Manifolds with small Dirac eigenvalues are nilmanifolds
Bernd Ammann and Chad Sprouse
Consider the class of $n$-dimensional Riemannian spin manifolds with bounded sectional curvatures and diameter, and almost non-negative scalar curvature. Let $r=1$ if $n=2,3$ and $r=2^{[n/2]-1}+1$ if $n\geq 4$. We show that if the square of the Dirac operator on such a manifold has $r$ small eigenvalues, then the manifold is diffeomorphic to a nilmanifold and has trivial spin structure. If a manifold with almost nonnegative scalar curvature has one small Dirac eigenvalue, and if the volume is not too small, then we show that the metric is close to a Ricci-flat metric on $M$ with a parallel spinor.