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

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".