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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}" $\tildef\in\mathcal{H}(\Omega)$ 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; thisequivalence proves the converse implication from the Hartogs Kugelsatz. Theexistence of vector-valued Hartogs companions in any dimension yields aHartogs-type extension theorem for G\^ateaux holomorphic maps$f\in\mathcal{H}_\mathrm{G}(\Omega\setminus K,Y)$ on finitely open sets inarbitrary complex vector spaces. The equivalence is very similar to that for$K\subset\Omega\subset\mathbb{C}^n$ and leads to a corresponding HartogsKugelsatz in arbitrary dimension and to extension theorems for five types ofholomorphy (G\^ateaux, Mackey/Silva, hypoanalytic, Fr\'echet, locally bounded).We also show that the range $\tilde f(\Omega)$ of a vector-valued Hartogscompanion cannot leave a domain of holomorphy containing $f(\Omega\setminusK)$. We establish a boundary principle for maps$f\in\mathcal{H}_\mathrm{G}(\Omega,Y)\cap\mathcal{C}(\bar\Omega,Y)$ on finitelybounded 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 resultsrequire a new identity theorem, which yields a maximum norm principle and a"max-min" seminorm principle.

Hartogs companions and holomorphic extensions in arbitrary dimension