Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 054, 15 pages      arXiv:1502.05252      https://doi.org/10.3842/SIGMA.2015.054

Eigenvalue Estimates of the ${\rm spin}^c$ Dirac Operator and Harmonic Forms on Kähler-Einstein Manifolds

Roger Nakad a and Mihaela Pilca bc
a) Notre Dame University-Louaizé, Faculty of Natural and Applied Sciences, Department of Mathematics and Statistics, P.O. Box 72, Zouk Mikael, Lebanon
b) Fakultät für Mathematik, Universität Regensburg, Universitätsstraße 31, 93040 Regensburg, Germany
c) Institute of Mathematics ''Simion Stoilow'' of the Romanian Academy, 21, Calea Grivitei Str, 010702-Bucharest, Romania

Received March 03, 2015, in final form July 02, 2015; Published online July 14, 2015

Abstract
We establish a lower bound for the eigenvalues of the Dirac operator defined on a compact Kähler-Einstein manifold of positive scalar curvature and endowed with particular ${\rm spin}^c$ structures. The limiting case is characterized by the existence of Kählerian Killing ${\rm spin}^c$ spinors in a certain subbundle of the spinor bundle. Moreover, we show that the Clifford multiplication between an effective harmonic form and a Kählerian Killing ${\rm spin}^c$ spinor field vanishes. This extends to the ${\rm spin}^c$ case the result of A. Moroianu stating that, on a compact Kähler-Einstein manifold of complex dimension $4\ell+3$ carrying a complex contact structure, the Clifford multiplication between an effective harmonic form and a Kählerian Killing spinor is zero.

Key words: ${\rm spin}^c$ Dirac operator; eigenvalue estimate; Kählerian Killing spinor; parallel form; harmonic form.

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References

  1. Atiyah M.F., Singer I.M., The index of elliptic operators. I, Ann. of Math. 87 (1968), 484-530.
  2. Bär C., Real Killing spinors and holonomy, Comm. Math. Phys. 154 (1993), 509-521.
  3. Bär C., Extrinsic bounds for eigenvalues of the Dirac operator, Ann. Global Anal. Geom. 16 (1998), 573-596.
  4. Bourguignon J.-P., Hijazi O., Milhorat J.-L., Moroianu A., Moroianu S., A spinorial approach to Riemannian and conformal geometry, EMS Monographs in Mathematics, European Mathematical Society, 2015.
  5. Donaldson S.K., The Seiberg-Witten equations and $4$-manifold topology, Bull. Amer. Math. Soc. 33 (1996), 45-70.
  6. Friedrich Th., Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr. 97 (1980), 117-146.
  7. Friedrich Th., Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, Vol. 25, Amer. Math. Soc., Providence, RI, 2000.
  8. Gursky M.J., LeBrun C., Yamabe invariants and ${\rm Spin}^c$ structures, Geom. Funct. Anal. 8 (1998), 965-977, dg-ga/9708002.
  9. Herzlich M., Moroianu A., Generalized Killing spinors and conformal eigenvalue estimates for ${\rm Spin}^c$ manifolds, Ann. Global Anal. Geom. 17 (1999), 341-370.
  10. Hijazi O., A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986), 151-162.
  11. Hijazi O., Eigenvalues of the Dirac operator on compact Kähler manifolds, Comm. Math. Phys. 160 (1994), 563-579.
  12. Hijazi O., Montiel S., Urbano F., ${\rm Spin}^c$ geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds, Math. Z. 253 (2006), 821-853.
  13. Hijazi O., Montiel S., Zhang X., Dirac operator on embedded hypersurfaces, Math. Res. Lett. 8 (2001), 195-208, math.DG/0012262.
  14. Hijazi O., Montiel S., Zhang X., Eigenvalues of the Dirac operator on manifolds with boundary, Comm. Math. Phys. 221 (2001), 255-265, math.DG/0012261.
  15. Hijazi O., Montiel S., Zhang X., Conformal lower bounds for the Dirac operator of embedded hypersurfaces, Asian J. Math. 6 (2002), 23-36.
  16. Hijazi O., Zhang X., Lower bounds for the eigenvalues of the Dirac operator. I. The hypersurface Dirac operator, Ann. Global Anal. Geom. 19 (2001), 355-376.
  17. Hijazi O., Zhang X., Lower bounds for the eigenvalues of the Dirac operator. II. The submanifold Dirac operator, Ann. Global Anal. Geom. 20 (2001), 163-181.
  18. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
  19. Kirchberg K.-D., 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.
  20. Kirchberg K.-D., The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys. 7 (1990), 449-468.
  21. Kirchberg K.-D., Killing spinors on Kähler manifolds, Ann. Global Anal. Geom. 11 (1993), 141-164.
  22. Kirchberg K.-D., Eigenvalue estimates for the Dirac operator on Kähler-Einstein manifolds of even complex dimension, Ann. Global Anal. Geom. 38 (2010), 273-284, arXiv:0912.1451.
  23. Kirchberg K.-D., Semmelmann U., Complex contact structures and the first eigenvalue of the Dirac operator on Kähler manifolds, Geom. Funct. Anal. 5 (1995), 604-618.
  24. Lawson H.B., Michelsohn M.-L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  25. LeBrun C., Einstein metrics and Mostow rigidity, Math. Res. Lett. 2 (1995), 1-8, dg-ga/9411005.
  26. LeBrun C., Four-manifolds without Einstein metrics, Math. Res. Lett. 3 (1996), 133-147, dg-ga/9511015.
  27. Lichnerowicz A., Spineurs harmoniques, C. R. Acad. Sci. Paris 257 (1963), 7-9.
  28. Moroianu A., La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes, Comm. Math. Phys. 169 (1995), 373-384.
  29. Moroianu A., Formes harmoniques en présence de spineurs de Killing kählériens, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 679-684.
  30. Moroianu A., On Kirchberg's inequality for compact Kähler manifolds of even complex dimension, Ann. Global Anal. Geom. 15 (1997), 235-242.
  31. Moroianu A., Parallel and Killing spinors on ${\rm Spin}^c$ manifolds, Comm. Math. Phys. 187 (1997), 417-427.
  32. Moroianu A., Lectures on Kähler geometry, London Mathematical Society Student Texts, Vol. 69, Cambridge University Press, Cambridge, 2007.
  33. Moroianu A., Semmelmann U., Kählerian Killing spinors, complex contact structures and twistor spaces, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 57-61, dg-ga/9502003.
  34. Nakad R., Special submanifolds of ${\rm Spin}^c$ manifolds, Ph.D. Thesis, Institut Élie Cartan, France, 2011.
  35. Nakad R., Roth J., Hypersurfaces of ${\rm Spin}^c$ manifolds and Lawson type correspondence, Ann. Global Anal. Geom. 42 (2012), 421-442, arXiv:1203.3034.
  36. Pilca M., Kählerian twistor spinors, Math. Z. 268 (2011), 223-255, arXiv:0812.3315.
  37. Seiberg N., Witten E., Monopoles, duality and chiral symmetry breaking in $N=2$ supersymmetric QCD, Nuclear Phys. B 431 (1994), 484-550, hep-th/9408099.
  38. Wang M.Y., Parallel spinors and parallel forms, Ann. Global Anal. Geom. 7 (1989), 59-68.
  39. Witten E., Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), 769-796, hep-th/9411102.

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