Włodzimierz Zwonek

The Lempert function of the symmetrized polydisc in higher dimensions is not a distance

We prove that the Lempert function of the symmetrized polydisc in dimension greater than two is not a distance.

DESCRIPTION OF ALL COMPLEX GEODESICS IN THE SYMMETRIZED BIDISC

In the paper we find effective formulas for the complex geodesics in the symmetrized bidisc.

Invariance of the pluricomplex green function under proper mappings with applications

In the paper we utilize the formula describing the behaviour of the pluricomplex Green function with many poles under proper holomorphic mappings to find the effective for-mulas for the function in the unit ball in with two poles.

Schwarz lemma for the tetrablock

We describe all complex geodesics in the tetrablock passing through the origin thus obtaining the form of all extremals in the Schwarz Lemma for the tetrablock. Some other extremals for the Lempert function and geodesics are also given. The paper may be seen as a continuation of the results from (2). The proofs rely on a necessary form of complex geodesics in general domains which is also proven in the paper.

Exhausting domains of the symmetrized bidisc

We show that the symmetrized bidisc may be exhausted by strongly linearly convex domains. It shows in particular the existence of a strongly linearly convex domain that cannot be exhausted by domains biholomorphic to convex ones.

An example concerning Bergman completeness

We construct a bounded plane domain which is Bergman complete but for which the Bergman kernel does not tend to infinity as the point approaches the boundary.

The kobayashi metric for non-convex complex ellipsoids

In the paper we give some necessary conditions for a mapping to be a k-geodesic in non-convex complex ellipsoids. Using these results we calculate explicitly the Kobavashi metric in the ellipsoids . where m< ½.

Proper holomorphic mappings of the spectral unit ball

We prove an Alexander type theorem for the spectral unit ball Ω n showing that there are no non-trivial proper holomorphic mappings in Ω n , n > 2.

On the product property for the Lempert function

In this article, we study the problem of the product property for the Lempert function with many poles and consider some properties of this function mostly for plane domains.

ON THE SUITA CONJECTURE FOR SOME CONVEX ELLIPSOIDS IN C 2

It was recently shown that for a convex domain Ω in and w ∈ Ω, the function , where is the Bergman kernel on the diagonal and the Kobayashi indicatrix, satisfies . While the lower bound is optimal, not much more is known about the upper bound. In general, it is quite difficult to compute even numerically, and the largest value of it obtained so far is 1.010182… . In this article, we present precise, although rather complicated, formulas for the ellipsoids Ω = {|z1|2m + |z2|2 < 1} (with m ≥ 1/2) and all w, as well as for Ω = {|z1| + |z2| < 1} and w on the diagonal. The Bergman kernel for those ellipsoids was already known; the main point is to compute the volume of the Kobayashi indicatrix. It turns out that in the second case, the function is not C3, 1.