Analytic Continuation , Envelopes of Holomorphy , and Projective and Direct Limit Spaces

For a Riemann domain Í2, a connected complex manifold where n (n = dimension) globally defined functions form a local system of coordinates at every point, and an arbitrary holomorphic function / in Í2, the "Riemann surface" Slf, a maximal holomorphic extension Riemann domain for /, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for n, a maximal holomorphic extension Riemann domain for all holomorphic functions in il, which is shown to be the projective limit space of the "Riemann surfaces" il*. Then it is shown that the generalized notion of envelope of holomorphy of an arbitrary subset of a Riemann domain can also be characterized in a natural way as the projective limit space of a family of "Riemann surfaces". Introduction. Unlike the case in C1, it was observed by Hartogs [10] that there are special domains in C" such that all holomorphic functions in such a special domain can be holomorphically extended to a larger domain. Thullen in [16] introduced the notion of "Regularitatshulle" of a "bereich" (domain) and refuted a conjecture of Aimer by giving an example of a Hartogs domain in C2 such that every holomorphic function could be holomorphically continued, but not always in a univalent manner, to a larger domain in C2. Later that year Cartan and Thullen in [6] defined the "Regularitatshulle" of a "domain" to be the "durchschnitt" of the family of "domains of holomorphy" of functions holomorphic in the given "domain". The idea of "durchschnitt" was generalized in [1] (see also [3, p. 179]) and is similar to the notion of a projective (inverse) limit space. The "durchschnitt" of a family of sets is a subset of the projective limit of this family, but its topology is stronger than the induced topology on it from the projective limit space. Given a family of domains in the plane with a point in common, the "durchschnitt" of this family is the interior of the intersection, while the projective limit space is homeomorphic to the intersection. Received by the editors November 30, 1973 and, in revised form, April 19, 1974. AMS (MOS) subject classifications (1970). Primary 32D10, 32D0S; Secondary 54B25, 32E99, 52A20.