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The first Dirac eigenvalues on manifolds with positive scalar curvature


Authors: Christian Bär and Mattias Dahl
Journal: Proc. Amer. Math. Soc. 132 (2004), 3337-3344
MSC (2000): Primary 53C27
DOI: https://doi.org/10.1090/S0002-9939-04-07427-1
Published electronically: May 21, 2004
MathSciNet review: 2073310
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Abstract: We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.


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  • 1. B. Alexandrov, G. Grantcharov, and S. Ivanov, An estimate for the first eigenvalue of the Dirac operator on compact riemannian spin manifold admitting parallel one-form, J. Geom. Phys. 28, (1998), 263-270. MR 99k:58182
  • 2. C. Bär, Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993), 509-521. MR 94i:53042
  • 3. -, Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Glob. Anal. Geom. 16 (1998), 573-596. MR 99k:58183
  • 4. T. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nicht-negativer Krümmung, Math. Nachr. 97 (1980), 117-146. MR 82g:58088
  • 5. M. Gromov and H. B. Lawson, The classification of simply connected manifolds of positive scalar curvature, Ann. Math., II. Ser. 111 (1980), 423-434. MR 81h:53036
  • 6. K.-D. Kirchberg, An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature, Ann. Global Anal. Geom. 4 (1986), 291-325. MR 89b:58221
  • 7. -, The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys. 7 (1990), 449-468. MR 92h:58199
  • 8. W. Kramer, U. Semmelmann, and G. Weingart, Eigenvalue estimates for the Dirac operator on quaternionic Kähler manifolds, Math. Z. 230 (1999), 727-751. MR 2000d:58046
  • 9. A. Moroianu and L. Ornea, Eigenvalue estimates for the Dirac operator and harmonic 1-forms of constant length, C. R. Acad. Sci. Paris Série I 338 (2004), 561-564.
  • 10. J. Rosenberg and S. Stolz, Metrics of positive scalar curvature and connections with surgery, Surveys on surgery theory, Vol. 2, Ann. of Math. Stud. 149, Princeton Univ. Press, 2001, 353-386. MR 2002f:53054

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Additional Information

Christian Bär
Affiliation: Institut für Mathematik, Universität Potsdam, PF 601553, 14415 Potsdam, Germany
Email: baer@math.uni-potsdam.de

Mattias Dahl
Affiliation: Institutionen för Matematik, Kungliga Tekniska Högskolan, 100 44 Stockholm, Sweden
Email: dahl@math.kth.se

DOI: https://doi.org/10.1090/S0002-9939-04-07427-1
Keywords: Dirac operator, eigenvalue, positive scalar curvature, Friedrich's estimate
Received by editor(s): July 2, 2003
Published electronically: May 21, 2004
Additional Notes: The first author has been partially supported by the Research and Training Networks HPRN-CT-2000-00101 “EDGE” and HPRN-CT-1999-00118 “Geometric Analysis” funded by the European Commission.
Communicated by: Józef Dodziuk
Article copyright: © Copyright 2004 American Mathematical Society

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