An Obata-type Theorem in CR Geometry

We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudo-hermitian manifold of dimension $2m+1\ge 5$. We prove that the equality holds iff the manifold is equivalent to the CR sphere up to a scaling. The essential step is a characterization of the CR sphere when there is a nonzero function satisfying a certain overdetermined system.