Some remarks about a theorem of Hartogs

Abstract

It is a theorem of Hartogs, [4, p. 231 and p. 239] and [2, Theorem 5, p. 660], that if a bounded domain GCCn, n_2, has a connected boundary, then any function defined on a connected neighborhood of the boundary of G has a unique extension to all of G. In this paper, we derive a similar theorem (Theorem 1) for a larger class of sets, the holomorphic (see Definition 2) introduced by the author in [5, Definition 4.1]. Also, using a theorem of Rossi [7, Theorem 6.6, p. 464], we can show that any compact set K in a connected normal Stein space S of dimension ? 2 such that S-K is connected, is a removable singularity (Theorem 3). Other new results concern various properties of deficiencies and envelopes of holomorphy. Perhaps the most interesting is Theorem 5: if D is an open subset of M, a Stein manifold, and X=M-D is a deficiency, then M is the envelope of holomorphy of D. Cartan and Schwartz have shown that H*'(M, 0) : 0, where M is a Stein manifold of dimension at least 2 and H*'(M, 0) is the first cohomology with compact supports of M with coefficients in 0, the sheaf of germs of functions [8, Theorem 4, p. 63]. Using this result, we shall give a direct proof of Theorem 3 for manifolds. See [8, p. 66] for another, closely related consequence of Cartan and Schwartz' result. The author would like to thank Robert C. Gunning for his help and encouragement in writing this article. Our notation is that of [3]. [6] develops the theory of envelopes of holomorphy for Riemann domains over Stein manifolds. There is a brief summary of needed theorems about such domains in [5]. DEFINITION 1. Let X and Y be point sets in M and N respectively, where M and N are complex manifolds. X and Y are holomorphically equivalent if they have open neighborhoods X' and Y' in M and N such that there is a biholomorphic map r: X'-* Y' which maps X onto Y. DEFINITION 2. A point set X is a complex manifold is a deficiency if it is holomorphically equivalent to a set Y such that