arXiv subjects
Publications and source records attributed to Pawel Zapalowski.
The explicit form of proper holomorphic mappings between complex ellipsoids is given. Using this description, we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, the automorphism group of such domains is found.
Answering all questions---concerning proper holomorphic mappings between generalized Hartogs triangles---posed by Jarnicki and Plfug (First steps in several complex variables: Reinhardt domains, 2008) we characterize the existence of proper holomorphic mappings between generalized Hartogs triangles and give their explicit form. In particular, we completely describe the group of holomorphic automorphisms of such domains and establish rigidity of proper holomorphic self-mappings on them.
In the paper we study the geometric properties of a large family of domains, called the generalized tetrablocks, related to the $μ$-synthesis, containing both the family of the symmetrized polydiscs and the family of the $μ_{1,n}$-quotients $\mathbb E_n$, $n\geq2$, introduced recently by G. Bharali. It is proved that the generalized tetrablock cannot be exhausted by domains biholomorphic to convex ones. Moreover, it is shown that the Carathéodory distance and the Lempert function are not equal on a large subfamily of the generalized tetrablocks, containing i.a. $\mathbb E_n$, $n\geq4$. We also derive a number of geometric properties of the generalized tetrablocks as well as the $μ_{1,n}$-quotients. As a by-product, we get that the pentablock, another domain related to the $μ$-synthesis problem introduced recently by J. Agler, Z. A. Lykova, and N. J. Young, cannot be exhausted by domains biholomorphic to convex ones.
We characterize the existence of proper holomorphic mappings in the special class of bounded $(1,2,...,n)$-balanced domains in $\mathbb{C}^n$, called the symmetrized ellipsoids. Using this result we conclude that there are no non-trivial proper holomorphic self-mappings in the class of symmetrized ellipsoids. We also describe the automorphism groupof these domains.