Notions of Visibility with respect to the Kobayashi distance: Comparison and Applications

TL;DR

This paper explores various notions of visibility related to the Kobayashi distance in complex analysis, introduces new classes of domains with these properties, and applies these concepts to extend Kobayashi isometries and prove a generalized Wolff--Denjoy theorem.

Contribution

It introduces a new sufficient condition for visibility, defines visibility subspaces, and generalizes existing theorems on Kobayashi isometries and Wolff--Denjoy type results.

Findings

New classes of domains with the visibility property are identified.

A theorem on the continuous extension of Kobayashi isometries is proved.

A generalized Wolff--Denjoy theorem is established.

Abstract

In this article, we study notions of visibility with respect to the Kobayashi distance for relatively compact complex submanifolds in Euclidean spaces. We present a sufficient condition for a domain to possess the visibility property relative to Kobayashi almost-geodesics introduced by Bharali--Zimmer (we call this simply the visibility property). As an application, we produce new classes of domains having this kind of visibility. Next, we introduce and study the notion of visibility subspaces of relatively compact complex submanifolds. Using this notion, we generalize to such submanifolds a recent result of Bracci--Nikolov--Thomas. The utility of this generalization is demonstrated by proving a theorem on the continuous extension of Kobayashi isometries. Finally, we prove a Wolff--Denjoy-type theorem that is a generalization of recent results of this kind by Bharali--Zimmer and Bharali--Maitra and that, owing to the new classes of domains mentioned, is a proper generalization. Along the way, we note that what is needed for the proof of this sort of theorem to work is a form of visibility that seems to be intermediate between what we are calling visibility and visibility with respect to ordinary Kobayashi geodesics.