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Annals of Global Analysis and Geometry

, Volume 15, Issue 3, pp 235–242 | Cite as

On Kirchberg's Inequality for Compact Kähler Manifolds of Even Complex Dimension

  • Andrei Moroianu
Article

Abstract

In 1986 Kirchberg showed that each eigenvalue of the Dirac operator on a compact Kähler manifold \(\left( {M^{2m} ,g} \right)\) of even complex dimension satisfies the inequality \(\left( {M^{2m} ,g} \right)\), where by S we denote the scalar curvature. It is conjectured that the manifolds for the limiting case of this inequality are products T2×N, where T2 is a flat torus and N is the twistor space of a quaternionic Kähler manifold of positive scalar curvature. In 1990 Lichnerowicz announced an affirmative answer for this conjecture (cf. [11]), but his proof seems to work only when assuming that the Ricci tensor is parallel. The aim of this note is to prove several results about manifolds satisfying the limiting case of Kirchberg's inequality and to prove the above conjecture in some particular cases.

Dirac operator Kirchberg's inequality 

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References

  1. 1.
    Bär, C.: Real Killing spinors and holonomy, Commun. Math. Phys. 154 (1993), 509–521.Google Scholar
  2. 2.
    Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nach. 97 (1980), 117–146.Google Scholar
  3. 3.
    Friedrich, Th.: The classification of 4-dimensional Kähler manifolds with small eigenvalue of the Dirac operator, Math. Ann. 295(3) (1993), 565–574.Google Scholar
  4. 4.
    Gauduchon, P.: L'opérateur de Penrose kählérien et les inégalités de Kirchberg. Unpublished.Google Scholar
  5. 5.
    Hijazi, O.: Opérateurs de Dirac sur les variétés riemanniennes: Minoration des valeurs propres, Thèse de 3ème Cycle, Ecole Polytechnique, 1984.Google Scholar
  6. 6.
    Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Commun. Math. Phys. 104 (1986), 151–162.Google Scholar
  7. 7.
    Hijazi, O.: Eigenvalues of the Dirac operator on compact Kähler manifolds, Commun. Math. Phys. 160 (1994), 563–579.Google Scholar
  8. 8.
    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. 3 (1986), 291–325.Google Scholar
  9. 9.
    Kirchberg, K.-D.: The first eigenvalue of the Dirac operator on Kähler manifolds, J. Geom. Phys. 7 (1990), 449–468.Google Scholar
  10. 10.
    Lichnerowicz, A.: Spineurs harmoniques, C. R. Acad. Sci. Paris, Série A–B 257 (1963), 7–9.Google Scholar
  11. 11.
    Lichnerowicz, A.: La première valeur propre de l'opérateur de Dirac pour une variété kählérienne et son cas limite, C. R. Acad. Sci. Paris, Série I 311 (1990), 717–722.Google Scholar
  12. 12.
    Moroianu, A.: La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes, Commun. Math. Phys. 169 (1995), 373–384.Google Scholar
  13. 13.
    Moroianu, A.: Opérateur de Dirac et submersions riemanniennes, Thesis, Ecole Polytechnique, 1996.Google Scholar

Copyright information

© Kluwer Academic Publishers 1997

Authors and Affiliations

  • Andrei Moroianu
    • 1
  1. 1.Centre de Mathématiques de l'Ecole Polytechnique, URA 169 du CNRSPalaiseau CedexFrance

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