Oberseminar: Nikolay Shcherbina (G.15.25)
Our main motivation in this talk is the study of existence of the Kobayashi and Bergman metrics for pseudoconvex domains of the form $\mathfrak{A} = \{(z, w) \in \mathbb{C}^n_z \times \mathbb{C}_w : v > F(z, u)\}$, where $w = u + iv$ and $F$ is a continuous function on $\mathbb{C}^n_z \times \mathbb{R}_u$. This classical type of domains is called for $\it{Model domains}$. In the case when $n = 1$ this question was completely answered in our paper in Math. Ann. few years ago, but the general case when $n>1$ remains completely open even for smooth rigid strictly pseudoconvex domains $\mathfrak{A}$. The main purpose of this presentation is to construct a smooth non-negative strictly plurisubhamonic function in $\mathbb{C}^2$ such that "its mean value" remains bounded. This shows that the initial question even for rigid domains is more complicated than one can expect.
