Holomorphic mappings and Kobayashi geometry of domains

Abstract

In this thesis, we study certain aspects of the geometry of the Kobayashi (pseudo)distance and the Kobayashi (pseudo)metric for domains in $\mathbb{C}^n$. We focus on the following themes: on the interaction between Kobayashi geometry and the extension of holomorphic mappings, and on certain negative-curvature-type properties of Kobayashi hyperbolic domains equipped with their Kobayashi distances.

In the initial part of this thesis, we prove a couple of results on local continuous extension of proper holomorphic mappings $F:D \rightarrow \Omega$, $D, \Omega \varsubsetneq \mathbb{C}^n$, making local assumptions on $\partial D$ and $\partial \Omega$. These results are motivated by a well-known work by Forstneri\v{c}--Rosay. However, our results allow us to have much lower regularity, for the patches of $\partial D, \partial \Omega$ that are relevant, than in earlier results in the literature. Moreover, our assumptions allow $\partial \Omega$ to contain boundary points of infinite type.

We also study another type of extension phenomenon for holomorphic mappings, namely, Picard-type extension theorems. Well-known works by Kobayashi, Kiernan, and Joseph--Kwack have showed that Picard-type extension results hold true when the target spaces of the relevant holomorphic mappings belong to a class of Kobayashi hyperbolic complex manifolds---viewed as complex submanifolds embedded in some ambient complex manifold---with certain analytical properties. Beyond some classical examples, identifying such a target manifold by its geometric properties is, in general, hard. Restricting to $\mathbb{C}^n$ as the ambient space, we provide some geometric conditions on $\partial \Omega$, for any unbounded domain $\Omega \varsubsetneq \mathbb{C}^n$, for a Picard-type extension to hold true for holomorphic mappings into $\Omega$. These conditions are suggested, in part, by an explicit lower bound for the Kobayashi metric of a certain class of bounded domains. We establish the latter estimates using the regularity theory for the complex Monge--Amp{\`e}re equation. The notion that allows us to connect these estimates with Picard-type extension theorems is called ``visibility''.

In the concluding part of this thesis, we explore the notion of visibility for its own sake. For a Kobayashi hyperbolic domain $\Omega \varsubsetneq \mathbb{C}^n$, $\Omega$ being a visibility domain is a notion of negative curvature of $\Omega$ as a metric space equipped with the Kobayashi distance $K_{\Omega}$ and encodes a specific way in which $(\Omega, K_{\Omega})$ resembles the Poincar{\'e} disc model of the hyperbolic plane. The earliest examples of visibility domains, given by Bharali--Zimmer, are pseudoconvex. In fact, all examples of visibility domains in the literature are, or are conjectured to be, pseudoconvex. We show that there exist non-pseudoconvex visibility domains. We supplement this proof by a general method to construct a wide range of non-pseudoconvex, hence non-Kobayashi complete, visibility domains.