Effective formulas for invariant functions -- case of elementary Reinhardt domains

We describe all possibilities of existence of non-elementary proper holomorphic maps between non-hyperbolic Reinhardt domains in $\mathbb C^2$ and the corresponding pairs of domains.

We study the Bergman kernel of certain domains in $\mathbb{C}^n$, called elementary Reinhardt domains, generalizing the classical Hartogs triangle. For some elementary Reinhardt domains, we explicitly compute the kernel, which is a rational function of the coordinates. For some other such domains, we show that the kernel is not a rational function. For a general elementary Reinhardt domain, we obtain a representation of the kernel as an infinite series.

We characterize pairs of bounded Reinhardt domains in $\CC^2$ between which there exists a proper holomorphic map and find all proper maps that are not elementary algebraic.

We present an effective formula for the Sibony function for all Reinhardt domains.

The Riemann Theorem states, that for any nontrivial connected and simply connected domain on the Riemann sphere there exists some its conformal bijection to the exterior of the unit disk. In this paper we find an explicit form of this map for a broad class of domains with analytic boundaries.

We present an elementary proof of the cross theorem in the case of Reinhardt domains. The results illustrates the well-known interrelations between the holomorphic geometry of a Reinhardt domain and the convex geometry of its logarithmic image.

A characterization of non-hyperbolic pseudoconvex Reinhardt domains in $\mathbb C^2$ for which the answer to the Serre problem is positive is given.

In this paper we obtain the closed forms of some hypergeometric functions. As an application, we obtain the explicit forms of the Bergman kernel functions for Reinhardt domains $\{|z_3|^{\lambda} < |z_1|^{2p} + |z_2|^2, \quad |z_1|^{2p} + |z_2|^2 < |z_1|^{p} \}$ and $\{|z_4|^{\lambda} < (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2, \quad (|z_1|^2 + |z_2|^2)^{p} + |z_3|^2 < (|z_1|^2 + |z_2|^2 )^{p/2} \}$.

The Serre problem for a class of hyperbolic pseudoconvex Reinhardt domains in $\Bbb C^2$ as fibers is solved.

We study the possible dimensions that the groups of holomorphic automorphisms of hyperbolic Reinhardt domains can have. We are particularly interested in the problem of characterizing Reinhardt domains with automorphism group of prescribed dimension.