Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Hartogs companions and holomorphic extensions in arbitrary dimension   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.

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The categories of this paper are: math-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: ao-sci
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The categories of this paper are: q-alg
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The categories of this paper are: hep-ex
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The categories of this paper are: bayes-an
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The categories of this paper are: q-fin
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The categories of this paper are: hep-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: q-bio
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The categories of this paper are: nlin
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The categories of this paper are: alg-geom
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The categories of this paper are: stat
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The categories of this paper are: cond-mat
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The categories of this paper are: atom-ph
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The categories of this paper are: funct-an