We wish to study those domains in $\mathbb{C}^n$, for $n\geq 2$, the so-called domains of holomorphy, which are in some sense the maximal domains of existence of the holomorphic functions defined on them. We shall demonstrate that this study is radically different from that of domains in $\mathbb{C}$ by discussing some examples of special types of domains in $\mathbb{C}^n$, $n\geq 2$, such that every function holomorphic on them extends to a strictly larger domain. This leads to Thullen’s construction of a domain (not necessarily in $\mathbb{C}^n$) spread over $\mathbb{C}^n$, the so-called envelope of holomorphy, which fulfills our criteria. With the help of this abstract approach we shall give a characterization of the domains of holomorphy in $\mathbb{C}^n$.The aforementioned characterization (holomorphic convexity) is very difficult to check. This calls for other (equivalent) criteria for a domain in $\mathbb{C}^n$, $n\geq 2$, to be a domain of holomorphy. We shall survey these criteria. We shall sketch those proofs of equivalence that rely on the first part of our survey: namely, on analytic continuation theorems. If a domain $\Omega\subset \mathbb{C}^n$, is not a domain of holomorphy, we would still like to explicitly describe a domain strictly larger than $\Omega$ to which all functions holomorphic on $\Omega$ continue analytically. One tool that is used most often in such constructions is called “Kontinuitaetssatz”. It has been invoked, without any clear statement, in many works on analytic continuation. The basic (unstated) principle that seems to be in use in these works appears to be a folk theorem. We shall provide a precise statement of this folk Kontinuitaetssatz and give a proof of it.

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Last updated: 29 Feb 2024