Abstract
We discuss a sharp lower bound for the first positive eigenvalue of the sublaplacian on a closed, strictly pseudoconvex pseudohermitian 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. For this purpose, we establish an Obata-type theorem in CR geometry that characterizes the CR sphere in terms of a nonzero function satisfying a certain overdetermined system. Similar results are proved in dimension 3 under an additional condition.
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