In this paper, we first prove the CR analogue of M. Obata's theorem on a closed pseudohermitian (2n + 1)-manifold with free pseudohermitian torsion. Secondly, we have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian on a closed pseudohermitian (2n + 1)-manifold with a more general curvature condition for n ≥ 2. The key step is a discovery of CR analogue of Bochner formula which involving the CR Paneitz operator and nonnegativity of CR Paneitz operator P 0 for n ≥ 2.

Abstract. Let (M 3 , J, θ 0 ) be a closed pseudohermitian 3-manifold. Suppose the associated torsion vanishes and the associated Q-curvature has no kernel part with respect to the associated Paneitz operator. On such a background pseudohermitian 3-manifold, we study the change of the contact form according to a certain version of normalized Q-curvature flow. This is a fourth order evolution equation. We prove that the solution exists for all time and converges smoothly to a contact form of zero Q-curvature. We also consider other background conditions and obtain a priori bounds up to high orders for the solution.

In this paper, we first prove the CR analogue of Obata's theorem on a closed pseudohermitian 3-manifold with zero pseudohermitian torsion. Secondly, instead of zero torsion, we have the CR analogue of Li-Yau's eigenvalue estimate on the lower bound estimate of positive first eigenvalue of the sub-Laplacian in a closed pseudohermitian 3-manifold with nonnegative CR Paneitz operator P 0 . Finally, we have a criterion for the positivity of first eigenvalue of the sub-Laplacian on a complete noncompact pseudohermitian 3-manifold with nonnegative CR Paneitz operator. The key step is a discovery of integral CR analogue of Bochner formula which involving the CR Paneitz operator.

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In this paper, we study a lower bound estimate of the first positive eigenvalue of the sublaplacian on a three-dimensional pseudohermitian manifold. S.-Y. Li and H.-S. Luk derived the lower bound estimate under certain conditions for curvature tensors bounded below by a positive constant. By using the Li-Yau gradient estimate, we are able to get an effective lower bound estimate under a general curvature condition. The key is the discovery of a new CR version of the Bochner formula which involves the CR Paneitz operator.