The sharp lower bound for the first positive eigenvalue of the sublaplacian on a pseudohermitian 3-manifold

In this paper, we study a sharp lower bound of the first eigenvalue of the sublaplacian on a 3-dimensional pseudohermitian manifold with the CR Paneitz operator positive. In general cases, S.-Y. Li and H.-S. Luk ({Proc. Am. Math. Soc.} 132(3), 789-798) (2004) proved the lower bound under a condition on a covariant derivative of the torsion as well as the Ricci curvature and the torsion. We show that if the CR Paneitz operator is positive, then the sharp lower bound is obtained under one simpler condition on only the Ricci curvature and the torsion itself; which is similar to the condition given in high-dimensional cases in ({Commun. Partial Differential Equations}, 10(2/3), 191-217) (1985). We also show examples where our theorem applies, but Theorem 1.2 in ({Proc. Am. Math. Soc.} 132(3), 789-798) (2004) does not.