Observations regarding compactness in the $\overline{\partial}$-Neumann problem

TL;DR

This paper demonstrates that the compactness of the $ar{ ext{d}}$-Neumann operator is metric-independent, introduces an abstract obstruction concept, and characterizes this obstruction for specific domain classes in complex analysis.

Contribution

It provides a new proof of metric independence for compactness and defines an abstract obstruction, identifying it explicitly for convex and Hartogs domains.

Findings

Compactness is metric-independent.

Introduces an abstract obstruction to compactness.

Identifies the obstruction for convex and Hartogs domains.

Abstract

We show that compactness of the $\overline{\partial}$-Neumann operator is independent of the metric, and we give a new proof of this independence for subellipticity. We define an abstract obstruction to compactness, namely the common zero set of all the compactness multipliers, and we identify this subset of the boundary for convex domains in $\mathbb{C}^{n}$ and for complete Hartogs domains in $\mathbb{C}^{2}$.