A weak notion of visibility, a family of examples, and Wolff-Denjoy theorems

Abstract

We investigate a form of visibility introduced recently by Bharali and Zimmerand shown to be possessed by a class of domains called Goldilocks domains. The range of theorems established for these domains stem from this form of visibility together with certain quantitative estimates that define Goldilocks domains. We show that some of the theorems alluded to follow merely from the latter notion of visibility. We call those domains that possess this property visibility domains with respect to the Kobayashi distance. We provide a sufficient condition for a domain in C n to be a visibility domain. A part of this paper is devoted to constructing a family of domains that are visibility domains with respect to the Kobayashi distance but are not Goldilocks domains. Our notion of visibility is reminiscent of uniform visibility in the context of CAT(0) spaces. However, this is an imperfect analogy because, given a bounded domain Ω in C n , n 2, it is, in general, not even known whether the metric space (Ω, kΩ) (where kΩ is the Kobayashi distance) is a geodesic space. Yet, with just this weak property, we establish two new Wolff-Denjoy-type theorems.2010 Mathematics Subject Classification. Primary: 32F45, 32H50, 53C23; Secondary: 32U05.

Visibility of Kobayashi geodesics in convex domains and related properties

“…It is proved in [BZ,Theorem 1.4] that a Goldilocks domain has an extended visibility property which implies in particular the one in Definition 2.2 when the Kobayashi distance is geodesic. A more general result implying visibility is given in [BM,Theorem 1.5 (General Visibility Lemma)], which essentially reduces to the Goldilocks case when the domain is convex.…”

“…Note that the presence of a point on the boundary of the domain where the boundary "approaches" very fast a complex tangent line is enough to prevent the first (and crucial) condition in [BZ,Definition 1.1] (see also Definition 2.6 below) to hold. So Theorem 1.2 provides new examples of bounded convex domains having the visibility property without being Goldilocks nor satisfying the weaker hypotheses of [BM,Theorem 1.5].…”

Local and global visibility and Gromov hyperbolicity of domains with respect to the Kobayashi distance

“…Roughly speaking this means that geodesic lines that converges to different points in the Gromov boundary bend inside the space. However, visibility (with respect to the Euclidan boundary) has been exhibited for domains which are not Gromov hyperbolic in [3,2,10,17], and turns out to be a key notion for continuous extension of biholomorphisms and Denjoy-Wolff type theorems. In [13], this notion has been extended to embedded submanifolds of C d .…”

Abstract boundaries and continuous extension of biholomorphisms

“…In this survey, we consider extension property which comes from "visible geometry", as introduced in [BB,Ka,Z1,Z2,Z3,BZ,BM,BG,BGZ1,CL,BNT].…”