Upper bounds of small eigenvalues of the Dirac operator and isometric immersions (Q912404)

It is shown that a topologically determined number of eigenvalues \(\lambda\) of the Dirac operator D of a closed Riemannian spin manifold M of even dimension n can be bounded by \(| \lambda | \leq 2^{n/4}\| h_ f\|,\) where \(\| h_ f\|\) is the maximum of the second fundamental form \(h_ f\) of an isometric immersion f of M into \({\mathbb{R}}^ N\). This leads to similar bounds in terms of the maximum of the scalar curvature if the manifold admits a minimal immersion into a sphere, or if M is complex, an isometric holomorphic immersion into a complex projective space. In the appendix the spectrum of the Dirac operator on the quaternionic projective space \({\mathbb{H}}P^ 2\) is calculated.