Some Remarks on the Total CR $Q$ and $Q^\prime$-Curvatures

TL;DR

This paper proves the vanishing of total CR Q-curvature on compact strictly pseudoconvex CR manifolds, establishes the self-adjointness of the P'-operator, and confirms the invariance of total Q'-curvature without Stein boundary assumptions.

Contribution

It demonstrates the vanishing of total CR Q-curvature and the invariance of total Q'-curvature in broader settings, extending previous results.

Findings

Total CR Q-curvature vanishes on compact strictly pseudoconvex CR manifolds.

The P'-operator is formally self-adjoint.

Total Q'-curvature is CR invariant for pseudo-Einstein manifolds without Stein boundary assumptions.

Abstract

We prove that the total CR $Q$-curvature vanishes for any compact strictly pseudoconvex CR manifold. We also prove the formal self-adjointness of the $P^\prime$-operator and the CR invariance of the total $Q^\prime$-curvature for any pseudo-Einstein manifold without the assumption that it bounds a Stein manifold.