Horosphere topology - Archive ouverte HAL Access content directly
Preprints, Working Papers, ... Year : 2016

Horosphere topology

Hervé Gaussier

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.
Fichier principal
Vignette du fichier
horosphere-final.pdf (356.17 Ko) Télécharger le fichier
Origin : Files produced by the author(s)
Loading...

Dates and versions

hal-01315332 , version 1 (13-05-2016)
hal-01315332 , version 2 (24-06-2016)
hal-01315332 , version 3 (18-09-2016)
hal-01315332 , version 4 (17-01-2017)
hal-01315332 , version 5 (04-10-2017)
hal-01315332 , version 6 (05-04-2018)

Identifiers

Cite

Filippo Bracci, Hervé Gaussier. Horosphere topology. 2016. ⟨hal-01315332v6⟩

Collections

UGA CNRS FOURIER
305 View
249 Download

Altmetric

Share

Gmail Facebook X LinkedIn More