Kobayashi geometry and a metric notion of negative curvature

Abstract

In this thesis, we study some problems that represent various aspects of Kobayashi geometry. Our problems are, in some sense, motivated by a close examination---which forms a part of this thesis---of a property that may be viewed as a metric notion of negative curvature, for domains $\Omega$ in an arbitrary complex manifold, in terms of its Kobayashi distance $K_\Omega$. The latter property is known as "visibility''. Roughly speaking, the visibility property is that all geodesics originating sufficiently close to and terminating sufficiently close to two distinct points in $\bdy\Omega$ must bend uniformly into $\Omega$. Even for pseudoconvex domains $\Omega\varsubsetneq\mathbb{C}^n$, $n\geq 3$, let alone domains in complex manifolds, it is unclear when the metric space $(\Omega,K_\Omega)$ is a geodesic space. However, this property follows whenever $(\Omega,K_\Omega)$ is Cauchy-complete. This motivates our first set of problems. We give sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains (not necessarily relatively compact) in complex manifolds. We also provide a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. Next, we extend the notion of visibility relative to the Kobayashi distance to domains in arbitrary complex manifolds. Since it is difficult to determine whether domains are Cauchy-complete with respect to the Kobayashi distance, we do not assume so here. We establish some consequences of visibility and provide many sufficient conditions for visibility. We establish a Wolff--Denjoy-type theorem in a very general setting as an application. Furthermore, we explore some connections between visibility and Gromov hyperbolicity for Kobyashi hyperbolic domains in the above setting. We also include a short chapter that combines the two themes above. One may ask whether, with $(\Omega,K_\Omega)$ as above, it is the case that if $\Omega$ has the visibility property, then $(\Omega,K_\Omega)$ is automatically Cauchy-complete. This proves to be too naive and the "right'' question in $\mathbb{C}^n$ was raised by Banik, which we answer in the negative by showing that for each $n\geq 2$, there exists a taut visibility domain $\Omega$ in $\mathbb{C}^n$, with rather nice boundary, such that $(\Omega,K_\Omega)$ is not Cauchy-complete. The visibility framework is optimised for continuous quasi-isometries for the Kobayashi distance between domains, but it is silent about the extension of proper holomorphic maps. It should be possible to combine Kobayashi geometry with other techniques to study proper holomorphic maps. To this end, we explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge--Amp{\`e}re equation. Among the results given by these connections: $(i)$~a theorem on the continuous extension to $\overline{D}$ of a proper holomorphic map $F: D\lrarw \Omega$ between domains with $\dim_{\mathbb{C}}(D) < \dim_{\mathbb{C}}(\Omega)$, and $(ii)$~a result that establishes the existence of bounded domains with "nice'' boundary geometry where H{\"o}lder regularity of the solutions to the complex Monge--Amp{\`e}re equation fails.