Nonnegativity of CR Paneitz operator and its application to the CR Obata's theorem (Q836229)

Recall that a CR structure on a smooth manifold of odd dimension \(2n+1\) is given by a contact structure together with a complex structure on the contact subbundle. Given such a geometry, a choice of contact form is referred to as a pseudo-Hermitian structure. Having chosen a contact form, CR-invariant differential operators which initially map between densities of different weights can be viewed as operators mapping functions to functions, so questions on eigenvalues of such operators make sense. In particular, this applies to the sub-Laplacian associated to the CR-structure. By work of \textit{A.~Greenleaf} [Commun. Partial Differ. Equations 10, 191--217 (1985; Zbl 0563.58034)], a restriction on the Webster-Ricci curvature and torsion implies a lower bound on the first eigenvalue of the sub-Laplacian. The paper under review studies the question of when this estimate is sharp. The authors conjecture an analog of the Obata theorem in this case, namely that this can only happen on the standard sphere. This paper only discusses the case \(n\geq 2\), the three-dimensional case is different in some respects and has been studied in the article [Math. Ann. 345, No. 1, 33--51 (2009; Zbl 1182.32012)] of the authors. The authors prove their conjecture in the case of vanishing pseudo-Hermitian torsion. Moreover, using Li-Yau's first eigenvalue estimate, they prove that the first eigenvalue of the sub-Laplacian is bounded below by a positive constant under a weaker curvature condition than the one used by Greenleaf. One key ingredient for the study of the eigenvalues is a new Bochner-formula for the sub-Laplacian which involves the CR Paneitz-operator, a CR-invariant operator with principal part the square of the sub-Laplacian. In contrast to the three-dimensional case, the CR Paneitz-operator is shown to be always non-negative in higher dimensions. The second important ingredient is a family of Riemannian metrics associated to a pseudo-Hermitian structure. The authors compare the Ricci curvatures of these metrics to the Webster-Ricci curvature and prove an eigenvalue estimate for the Laplacians of these metrics.