Hartogs companions and holomorphic extensions in arbitrary dimension

TL;DR

This paper establishes the existence and uniqueness of Hartogs companions for holomorphic maps in multiple complex variables, providing new extension theorems and principles applicable to various types of holomorphy in arbitrary dimensions.

Contribution

It introduces the concept of Hartogs companions for holomorphic maps in higher dimensions and proves their properties, extending classical results to vector-valued and Gâteaux holomorphic maps.

Findings

Existence and uniqueness of Hartogs companions for holomorphic maps

Extension of holomorphic maps based on domain connectivity

Boundary and maximum principles for holomorphic maps

Abstract

We show that every holomorphic map $f\in\mathcal{H}(\Omega\setminus K)$ ($K\subset\Omega\subset\mathbb{C}^n$, with $K$ compact, $\Omega$ open, and $n\ge2$), has a unique "\emph{Hartogs companion}" $\tilde f\in\mathcal{H}(\Omega)$ matching $f$ on an open subset $C_{K,\Omega}\subset\Omega\setminus K$. Furthermore, $\tilde f$ extends $f$, \emph{if and only if} $\mathbb{C}^n\setminus K$ is a connected set; this equivalence proves the converse implication from the Hartogs Kugelsatz. The existence of vector-valued Hartogs companions in any dimension yields a Hartogs-type extension theorem for G\^ateaux holomorphic maps $f\in\mathcal{H}_\mathrm{G}(\Omega\setminus K,Y)$ on finitely open sets in arbitrary complex vector spaces. The equivalence is very similar to that for $K\subset\Omega\subset\mathbb{C}^n$ and leads to a corresponding Hartogs Kugelsatz in arbitrary dimension and to extension theorems for five types of holomorphy (G\^ateaux, Mackey/Silva, hypoanalytic, Fr\'echet, locally bounded). We also show that the range $\tilde f(\Omega)$ of a vector-valued Hartogs companion cannot leave a domain of holomorphy containing $f(\Omega\setminus K)$. We establish a boundary principle for maps $f\in\mathcal{H}_\mathrm{G}(\Omega,Y)\cap\mathcal{C}(\bar\Omega,Y)$ on finitely bounded open sets. For $Y=\mathbb{C}$, the principle states that $f\big(\bar\Omega\big)=f(\partial\Omega)$ (hence $\sup_{x\in\Omega}|f(x)|=\sup_{x\in\partial\Omega}|f(x)|$). Several results require a new identity theorem, which yields a maximum norm principle and a "max-min" seminorm principle.