Quantitative localization and comparison of invariant distances of domains in $mathbb C^n$

We obtain explicit bounds on the difference between local and global Kobayashi distances in a domain of $mathbb C^n$ as the points go toward a boundary point with appropriate geometric properties. We use this for the global comparison of various invariant distances. We provide some sharp estimates in dimension $1$.

Visibility of Kobayashi geodesics in convex domains and related properties

Let $Dsubset mathbb C^n$ be a bounded domain. A pair of distinct boundary points ${p,q}$ of $D$ has the visibility property provided there exist a compact subset $K_{p,q}subset D$ and open neighborhoods $U_p$ of $p$ and $U_q$ of $q$, such that the real geodesics for the Kobayashi metric of $D$ which join points in $U_p$ and $U_q$ intersect $K_{p,q}$. Every Gromov hyperbolic convex domain enjoys the visibility property for any couple of boundary points. The Goldilocks domains introduced by Bharali and Zimmer and the log-type domains of Liu and Wang also enjoy the visibility property. In this paper we relate the growth of the Kobayashi distance near the boundary with visibility and provide new families of convex domains where that property holds. We use the same methods to provide refinements of localization results for the Kobayashi distance, and give a localized sufficient condition for visibility. We also exploit visibility to study the boundary behavior of biholomorphic maps.

Growth of Sibony metric and Bergman kernel for domains with low regularity

It is shown that even a weak multidimensional Suita conjecture fails for any bounded non-pseudoconvex domain with $mathcal C^1$ boundary: the product of the Bergman kernel by the volume of the indicatrix of the Azukawa metric is not bounded below. This is obtained by finding a direction along which the Sibony metric tends to infinity as the base point tends to the boundary. The analogous statement fails for a Lipschitz boundary. For a general $mathcal C^1$ boundary, we give estimates for the Sibony metric in terms of some directional distance functions. For bounded pseudoconvex domains, the Blocki-Zwonek Suita-type theorem implies growth to infinity of the Bergman kernel; the fact that the Bergman kernel grows as the square of the reciprocal of the distance to the boundary, proved by S. Fu in the $mathcal C^2$ case, is extended to bounded pseudoconvex domains with Lipschitz boundaries.

An analogue of the squeezing function for projective maps

In the spirit of Kobayashis applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankels work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong-Rosay theorem.

Comparison of the Bergman kernel and the Caratheodory--Eisenman volume

It is proved that for any domain in $mathbb C^n$ the Caratheodory--Eisenman volume is comparable with the volume of the indicatrix of the Caratheodory metric up to small/large constants depending only on $n.$ Then the multidimensional Suita conjecture theorem of Blocki and Zwonek implies a comparable relationship between these volumes and the Bergman kernel.