Complex tangential flows and compactness of the $\bar{\partial }$- Neumann operator

Abstract

We provide geometric conditions on the set of boundary points of infinite type of a smooth bounded pseudoconvex domain in ${\mathbb{C}}^n$ which imply that the $\bar{\partial }$-Neumann operator is compact. These conditions are formulated in terms of certain short time flows in suitable complex tangential directions. It is noteworthy that compactness is \emph{not} established via the known potential theoretic sufficient conditions. Our results generalize to ${\mathbb{C}}^n$ the corresponding ${\mathbb{C}}^2$ results due to the second author.

9 pages