An Alternative for Constant Mean Curvature Hypersurfaces

Let M^n+1 be a closed manifold of dimension 3 n+1 7 equipped with a generic Riemannian metric g. Let c be a positive number. We show that, either there exist infinitely many distinct closed hypersurfaces with constant mean curvature equal to c, or there exist infinitely many distinct closed hypersurfaces with constant mean curvature less than c but enclosing half the volume of M.