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THE ENVELOPE OF HOLOMORPHY
OF A TWO-MANIFOLD IN C²*

by L. R. Hunt and R. O. Wells, Jr.**

I. Introduction

There has been a large amount of study devoted to the problem of ana-
lytic continuation. If K is a subset of C", n > 1, then one wishes to know
if there is a larger set K', containing K, such that all holomorphic func-
tions on K can be extended to holomorphic functions on K', It has been
known for some time that the envelope of holomorphy of a domain in
Cⁿ is a Stein manifold spread over C". One can then consider the case where
K is a lower dimensional set in C", n > 1. For example, Hartogs proved
that every function holomorphic in a neighborhood of the boundary of
the unit ball in C" can be extended to a holomorphic function in the interior
of the ball. In [72] it was shown that if M is a real (n + l)-dimensional
differentiable submanifold embedded in C", one obtains local extendibility
over a manifold of real dimension n + 2, provided the so-called Levi form
does not vanish. Greenfield [5] has proven a similar result for an (n + ∕<)-
dimensional submanifold of C" with 1 ≤ к ≤ n— 1.

Denote by Φ( = C^,_cj the structure sheaf of germs of holomorphic func-
tions on C". If TC is a subset of C", let <9(K) be the algebra of sections of 0
over K (germs of holomorphic functions defined near K). If K ≠ K'
(where K and K' are connected sets), we say that K is extendible to K'
if the natural restriction map

r∙.β(K') → O(K)

is onto.

If K is a compact subset of C", then <9(7C) is, in a natural way, the induc-
tive limit of Frechet algebras of the form O(U) where U is an open set
containing K. As such it has a natural locally convex inductive limit topol-
ogy and becomes a topological algebra. The spectrum (or maximal ideal
space) of the topological algebra O(K) is defined to be the envelope of holo-

* This research was supported by NSF Grant No. GP-5951, at Rice University.

** Author who presented paper.

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