Let

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.

Abstract. We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.

We study the horizontally regular curves in the Heisenberg groups Hn. We show the fundamental theorem of curves in Hn (n ≥ 2) and define the orders of horizontally regular curves. We also show that the curve γ is of order k if and only if, up to a Heisenberg rigid motion, γ lies in H k but not in H k−1 ; moreover, two curves with the same order differ from a rigid motion if and only if they have the same invariants: p-curvatures and contact normality. Thus, combining with our previous work [1] we have completed the classification of horizontally regular curves in Hn for n ≥ 1.