On the Hans Lewy extension phenomenon in higher codimension

Abstract:

In this work the authors extend the results of their paper entitled Families of analytic discs in ${{\mathbf {C}}^n}$ with boundaries on a prescribed $CR$ submanifold. It is proven that a nongeneric CR manifold $M$ whose Lewy form is not identically zero can be extended to a manifold $\tilde M$ of one higher dimension, which is foliated by analytic discs. Moreover this result is used to prove that a sufficiently smooth CR function $f$ on $M$ extends to a function $\tilde f$ which is CR on $\tilde M$.