In this talk, we will consider 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 my 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 $\partial\Omega$ must bend uniformly into $\Omega$.

Even for pseudoconvex domains $\Omega\subset \mathbb{C}^n$, $n\geq 3$, 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 will discuss a sufficient condition for completeness for relatively compact domains in several large classes of manifolds. If time permits, we will discuss sufficient conditions for Cauchy-completeness of Kobayashi hyperbolic domains (not necessarily relatively compact) in arbitrary complex manifolds.

Next, we will discuss extending 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 will see some consequences of visibility and several sufficient conditions for visibility. I will also establish a Wolff–Denjoy-type theorem in a very general setting as an application. If time permits, we will explore some connections between visibility and Gromov hyperbolicity for Kobayashi hyperbolic domains in the above setting.

We also have a short discussion 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 will answer in the negative by seeing 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 shall explore some connections between Kobayashi geometry and the Dirichlet problem for the complex Monge–Ampere equation. Among the results given by these connections: (i) a theorem on the continuous extension to $\overline{D}$ of a proper holomorphic map $F: D\to \Omega$ between domains with $\dim(D) < \dim(\Omega)$, and (ii) a result that establishes the existence of bounded domains with “nice” boundary geometry where Hoelder regularity of the solutions to the complex Monge–Ampere equation fails.