The upper bounds for eigenvalues of Dirac operators (Q5954669)

Let \(D\) be a Dirac operator on a compact oriented Riemannian manifold \(M\) of dimension \(2m.\) Let \(\lambda _k^2\) be the \(k\)-th nonzero eigenvalue of the operator \(D^2\) counting with multiplicity. NEWLINENEWLINENEWLINEThe purpose of this article is to obtain for \(\lambda _k^2\) some specific universal upper bounds of the type: \(\lambda _k^2\leq \text{Const} (m,\text{dimker} (D^2), \text{Vol} (M), k_D, k_M)\) where \(k_D\) is an integer defined by the operator \(D\) and \(k_M\) is an integer determined by the geometry of \(M.\)