The name is absent



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.

51



More intriguing information

1. Surveying the welfare state: challenges, policy development and causes of resilience
2. The name is absent
3. Gerontocracy in Motion? – European Cross-Country Evidence on the Labor Market Consequences of Population Ageing
4. Connectionism, Analogicity and Mental Content
5. The name is absent
6. Industrial Cores and Peripheries in Brazil
7. Ein pragmatisierter Kalkul des naturlichen Schlieβens nebst Metatheorie
8. Prevalence of exclusive breastfeeding and its determinants in first 6 months of life: A prospective study
9. The Macroeconomic Determinants of Volatility in Precious Metals Markets
10. Indirect Effects of Pesticide Regulation and the Food Quality Protection Act