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Higher dimensional worm domains

Simone Calamai, Gian Maria Dall'Ara

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.

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 -Neumann problem are discussed in the setting of these domains.

Paper Structure

This paper contains 5 sections, 3 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. A class of domains
  3. Two lemmas
  4. Proof of Theorem \ref{['thm:main']}
  5. Further properties and some open problems

Key Result

Theorem 1

Let $X$ be a Stein manifold. Let $Y \subset X$ be a precompact domain with strongly pseudoconvex boundary of class $C^k$, where $k\geq 2\dim_\mathbb{C} Y$ (possibly $\infty$). Let $\gamma\in H^1_{\mathrm{dR}}(Y,\mathbb{R})$. Let $d\geq 1$ be an integer. Then there exists a domain $W$ in $X\times \ma

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof