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Manifolds with small Dirac eigenvalues are nilmanifolds

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Abstract

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 ≥ 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 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.

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Authors and Affiliations

  1. Institut Élie Cartan, Université Henri Poincaré, Nancy I, B.P. 239, 54506, Vandoeuvre-Lès-Nancy, Cedex, France

    Bernd Ammann

  2. Department of Mathematics, CSU Northridge, 18111 Nordhoff Street, Northridge, CA, 91330-8313, USA

    Chad Sprouse

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  1. Bernd Ammann
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  2. Chad Sprouse
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Correspondence to Bernd Ammann.

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Ammann, B., Sprouse, C. Manifolds with small Dirac eigenvalues are nilmanifolds. Ann Glob Anal Geom 31, 409–425 (2007). https://doi.org/10.1007/s10455-006-9048-2

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  • DOI: https://doi.org/10.1007/s10455-006-9048-2

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