The CR Paneitz Operator and the Stability of CR Pluriharmonic Functions

TL;DR

This paper establishes conditions under which the CR Paneitz operator on embedded three-dimensional CR manifolds is nonnegative with a kernel of CR pluriharmonic functions, using curvature positivity and stability under deformation.

Contribution

It provides new criteria linking Webster scalar curvature positivity and stability to the nonnegativity and kernel characterization of the CR Paneitz operator.

Findings

CR Paneitz operator is nonnegative under certain conditions.

Kernel of the operator consists solely of CR pluriharmonic functions.

Real ellipsoids in ^2 satisfy these conditions.

Abstract

We give a condition which ensures that the Paneitz operator of an embedded three-dimensional CR manifold is nonnegative and has kernel consisting only of the CR pluriharmonic functions. Our condition requires uniform positivity of the Webster scalar curvature and the stability of the CR pluriharmonic functions for a real analytic deformation. As an application, we show that the real ellipsoids in $\mathbb{C}^2$ are such that the CR Paneitz operator is nonnegative with kernel consisting only of the CR pluriharmonic functions.