Higher dimensional worm domains

Abstract

We show how to construct a class of smooth bounded pseudoconvex domains whose boundary contains a given Stein manifold with strongly pseudoconvex boundary, having a prescribed codimension and D'Angelo class (a cohomological invariant measuring the "winding" of the boundary of the domain around the submanifold). Some open questions in the regularity theory of the $\overline{\partial }$-Neumann problem are discussed in the setting of these domains.

12 pages. Minor corrections/clarifications: higher regularity of the submanifold needed for Sard theorem to work; domains are smooth in Section 5