The diameter estimate and its application to CR Obata's theorem on closed pseudohermitian \((2n+1)\)-manifolds (Q2839936)

On a pseudo-Hermitian manifold \((M,J,\theta)\) of dimension \(2n+1\), one can naturally define a sub-Laplacian. The study of the first positive eigenvalue of this sub-Laplacian has a longer history. In particular, assuming a certain inequality involving the Ricci curvature and the torsion, as well as non-negativity of the CR-Paneitz operator if \(n=1\), one obtains a universal lower bound on this first eigenvalue. This bound is attained in the case of the standard pseudo-Hermitian structure on the unit sphere in \(\mathbb C^{n+1}\) (for which the torsion vanishes). The pseudo-Hermitian analog of Obata's theorem would be that this is the only example in which the bound is attained. This has been proved in the case of vanishing torsion by the first author of this article and \textit{H.-L. Chiu} [J. Geom. Anal. 19, No. 2, 261--287 (2009; Zbl 1205.32026); Math. Ann. 345, No. 1, 33--51 (2009; Zbl 1182.32012)], but the general case is open.NEWLINENEWLINEIn this article, the CR analog of Obata's theorem is proved in the presence of torsion, but imposing additional assumptions on covariant derivatives of the torsion and a further technical assumption in the case \(n=1\). The main ingredient in the proof are diameter estimates for the Webster-metrics associated to the pseudo-Hermitian structure, which are used to identify the unit sphere.