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LOCAL HOLOMORPHIC CONVEXITY

OF A TWO-MANIFOLD IN C²*

by Michael Freeman

1. Introduction

A compact set M in C" is holomorphically convex [3, Prop. 2.4] if the
natural injection of it into the maximal ideal space [2, p. 2] of A(M) is a
homeomorphism. A(M) is the Banach algebra obtained as the uniform
closure on M of the continuous functions on M which admit holomorphic
extensions to some neighborhood of M. Polynomially convex [2, p. 66]
sets are holomorphically convex, as are sets M which are convex with
respect to the holomorphic functions on a fixed domain of holomorphy
U containing M. So are the rationally convex sets defined below.

Aside from simple sufficient conditions such as ordinary geometric
convexity or containment in R" (as the “real part” of C"), very few criteria
for determining holomorphic convexity are known. Yet this property is
extremely important as a condition for the solution of many kinds of
function-theoretic approximation problems [2, Ch. 111].

This paper discusses the problem of finding differential conditions for
holomorphic convexity. The idea is to find conditions on the tangent space,
curvature, etc., of a real submanifold of C" near one of its points p to dis-
cern which small compact subsets of it near p are holomorphically convex.
This has been done for one-dimensional manifolds by Stolzenberg [<⅛]
and others who show that a compact subset of a smooth simple arc in C" is
polynomially convex, and in fact that any continuous function on such
a set can be uniformly approximated by polynomials.

The discussion here is restricted to two-manifolds in C², although some
of the results obtained have ready generalizations to a two-manifold in C".
Something is already known in this situation, mainly through the efforts of
E. Bishop [7] (whose results also apply to many cases of higher dimension).
By attaching the boundaries of small analytic discs in C² to a two-manifold
in a neighborhood of an “elliptic” point, Bishop showed that no neigh-

* This work was supported by NSF Grant GP-8997.

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