A Two-Sided H. Lewy Extension Phenomenon

Abstract

Let M be a ${C^\infty }$ real $(n + k)$-dimensional CR-manifold in ${{\mathbf {C}}^n}$. We are interested in finding conditions on M near a point $p \in M$ which imply that all CR-function on M extend to holomorphic functions in some fixed neighborhood of p in ${{\mathbf {C}}^n}$. Of course if M is a real hypersurface, it is known that M having eigenvalues of opposite sign in its Levi form at p will give us such an extension result. If we view the Levi form at a point on a general CR-manifold M as a quadratic map from the holomorphic tangent space to the normal space of the real tangent space in ${{\mathbf {C}}^n}$, and if this map is surjective, then we prove our CR-functions extend to holomorphic functions in an open neighborhood of the point. We also show that if the real codimension of M in ${{\mathbf {C}}^n}$ is 2, and if the Levi form is totally indefinite, then the Levi form is onto ${{\mathbf {R}}^2}$ as a quadratic map, and hence we have our extension theory.