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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Analytic continuation, envelopes of holomorphy, and projective and direct limit spaces


Author: Robert Carmignani
Journal: Trans. Amer. Math. Soc. 209 (1975), 237-258
MSC: Primary 32D10
DOI: https://doi.org/10.1090/S0002-9947-1975-0385165-2
MathSciNet review: 0385165
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Abstract: For a Riemann domain $\Omega$, 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 $f$ in $\Omega$, the “Riemann surface” ${\Omega _f}$, a maximal holomorphic extension Riemann domain for $f$, is formed from the direct limit of a sequence of Riemann domains. Projective limits are used to construct an envelope of holomorphy for $\Omega$, a maximal holomorphic extension Riemann domain for all holomorphic functions in $\Omega$, which is shown to be the projective limit space of the “Riemann surfaces” ${\Omega _f}$. 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".


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Keywords: Riemann domain, direct limit space, domain of holomorphy, projective limit, projective limit spaces, envelope of holomorphy, Stein manifold, holomorphically convex sets, convex hull
Article copyright: © Copyright 1975 American Mathematical Society