. We prove that a bounded strongly pseudoconvex domain endowed with the horosphere topology is homeomorphic to its Euclidean closure, while for the polydisc such a horosphere topology is not even Hausdorff and is different from the Gromov topology. We use this theory to study boundary behavior of univalent maps from bounded strongly pseudoconvex domains. Among other things, we prove that every univalent map of the unit ball whose image is bounded and convex, extends as a homeomorphism up to the closure. Such a result, relying in an essential way on our theory and on the Gromov hyperbolicity theory, is completely new, dealing with non smooth domains.