Horosphere topology

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.

Homeomorphic extension of quasi-isometries for convex domains in $${\mathbb {C}}^d$$ and iteration theory

“…In case D is the unit ball (or more generally a strongly convex domain with smooth boundary), the previous extension result for biholomorphisms has been proved in [15] in case Ω is bounded, as an application of a Carathéodory prime ends type theory in higher dimension called "horosphere topology", and in [16] for the case Ω is unbounded by using a direct argument relying on the dynamics of semigroups of holomorphic self-maps.…”

A Characterization of Orthogonal Convergence in Simply Connected Domains

“…In this paper we give a complete answer to the above question in terms of hyperbolic geometry and "horocycles" in the spirit of [7], explaining also more geometrically Betsakos's theorem.…”

“…In this section we define horocycles (also called "horospheres") in simply connected domains, using in this context an abstract approach introduced in [7], and inspired on the general definition of horospheres in several complex variables introduced by M. Abate [1,2].…”

Regularity of complex geodesics and (non-)Gromov hyperbolicity of convex tube domains

“…For a domain Ω ⊂ R n denote by T Ω the tube domain with the base Ω as follows (4) T Ω := {z ∈ C n : Re z ∈ Ω}.…”