Manifolds with small Dirac eigenvalues are nilmanifolds

This study shows 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 exi...

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Bibliographic Details
Published in:Annals of global analysis and geometry 2007-06, Vol.31 (4), p.409-425
Main Authors: Ammann, Bernd, Sprouse, Chad
Format: Article
Language:English
Subjects:
Curvature
Diameters
Eigenvalues
Euclidean space
Lower bounds
Manifolds (mathematics)
Mathematical models
Spin structure
Studies
Topological manifolds
Toruses
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Summary:This study shows 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 non-negative 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. In dimension 4 this implies that M is either a torus or a K3-surface.
ISSN:0232-704X
1572-9060
DOI:10.1007/s10455-006-9048-2