Semi periodic maps on complex manifolds

TL;DR

This paper proves that a holomorphic map from a tube domain to a complex manifold, which is semi periodic on a hyperplane, extends this semi periodicity to the entire domain under certain conditions.

Contribution

It establishes a new semi periodicity extension theorem for holomorphic maps from tube domains to complex manifolds.

Findings

Semi periodicity on a hyperplane implies semi periodicity on the whole domain.

Holomorphic maps with relatively compact images are considered.

The theorem applies to mappings with uniform continuity on compact subsets.

Abstract

In this letter we proved this theorem: \emph{if $F$ be a holomorphic mapping of $T_{\Omega}$ to a mapping manifold $X$ such that for every compact subset $K\subset \Omega$ the mapping $F$ is uniformly continues on $T_{K}$ and $F(T_{K})$ is a relatively compact subset of $X$. If the restriction of $F(z)$ to some hyperplane $\mathbb{R}^{m}+iy'$ is semi periodic, then $F(z)$ is an semi mapping of $T_{\Omega}$ to $X$.}