Higher dimensional worm domains

TL;DR

This paper constructs smooth bounded pseudoconvex domains with boundaries containing a specified Stein manifold, exploring implications for the regularity of the $ar{ ext{d}}$-Neumann problem and related invariants.

Contribution

It introduces a method to build pseudoconvex domains with prescribed boundary submanifolds and invariants, advancing understanding of boundary geometry and regularity issues.

Findings

Constructed domains with specified Stein submanifolds

Analyzed boundary invariants like D'Angelo class

Discussed open problems in $ar{ ext{d}}$-Neumann regularity

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.