On smooth and analytic disks in 𝐂² with common boundary

Abstract

We construct explicitly a real analytic embedded real two-dimensional disk in C 2 {{\mathbf {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 {{\mathbf {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 {{\mathbf {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 {{\mathbf {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 {{\mathbf {C}}^2} or a subset of any strongly pseudoconvex graph in C 2 {{\mathbf {C}}^2} in the sense of Bedford and Gaveau.