Dirac eigenvalues and total scalar curvature

Abstract

It has recently been conjectured that the eigenvalues $\lambda$ of the Dirac operator on a closed Riemannian spin manifold $M$ of dimension $n\ge 3$ 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.