Manifolds with small Dirac eigenvalues are nilmanifolds

Bernd Ammann (1) , Chad Sprouse (2)

1 IECN - Institut Élie Cartan de Nancy
2 Department of Mathematics [CSUN]

Consider the class of $n$-dimensional Riemannian spin manifolds with bounded sectional curvatures and bounded 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. Equivalently, if $M$ is not a nilmanifold or if $M$ is a nilmanifold with a non-trivial spin structure, then there exists a uniform lower bound on the $r$-th eigenvalue of the square of the Dirac operator. If a manifold with almost nonnegative scalar curvature has one small Dirac eigen value, 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. In dimension $4$ this implies that $M$ is either a torus or a $K3$-surface.

Mots clés: Dirac operator; nilmanifold; small eigenvalues; almost positive scalar curvature

Domaines: Géométrie différentielle [math.DG]

Bernd Ammann, Chad Sprouse. Manifolds with small Dirac eigenvalues are nilmanifolds. 2006. ⟨hal-00091177⟩