Manifolds with small Dirac eigenvalues are nilmanifolds
| dc.creator | Ammann, Bernd | |
| dc.creator | Sprouse, Chad | |
| dc.date | 2004-03-08 | |
| dc.date | 2004-03-10 | |
| dc.date.accessioned | 2026-07-07T09:22:49Z | |
| dc.date.available | 2026-07-07T09:22:49Z | |
| dc.description | 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. 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 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. | |
| dc.identifier | https://arxiv.org/abs/math/0403142 | |
| dc.identifier | http://arxiv.org/abs/math/0403142 | |
| dc.identifier | Ann. Glob. Anal. Geom. 31, 409-425 (2007) | |
| dc.identifier | doi:10.1007/s10455-006-9048-2 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/155509 | |
| dc.subject | Differential Geometry | |
| dc.subject | Spectral Theory | |
| dc.subject | 53C27 (Primary), 58J50, 53C20, 53C21. (Secondary) | |
| dc.title | Manifolds with small Dirac eigenvalues are nilmanifolds | |
| dc.type | text |