On the CR analogue of Obata's theorem in a pseudohermitian 3-manifold
Let \((M, H^{10}, \theta)\) be a closed pseudo-Hermitian manifold of dimension \(2n+1\) that satisfies the following inequality for any vector field \(Z\) in the holomorphic distribution \(H^{10} \subset T^{\mathcal C} M\) \[ Ric_x(Z, Z) - \frac{n+1}{2} Tor_x(Z, Z) \geq k_0 d \vartheta(Z, \bar Z)\;,\tag{*} \] where \(Ric\) and \(Tor\) are the pseudo-Hermitian Ricci curvature and torsion, respectively, and \(k_0\) is some positive constant. Under some additional hypothesis, it has been proved that the first positive eigenvalue \(\mu_1\) of the sub-laplacian satisfies the inequality \(\mu_1 \geq \frac{n k_0}{n+1}\). Moreover, when \(M\) is the sphere \(S^{2n+1}\) with its standard pseudo-Hermitian structure, the equality holds. It is therefore conjectured the possibility of proving a CR analogue of Obata's theorem, namely that if \((M, H^{10}, \theta)\) satisfies (*) and \(\mu_1 = \frac{n k_0}{n+1}\), then \((M, H^{10}, \theta)\) is the standard pseudo-Hermitian sphere. In this paper, the authors prove an analogue of the Bochner formula on 3-dimensional CR manifolds, involving the CR Paneitz operators, and explore various of its consequences. In particular, for the pseudo-Hermitian closed 3-manifolds, they prove that the above conjecture is true when the torsion is vanishing and obtain a CR analogue of Li-Yau eigenvalue estimates for \(\mu_1\) when the torsion is non-vanishing and the CR Paneitz operator is nonnegative.
scientific article; zbMATH DE number 5598894
scientific article