The first Dirac eigenvalues on manifolds with positive scalar curvature

Bär, Christian; Dahl, Matthias

doi:10.1090/s0002-9939-04-07427-1

Cited by 8 publications

(4 citation statements)

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“…(8) On any compact manifold M that admits a metric of positive scalar curvature, and an arbitrary spin structure on M , C. Bär and M. Dahl [11] have constructed a sequence of metrics g i on M with scalar curvature ≥ n(n − 1), but with λ…”

Section: Examplesmentioning

confidence: 99%

Manifolds with small Dirac eigenvalues are nilmanifolds

Ammann

Sprouse

2006

Ann Glob Anal Geom

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Abstract. 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 ≥ 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.

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Section: Examplesmentioning

confidence: 99%

Manifolds with small Dirac eigenvalues are nilmanifolds

Ammann

Sprouse

2006

Ann Glob Anal Geom

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“…Indeed, motivated by a conjecture appearing in an earlier version of this note, Bär and Dahl [3] have constructed, on any compact spin manifold M n and for every positive real number ǫ, a metric g ǫ on M with the property that Scal gǫ ≥ n(n − 1) and such that the first eigenvalue of the Dirac operator satisfies λ 2 1 (D ǫ ) ≤ n 2 4 + ǫ. This construction clearly shows that no improvement of Friedrich's inequality can be obtained under purely topological restrictions.…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length

Moroianu

2004

Comptes Rendus Mathematique

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“…Later, Moroianu and Ornea [17] weakened the assumption on the 1-form from parallel to harmonic with constant length. Note the condition that the norm of the 1-form being constant is essential, in the sense that the topological constraint alone (the existence of a non-trivial harmonic 1-form) does not allow any improvement of Friedrich's inequality (see [4]). The generalization of [2] to locally reducible Riemannian manifolds was achieved by Alexandrov [1], extending earlier work by Kim [12].…”

Section: Introductionmentioning

confidence: 99%