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σk-Scalar curvature and eigenvalues of the Dirac operator

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Abstract

On a four-dimensional closed spin manifold (M 4, g), the eigenvalues of the Dirac operator can be estimated from below by the total σ2-scalar curvature of M 4 as follows: \(\lambda^4 \geq \frac{32}{3} \frac{\int_{M^4} \sigma_2 (g) {\rm d} {\rm vol} (g)}{{\rm vol} (M^4, g)}\) Equality implies that (M 4, g) is a round sphere and the corresponding eigenspinors are Killing spinors.

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References

  1. Ammann, B.: A spin-conformal lower bound of the first positive Dirac eigenvalue. Differ. Geom. Appl. 18 21–32 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ammann, B., Bär, C.: Dirac eigenvalues and total scalar curvature. J. Geom. Phys. 33, 229–234 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bär, C.: Lower eigenvalue estimates for Dirac operators. Math. Ann. 293, 39–46 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bär, C.: Metrics with harmonic spinors. Geom. Funct. Anal. 6, 899–942 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistors and Killing Spinors on Riemannian Manifolds. Teubner-Texte zur Mathematik, Stuttgart, Germany (1991)

  6. Chang, S.-Y. A., Gursky, M., Yang, P.: An equation of Monge–Ampére type in conformal geometry, and four-manifolds ofs positive Ricci curvature. Ann. Math. 155(2), 709–787 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chang, S.-Y. A., Gursky, M., Yang P.: An a priori estimates for a fully nonlinear equation on Four-manifolds. J. D'Anal. Math. 87, 151–186 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chang, S.-Y. A., Gursky, M., Yang, P.: A conformally invariant sphere theorem in four dimensions. Publ. Math. Inst. Hautes Études Sci. No. 98, 105–143 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Friedrich, T.: Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97, 117–146 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Friedrich, T., Kim, E. C.: Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors. J. Geom. Phys. 37, 1–14 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Friedrich, T., Kirchberg, K.-D.: Eigenvalue estimates of the Dirac operator depending on the Ricci tensor. Math. Ann. 324, 799–816 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Friedrich, T., Kirchberg, K.-D.: Eigenvalue estimates for the Dirac operator depending on the Weyl tensor. J. Geom. Phys. 41, 196–207 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Guan, P., Lin, C.-S., Wang, G.: Schouten tensor and some topological properties. Comm. Anal. Geom. 13, 845–860 (2005)

    MathSciNet  Google Scholar 

  14. Guan, P., Wang, G.: A fully nonlinear conformal flow on locally conformally flat manifolds. J. reine und angew. Math. 557, 219–238 (2003) Preliminary version, ArXiv: math.DG/0112256v1

  15. Gursky, M., Viaclovsky, J.: A fully nonlinear equation on 4-manifolds with positive scalar curvature. J. Diff. Geom. 63, 131–154 (2003)

    MathSciNet  MATH  Google Scholar 

  16. Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Comm. Math. Phys. 104, 151–162 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  17. Hijazi, O.: Première valeur propre de lòpérateur de Dirac et nombre de Yamabe. C. R. Acad. Sci. Paris Sér. I Math. 313, 865–868 (1991)

    MathSciNet  MATH  Google Scholar 

  18. Hijazi, O.: Lower bounds for eigenvalues of the Dirac operator through modified connections. J. Geom. Phys. 16, 27–38 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hitchin, N.: Harmonic spinors. Adv. Math. 14. 1–55 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. Lichnerowicz, A.: Spineurs harmoniques. C.R. Acad. Sci. Paris 257, 7–9 (1963)

    MathSciNet  MATH  Google Scholar 

  21. Lott, J.: Eigenvalue bounds for the Dirac operator. Pac. J. Math. 125, 117–126 (1986)

    MathSciNet  MATH  Google Scholar 

  22. Viaclovsky, J.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101, 283–316 (2000)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institute Of Mathematics, Fudan University, Shanghai, China

    Guofang Wang

  2. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22-26, 04103, Leipzig, Germany

    Guofang Wang

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  1. Guofang Wang
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Correspondence to Guofang Wang.

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Dedicated to Professor Wang Guangyin on the occasion of his 80th birthday.

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Wang, G. σk-Scalar curvature and eigenvalues of the Dirac operator. Ann Glob Anal Geom 30, 65–71 (2006). https://doi.org/10.1007/s10455-006-9022-z

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  • DOI: https://doi.org/10.1007/s10455-006-9022-z

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