Curvature, Harnack's inequality, and a spectral characterization of nilmanifolds

For closed n-dimensional Riemannian manifolds M with almost nonnegative Ricci curvature, the Laplacian on one-forms is known to admit at most n small eigenvalues. If there are n small eigenvalues, or if M is orientable and has n - 1 small eigenvalues, then M is diffeomorphic to a nilmanifold, and the metric is almost left invariant. We show that our results are optimal for n greater than or equal to 4.