Geometric sufficient conditions for compactness of the complex Green operator

TL;DR

This paper extends geometric conditions ensuring the compactness of the complex Green operator on certain CR-submanifolds, using flow-based criteria related to the $ar{ ext{d}}$-Neumann problem.

Contribution

It introduces new geometric sufficient conditions for the compactness of the complex Green operator on pseudoconvex CR-submanifolds, generalizing previous results.

Findings

Established compactness estimates for $ar{ ext{d}}_M$ on CR-submanifolds.

Formulated conditions in terms of short time flows in complex tangential directions.

Connected geometric flow conditions to the compactness of the Green operator.

Abstract

We establish compactness estimates for $\bar{\partial}_{M}$ on a compact pseudoconvex CR-submanifold $M$ of $\mathbb{C}^{n}$ of hypersurface type that satisfies the (analogue of the) geometric sufficient conditions for compactness of the $\bar{\partial}$-Neumann operator given by the authors earlier. These conditions are formulated in terms of certain short time flows in complex tangential directions.