Involutive isometries, eigenvalue bounds and a spectral property of Clifford tori

In this paper we consider a compact Riemannian manifold or submanifold M, with an involutive isometry which has no fixed point, and we derive some spectral properties of this geometric situation. The aim is to give an upper bound for the gap lambda(2) (D) - lambda(1) (D) of the first two eigenvalues of a Laplace-type operator D acting on sections of a vector bundle over M. In the first part, using the classical barycenter method, we derive sharp upper bounds for the gap of an antipodal symmetric submanifold of Euclidean space. Moreover, if equality holds, we prove that the submanifold is minimal in a sphere. In particular, we give a spectral characterization of the Clifford torus S-p (root p/n) x Sn-p (root n - p /n) as the unique maximizer for the gap of the Hodge Laplacian on p-forms, among all antipodal symmetric hypersurfaces of the sphere Sn+1. In the second part we give upper bounds in the general case. The main point is that these bounds do not depend on the particular operator D we consider, but only on the natural intrinsic or extrinsic distance on M and on the displacement of the action of the considered isometry group.