The Lichnerowicz theorem on CR manifolds

For any compact strictly pseudoconvex CR manifold $M$ endowed with a contact form $\theta$ we obtain the Bochner type formula $\frac{1}{2} \Delta_b(|\nabla^H f|^2) = |\pi_H \nabla^2 f|^2 + (\nabla^H f)(\Delta_b f) + \rho (\nabla^H f , \nabla^H f) + 2 L f$ (involving the sublaplacian $\Delta_b$ and the pseudohermitian Ricci curvature $\rho$). When $M$ is compact of CR dimension $n$ and $\rho (X,X) + 2 A (X, JX) \geq k G_\theta (X,X)$, $X \in H(M)$, 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 $\theta$.