We study questions broadly related to the Kobayashi (pseudo)distance and (pseudo)metric on domains in $\mathbb{C}^n$. Specifically, we study the following subjects:

**Estimates for holomorphic images of subsets in convex domains:**
Consider the following problem: given domains $\Omega_1\varsubsetneq
\mathbb{C}^n$ and $\Omega_2\varsubsetneq \mathbb{C}^m$, and points $a\in
\Omega_1$ and $b \in \Omega_2$, find an explicit lower bound for the
distance of $f(\Omega_1(r))$ from the complement of $\Omega_2$ in terms
of $r$, where $f:\Omega_1\to \Omega_2$ is a holomorphic map such that
$f(a)=b$, and $\Omega_1(r)$ is the set of all points in $\Omega_1$ that
are at a distance of at least $r$ from the complement of $\Omega_1$.
This is motivated by the classical Schwarz lemma (i.e., $\Omega_1 =
\Omega_2$ being the unit disk) which gives a sharp lower bound of the
latter form. We extend this to the case where $\Omega_1$ and $\Omega_2$
are convex domains. In doing so, we make crucial use of the Kobayashi
pseudodistance.

**Upper bounds for the Kobayashi metric:**
We provide new upper bounds for the Kobayashi metric on bounded
convex domains in $\mathbb{C}^n$. This bears relation to Graham’s
well-known big-constant/small-constant bounds from above and below on
convex domains. Graham’s upper bounds are frequently not sharp. Our
estimates improve these bounds.

**The continuous extension of Kobayashi isometries:**
We provide a new result in this direction that is based on the
properties of convex domains viewed as distance spaces (equipped
with the Kobayashi distance). Specifically, we sharpen certain
techniques introduced recently by A. Zimmer and extend a result of
his to a wider class of convex domains having lower boundary
regularity. In particular, all complex geodesics into any such
convex domain are shown to extend continuously to the unit circle.

**A weak notion of negative curvature for the Kobayashi distance on domains in $\mathbb{C}^n$:**
We introduce and study a property that we call “visibility with
respect to the Kobayashi distance”, which is an analogue of the
notion of uniform visibility in CAT(0) spaces. It abstracts an
important and characteristic property of Gromov hyperbolic spaces.
We call domains satisfying this newly-introduced property
“visibility domains”. Bharali–Zimmer recently introduced a class
of domains called Goldilocks domains, which are visibility domains,
and proved for Goldilocks domains a wide range of properties. We show
that visibility domains form a proper superclass of the Goldilocks
domains. We do so by constructing a family of domains that are
visibility domains but not Goldilocks domains. We also show that
visibility domains enjoy many of the properties shown to hold for
Goldilocks domains.

**Wolff–Denjoy-type theorems for visibility domains:**
To emphasise the point that many of the results shown to hold for
Goldilocks domains can actually be extended to visibility domains, we
prove two Wolff–Denjoy-type theorems for taut visibility domains, with
one of them being a generalization of a similar result for Goldilocks
domains. We also provide a corollary to one of these results to
demonstrate the sheer diversity of domains to which the Wolff–Denjoy
phenomenon extends.

- All seminars.
- Seminars for 2019

Last updated: 23 Jun 2024