Dirac eigenvalues and total scalar curvature
by
Bernd Ammann and Christian Bär

Dirac eigenvalues and total scalar curvature (.dvi, .ps, .ps.gz oder .pdf)
J. Geom. Phys. 33, 229-234 (2000)

Abstract

It has recently been conjectured that the eigenvalues \lambda of the Dirac operator on a closed Riemannian spin manifold M of dimension n \geq 3 can be estimated from below by the total scalar curvature:
\lambda^2 \ge \frac{n}{4(n-1)} \cdot \frac{\int_M S}{vol(M)}.

We show by example that such an estimate is impossible. The example contains a very long and thin cylinder and therefore looks like a manifold with a very long nose.
Bild

Mathematics Subject Classification

58G25

Keywords

eigenvalues of the Dirac operator, total scalar curvature, Pinocchio metric
Bernd Ammann, 24.6.1999