ON SMOOTH AND ANALYTIC DISKS IN C-2 WITH COMMON BOUNDARY

Abstract

We construct explicitly a real analytic embedded real two-dimensional disk in C-2 totally real except at exactly one elliptic complex tangent point, which shares the common boundary with an analytic disk in the same C-2, but does not contain this analytic disk in its envelope of holomorphy. The same proof further yields an explicit example of a holomorphic re-embedding of the standard two-sphere into C-2 in such a way that the new embedding shows some exceptional properties: It bounds a real three-dimensional Levi flat cell in C-2 foliated by analytic disks, which is not polynomially convex. In particular, this new embedding of the standard two-sphere cannot be a subset of any compact strongly pseudoconvex surface in C-2 or a subset of any strongly pseudoconvex graph in C-2 in the sense of Bedford and Gaveau.