The Lichnerowicz theorem on CR manifolds

For any compact strictly pseudoconvex CR manifold M endowed with a contact form θ we obtain the Bochner type formula 1 2 ∆b(|∇f |2) = |πH∇f |2 + (∇f)(∆bf) + ρ(∇Hf,∇Hf) + 2Lf (involving the sublaplacian ∆b and the pseudohermitian Ricci curvature ρ). WhenM is compact of CR dimension n and ρ(X,X)+ 2A(X, JX) ≥ k Gθ(X,X), X ∈ H(M), we derive the estimate −λ ≥ 2nk/(2n− 1) on each nonzero eigenvalue λ of ∆b satisfying Eigen(∆b;λ)∩Ker(T ) 6= (0) where T is the characteristic direction of dθ.