The Envelope of Holomorphy of a Two-manifold in C2*

I . Introduction There has been a large amount of study devoted to the problem of analytic continuation. If K is a subset of C" , 11 > l , then one wishes to know if there is a larger set K', containing K, such that all holomorphic functions 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 C" is a Stein manifold spread over C" . One can then consider the case where K is a lower dimensional set in C" , 11 > 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 hoIomorp11ic function in the interior of the ball. In [I21 it was shown that if M is a real (a + 1)-dimensional differentiable submanifold embedded in C", one obtains local extendibility over a manifold of real dime~lsion i . ~ f 2, provided the so-called Levi form does not vanish. Greenfield [5] has proven a similar result for an (n + k)dimensional submanifold of Cn with 1 5 k n 1. Denote by 9 (= 9,") the structure sheaf of germs of holomorphic functions on C". If K is a subset of C", let O(K) be the algebra of sections of O 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