Embedded Three Dimensional CR Manifolds and the Non-Negativity of Paneitz Operators

TL;DR

This paper proves that small deformations of certain CR structures in complex domains preserve the non-negativity of the Paneitz operator, and shows that Webster curvature is positive on ellipsoids, extending previous results in CR geometry.

Contribution

It establishes the non-negativity of the CR Paneitz operator under small real-analytic deformations of embeddable CR structures with zero Webster torsion.

Findings

CR Paneitz operator remains non-negative under small deformations.

Webster curvature is positive on ellipsoids in C^2.

Results complement earlier work and provide partial converses.

Abstract

Let $\Omega$ be a bounded strictly pseudoconvex domain in $C^2$ with a smooth, connected and compact boundary M and having a CR structure $J_0$ induced from $C^2$. Assume this CR structure has zero Webster torsion. Then if we deform the CR structure through real-analytic dependence on the deformation parameter and such that each deformed structure along the deformation path is smooth and embeddable in $C^2$, we show that for small deformations of the CR structure $J$ from $J_0$, the associated CR Paneitz operator for $J$ is non-negative. We also show that the Webster curvature for any ellipsoid in $C^2$ is positive. The results in this paper complement and provide partial converses to our earlier paper, (to appear Duke Math. J.) arxiv: 1007.5020.