A Paneitz-type operator for CR pluriharmonic functions

TL;DR

This paper introduces a new fourth order CR invariant operator on pluriharmonic functions in three-dimensional CR manifolds, extending previous work and establishing a scalar invariant analogous to the Q-curvature in conformal geometry.

Contribution

It generalizes the Paneitz operator to the CR setting and defines a new scalar invariant for a class of contact forms with vanishing Hirachi-Q curvature.

Findings

Defines a CR invariant fourth order operator on pluriharmonic functions.

Establishes a new scalar invariant analogous to Q-curvature.

Explores relations to problems in three-dimensional CR geometry.

Abstract

We introduce a fourth order CR invariant operator on pluriharmonic functions on a three-dimensional CR manifold, generalizing to the abstract setting the operator discovered by Branson, Fontana and Morpurgo. For a distinguished class of contact forms, all of which have vanishing Hirachi-$Q$ curvature, these operators determine a new scalar invariant with properties analogous to the usual $Q$-curvature. We discuss how these are similar to the (conformal) Paneitz operator and $Q$-curvature of a four-manifold, and describe its relation to some problems for three-dimensional CR manifolds.