Small eigenvalues of the Laplacians on vector bundles over towers of covers (Q1357875)

The author generalizes known results on the asymptotic behaviour of small eigenvalues of the Laplace-Beltrami operator on forms on towers of coverings to the case of differential forms with values in a Hermitian vector bundle. More precisely, let \(M\) be an \(m\)-dimensional closed Riemannian manifold with sufficiently pinched negative sectional curvature. Let \(V\to M\) be a Hermitian vector bundle with a connection of sufficiently small curvature. Let \(M_j\to M\) be a tower of finite normal coverings \((\pi_1 (M_j)\to \{1\})\). By pulling back \(V\) to \(M_j\) one can speak about \(V\)-valued differential forms on each \(M_j\). Let \(\widehat H(M_j)\) be the space of harmonic \(V\)-valued forms avoiding the middle degrees \((m/2\) if \(m\) is even, \((m\pm 1)/2\) if \(m\) is odd). Denote by \(\widehat N(M_j, \lambda)\) the sum of eigenspaces for eigenvalues \(\leq\lambda\), again avoiding middle degrees. It is shown that for sufficiently small \(\lambda\) \[ \dim \widehat N(M_j, \lambda) =o\bigl(\text{Vol} (M_j)\bigr) \quad \text{as} \quad j\to \infty \] and, in particular, \[ \dim \widehat H(M_j)= o\bigl(\text{Vol} (M_j)\bigr) \quad \text{as} \quad j\to \infty. \] If \(m\) is even, then one can also control the middle-dimensional forms.