Embeddability for Three-Dimensional Cauchy-Riemann Manifolds and CR Yamabe Invariants

TL;DR

This paper establishes sharp eigenvalue bounds for the Kohn Laplacian on CR 3-manifolds, linking embeddability to positivity conditions of the CR Paneitz operator and CR Yamabe constant, with implications for stability and non-embedability.

Contribution

It derives a new Bochner formula for the Kohn Laplacian that omits pseudo-hermitian torsion and connects eigenvalue bounds to embeddability criteria.

Findings

Eigenvalues of the Kohn Laplacian are bounded below by a positive constant under certain conditions.

All compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable.

The embeddability is stable under the given conditions, and the CR Paneitz operator's sign indicates embeddability status.

Abstract

Let M^3 be a closed CR 3-manifold. In this paper we derive a Bochner formula for the Kohn Laplacian in which the pseudo-hermitian torsion plays no role. By means of this formula we show that the non-zero eigenvalues of the Kohn Laplacian are bounded below by a positive constant provided the CR Paneitz operator is non-negative and the Webster curvature is positive. Our lower bound for the non-zero eigenvalues is sharp and is attained on S^3. A consequence of our lower bound is that all compact CR 3-manifolds with non-negative CR Paneitz operator and positive CR Yamabe constant are embeddable. Non-negativity of the CR Paneitz operator and positivity of the CR Yamabe constant are both CR invariant conditions and do not depend on conformal changes of the contact form. In addition we show that under the sufficient conditions above for embeddability, the embedding is stable in the sense of Burns and Epstein. We also show that for the Rossi example for non-embedability, the CR Paneitz operator is negative. For CR structures close to the standard structure on $S^3$ we show the CR Paneitz operator is positive on the space of pluriharmonic functions with respect to the standard CR structure on $S^3$.