The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold in dimension seven

Abstract

A version of Lichnerowicz’ theorem giving a lower bound of the eigenvalues of the sub-Laplacian on a compact seven dimensional quaternionic contact manifold is proved assuming a lower bound on the Sp(1)Sp(1)-components of the qc-Ricci curvature and the positivity of the P-function of any eigenfunction. The introduced P-function and nonlinearC-operator are motivated by the Paneitz operators defined previously in the Riemannian and CR settings and the P-function used in the theory of elliptic partial differential equations. It is shown that in the case of a seven dimensional compact 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is the round 3-Sasakian sphere.