We wish to study those domains in Cn,for n ≥ 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 demonstrate that this study is radically different from that of domains in C by discussing some examples of special types of domains in Cn , n ≥2, such that every function holomorphic on them extends to strictly larger domains. Given a domain in Cn , n ≥ 2, we wish to construct the maximal domain of existence for the holomorphic functions defined on the given domain. This leads to Thullen’s construction of a domain (not necessarily in Cn)spread overCn, the so-called envelope of holomorphy, which fulfills our criteria. Unfortunately this turns out to beavery abstract space, far from giving us sense in general howa domain sitting in Cn can be constructed which is strictly larger than the given domain and such that all the holomorphic functions defined on the given domain extend to it. But with the help of this abstract approach we can give a characterization of the domains of holomorphyin Cn , n ≥ 2. The aforementioned characterization is as follows: adomain in Cn is a domain of holomorphy if and only if it is holomorphically convex. However, holomorphic convexity is a very difficult property to check. This calls for other (equivalent) criteria for a domain in Cn , n ≥ 2, to be a domain of holomorphy. We survey these criteria. The proof of the equivalence of several of these criteria are very technical – requiring methods coming from partial differential equations. We provide those proofs that rely on the first part of our survey: namely, on analytic continuation theorems. If a domain Ω Cn , n ≥ 2, is not a domain of holomorphy, we would still like to explicitly describe a domain strictly larger than Ω to which all functions holomorphic on Ω continue analytically. Aspects of Thullen’s approach are also useful in the quest to construct an explicit strictly larger domain in Cn with the property stated above. The tool used most often in such constructions s called “Kontinuitatssatz”. It has been invoked, without a 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 provide a precise statement of this folk Kontinuitatssatz and give a proof of it.

It is well known that there exist domains Ω in Cn,n ≥ 2, such that all holomorphic functions in Ω continue analytically beyond the boundary. We wish to study this remarkable phenomenon. The first chapter seeks to motivate this theme by offering some well-known extension results on domains in Cn having many symmetries. One important result, in this regard, is Hartogs’ theorem on the extension of functions holomorphic in a certain neighbourhood of (D x {0} U (∂D x D), D being the open unit disc in C. To understand the nature of analytic continuation in greater detail, in Chapter 2, we make rigorous the notions of ‘extensions’ and ‘envelopes of holomorphy’ of a domain. For this, we use methods similar to those used in univariate complex analysis to construct the natural domains of definitions of functions like the logarithm. Further, to comprehend the geometry of these abstractly-defined extensions, we again try to deal with some explicit domains in Cn; but this time we allow our domains to have fewer symmetries. The subject of Chapter 3 is a folk result generalizing Hartogs’ theorem to the extension of functions holomorphic in a neighbourhood of S U (∂D x D), where S is the graph of a D-valued function Φ, continuous in D and holomorphic in D. This problem can be posed in higher dimensions and we give its proof in this generality. In Chapter 4, we study Chirka and Rosay’s proof of Chirka’s generalization (in C2) of the above-mentioned result. Here, Φ is merely a continuous function from D to itself. Chapter 5 — a departure from our theme of Hartogs-Chirka type of configurations — is a summary of the key ideas behind a ‘non-standard’ proof of the so-called Hartogs phenomenon (i.e., holomorphic functions in any connected neighbourhood of the boundary of a domain Ω Cn , n ≥ 2, extend to the whole of Ω). This proof, given by Merker and Porten, uses tools from Morse theory to tame the boundary geometry of Ω, hence making it possible to use analytic discs to achieve analytic continuation locally. We return to Chirka’s extension theorem, but this time in higher dimensions, in Chapter 6. We see one possible generalization (by Bharali) of this result involving Φ where is a subclass of C (D; Dn), n ≥ 2. Finally, in Chapter 7, we consider Hartogs-Chirka type configurations involving graphs of multifunctions given by “Weierstrass pseudopolynomials”. We will consider pseudopolynomials with coefficients belonging to two different subclasses of C(D; C), and show that functions holomorphic around these new configurations extend holomorphically to D2 .