Abstract
We prove that the largest first eigenvalue of the Dirac operator among all Hermitian metrics on the complex projective space of odd dimension m, larger than the Fubini-Study metric is bounded by (2m(m+1))1/2.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anghel, N.: On the first Vafa—Witten bound for two-dimensional tori. In: Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999). Sémin. Congr.(4), Soc. Math.: France, Paris, (2000), 1–16.
Atiyah, M.: Eigenvalues of the Dirac operator. In: Workshop Bonn 1984 (Bonn, 1984). Lecture Notes in Mathematics 1111, Springer, Berlin, 1985, 251–260.
Baum, H.: An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds. Math. Z. 206(3) (1991), 409–422.
Bourguignon, J. P. and Gauduchon, P.: Spineurs, opérateurs de Dirac et variations de métriques. Comm. Math. Phys. 144(3) (1992), 581–599.
Friedrich, Th.: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97 (1980), 117–146.
Griffiths, P. and Harris, J.:Principles of Algebraic Geometry, Pure and Appl. Math. New York: Wiley-Interscience/Wiley, Chapter 1, Section 4, 1978.
Herzlich, M.: Extremality for the Vafa—Witten bound on the sphere. ArXiv: math.DG/0407530. GAFA (2004) (in press).
Hijazi, O.: A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Comm. Math. Phys. 104(1) (1986), 151–162.
Kirchberg, K.-D.: The first eigenvalue of the Dirac operator on Kähler manifolds. J. Geom. Phys. 7(4) (1991) (1990), 449–468.
Kobayashi, S. and Nomizu, K.: In: Foundations of Differential Geometry, Vol. II Interscience Tracts in Pure and Applied Mathematics, Vol.2, No. 15. Interscience Publishers/Wiley, New York, London, Sydney, T.2, Chapter.9, Section 6, 7, 1969.
Lawson, H. B., Jr and Michelsohn, M. L.: Spin Geometry, In: Princeton Mathematical Series Vol.38 . Princeton, NJ: Princeton University Press, Section 4, p. 365, 1989 .
Moroianu, A.: La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes. Comm. Math. Phys. 169(2) (1995), 373–384.
Maeda, S. and Shimizu, Y.: Imbedding of a complex projective space defined by monomials of same degree. Math. Z. 179(3) (1982), 337–344.
Nakagawa, H. and Ogiue, K.: Complex space forms immersed in complex space formes. Trans. Am. Math. Soc. 219 (1976), 289–297.
Semmelmann, U.: A short proof of eigenvalue estimates for the Dirac operator on Riemannian and Kähler manifolds. In: Differential Geometry and applications (Brno, 1998), Brno: Masaryk University (1999), pp. 137–140.
Vafa, C. and Witten, E.: Eigenvalue inequalities for fermions in gauge theories. Comm. Math. Phys. 95(3) (1984), 257–276.
Author information
Authors and Affiliations
Corresponding author
Additional information
Mathematics Subject Classification (2000): 53C27, 58J50, 58J60.
Rights and permissions
About this article
Cite this article
Davaux, H., Min-Oo, M. Vafa-Witten bound on the complex projective space. Ann Glob Anal Geom 30, 29–36 (2006). https://doi.org/10.1007/s10455-005-9013-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-005-9013-5