Bochner transforms, perturbations and amoebae of holomorphic almost periodic mappings in tube domains

Local and Global Hartogs-Bochner Phenomenon in Tubes

A generalization of the Hartogs theorem is proved for a class of Tubes structures. We assume that the intervening commutative Lie algebra admits at least a number of globally solvable generators greater or equal to the structure codimension. We give necessary and sufficient conditions for the first cohomological group with compact support to be trivial. A such global result was previously obtained only when the supporting manifold was the Euclidean space. Further but canonical work is need to recover the Bochner extension theorem.

A brief proof of Bochner's tube theorem and a generalized tube

The aim of this note is firstly to give a new brief proof of classical Bochner's Tube Theorem (1938) by making use of K. Oka's Boundary Distance Theorem (1942), showing directly that two points of the envelope of holomorphy of a tube can be connected by a line segment. We then apply the same idea to show that if an unramified domain $\mathfrak{D}:=A_1+iA_2 \to \mathbf{C}^n$ with unramified real domains $A_j \to \mathbf{R}^n$ is pseudoconvex, then the both $A_j$ are univalent and convex (a generalization of Kajiwara's theorem). From the viewpoint of this result we discuss a generalization by M.…

Geometry of Hermitian symmetric spaces under the action of a maximal unipotent group

Let $\,G/K\,$ be a non-compact irreducible Hermitian symmetric space of rank $\,r\,$ and let $\,NAK\,$ be an Iwasawa decomposition of $\,G$. By the polydisc theorem, $\,AK/K\,$ can be regarded as the base of an $\,r$-dimensional tube domain holomorphically embedded in $\,G/K$. As every $\,N$-orbit in $\,G/K\,$ intersects $\,AK/K$ in a single point, there is a one-to-one correspondence between $\,N$-invariant domains in $\,G/K\,$ and tube domains in the product of $\,r\,$ copies of the upper half-plane in $\,\C$. In this setting we prove a generalization of Bochner's tube theorem. Namely, an $\…

关于刚性 $q$-多次调和函数与 $\nathbb{C}^n$ 中的 $q$-伪凸管状域

On rigid $q$-plurisubharmonic functions and $q$-pseudoconvex tube domains in $\mathbb{C}^n$

本文证明管状域上的 $q$-多次调和函数等价于基域上的实 $q$-凸函数，并由此刻画 $q$-伪凸管状域，应用于仿射线性映射图的补集和 Reinhardt 域。