Let be a closed Cauchy–Riemann (CR) 3-manifold. In this article, we derive a Bochner formula for the Kohn Laplacian in which the pseudo-Hermitian torsion does not play any role. By means of this formula we show that the nonzero eigenvalues of the Kohn Laplacian have a positive lower bound, provided that the CR Paneitz operator is nonnegative and the Webster curvature is positive. This means that is embeddable when the CR Yamabe constant is positive and the CR Paneitz operator is nonnegative. Our lower bound estimate is sharp. In addition, we show that the embedding is stable in the sense of Burns and Epstein.
Sagun Chanillo. Hung-Lin Chiu. Paul Yang. "Embeddability for 3-dimensional Cauchy–Riemann manifolds and CR Yamabe invariants." Duke Math. J. 161 (15) 2909 - 2921, 1 December 2012. https://doi.org/10.1215/00127094-1902154