Small Eigenvalues of the Laplacians on Vector Bundles over Towers of Covers

Abstract

Under the conditions that a compact Riemannian manifold is of sufficiently pinched negative sectional curvature and that a smooth Hermitian vector bundle over the manifold is also of sufficiently small curvature, we prove some pinching results on the asymptotic behavior of the numbers of small eigenvalues of the Laplacians on the induced Hermitian vector bundles over a tower of covers of the manifold. In the process we also obtain interesting results on the non-existence of square integrable 'almost harmonic' vector bundle-valued forms omitting the middle degree(s) on the universal cover.