On Bergman completeness of pseudoconvex Reinhardt domains

We give a precise description of Bergman complete bounded pseudoconvex Reinhardt domains.Comment: 13 page

Proper holomorphic mappings of the spectral unit ball

We prove an Alexander type theorem for the spectral unit ball $\Omega_n$ showing that there are no non-trivial proper holomorphic mappings in $\Omega_n$, $n\geq 2$.Comment: 6 page

Asymptotic behaviour of the sectional curvature of the Bergman metric for annuli

We extend and simplify results of \cite{Din~2009} where the asymptotic behavior of the holomorphic sectional curvature of the Bergman metric in annuli is studied. Similarly as in \cite{Din~2009} the description enables us to construct an infinitely connected planar domain (in our paper it is a Zalcman type domain) for which the supremum of the holomorphic sectional curvature is two whereas its infimum is equal to $-\infty$.Comment: 8 page

The Bergman kernel of the symmetrized polydisc in higher dimensions has zeros

We prove that the Bergman kernel of the symmetrized polydisc in dimension greater than two has zeros.Comment: ESI preprint 174

Geometric properties of the tetrablock

In this short note we show that the tetrablock is i \C-convex domain. In the proof of this fact a new class of (\C-convex) domains is studied. The domains are natural caniddates to study on them the behavior of holomorphically invariant functions

One dimensional estimates for the Bergman kernel and logarithmic capacity

Carleson showed that the Bergman space for a domain on the plane is trivial if and only if its complement is polar. Here we give a quantitative version of this result. One is the Suita conjecture, established by the first-named author in 2012, the other is an upper bound for the Bergman kernel in terms of logarithmic capacity. We give some other estimates for those quantities as well. We also show that the volume of sublevel sets for the Green function is not convex for all regular non simply connected domains, generalizing a recent example of Forn\ae ss.Comment: 8 page