The Lichnerowicz theorem on CR manifolds

Abstract

For any compact strictly pseudoconvex CR manifold $M$ endowed with a contact form $\theta$ we obtain the Bochner type formula $\frac{1}{2}\Delta_{b}(|\nabla^{H}f|^{2}) = |\pi_{H}\nabla^{2}f|^{2} + (\nabla ^{H}f)(\Delta_{b}f) + \rho(\nabla^{H}f, \nabla^{H}f) + 2Lf$ (involving the sublaplacian $\Delta_{b}$ and the pseudohermitian Ricci curvature $\rho$). When $M$ is compact of CR dimension $n$ and $\rho(X,X) + 2A(X,JX) \geq kG_{\theta}(X,X), X \in H(M)$, we derive the estimate $-\lambda \geq 2nk/(2n - 1)$ on each nonzero eigenvalue $\lambda$ of $\Delta_{b}$ satisfying $\mathrm{Eigen}(\Delta_{b}; \lambda) \cap \mathrm{Ker}(T) \neq (0)$ where $T$ is the characteristic direction of $d\theta$.