  We discuss a sharp lower bound for the first positive eigenvalue of the
sublaplacian on a closed, strictly pseudoconvex pseudo-hermitian manifold of
dimension $2m+1\geq 5$. 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.
