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

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.