no. 5
Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz
Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1637-1659.

Nous décrivons toutes les variétés kählériennes compactes de dimension complexe paire à courbure scalaire positive, admettant la plus petite valeur propre possible pour l’opérateur de Dirac.

We describe all compact spin Kähler manifolds of even complex dimension and positive scalar curvature with least possible first eigenvalue of the Dirac operator.

@article{AIF_1999__49_5_1637_0,
     author = {Moroianu, Andrei},
     title = {K\"ahler manifolds with small eigenvalues of the {Dirac} operator and a conjecture of {Lichnerowicz}},
     journal = {Annales de l'Institut Fourier},
     pages = {1637--1659},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {49},
     number = {5},
     year = {1999},
     doi = {10.5802/aif.1732},
     zbl = {0946.53040},
     mrnumber = {2001i:58062},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1732/}
}
TY  - JOUR
AU  - Moroianu, Andrei
TI  - Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz
JO  - Annales de l'Institut Fourier
PY  - 1999
SP  - 1637
EP  - 1659
VL  - 49
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1732/
DO  - 10.5802/aif.1732
LA  - en
ID  - AIF_1999__49_5_1637_0
ER  - 
%0 Journal Article
%A Moroianu, Andrei
%T Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz
%J Annales de l'Institut Fourier
%D 1999
%P 1637-1659
%V 49
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1732/
%R 10.5802/aif.1732
%G en
%F AIF_1999__49_5_1637_0
Moroianu, Andrei. Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz. Annales de l'Institut Fourier, Tome 49 (1999) no. 5, pp. 1637-1659. doi : 10.5802/aif.1732. https://aif.centre-mersenne.org/articles/10.5802/aif.1732/

[1] C. Bär, Real Killing spinors and holonomy, Commun. Math. Phys., 154 (1993), 509-521. | MR | Zbl

[2] H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistors and Killing Spinors on Riemannian Manifolds, Teubner-Verlag Stuttgart/Leipzig (1991). | Zbl

[3] Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., 97 (1980), 117-146. | Zbl

[4] Th. Friedrich, A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds, Math. Nachr., 102 (1981), 53-56. | MR | Zbl

[5] Th. Friedrich, The Classification of 4-dimensional Kähler Manifolds with Small Eigenvalue of the Dirac Operator, Math. Ann., 295 (1993), 565-574. | EuDML | Zbl

[6] Th. Friedrich, R. Grunewald, On the first eigenvalue of the Dirac operator on 6-dimensional manifolds, Ann. Global Anal. Geom., 3 (1985), 265-273. | MR | Zbl

[7] Th. Friedrich, I. Kath, Einstein manifolds of dimension five with small eigenvalues of the Dirac operator, J. Differential Geom., 29 (1989), 263-279. | MR | Zbl

[8] Th. Friedrich, I. Kath, Compact Seven-dimensional Manifolds with Killing Spinors, Commun. Math. Phys., 133 (1990), 543-561. | Zbl

[9] S. Gallot, Équations différentielles caractéristiques de la sphère, Ann. Sci. Ec. Norm. Sup. Paris, 12 (1979), 235-267. | EuDML | Numdam | MR | Zbl

[10] P. Gauduchon, L'opérateur de Penrose kählérien et les inégalités de Kirchberg, preprint (1993).

[11] O. Hijazi, Opérateurs de Dirac sur les variétés riemanniennes : Minoration des valeurs propres, Thèse de 3ème Cycle, École Polytechnique (1984).

[12] O. Hijazi, Caractérisation de la sphère par les premières valeurs propres de l'opérateur de Dirac en dimensions 3, 4, 7 et 8, C. R. Acad. Sci. Paris, 307, Série I (1986), 417-419. | MR | Zbl

[13] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Commun. Math. Phys., 104 (1986), 151-162. | MR | Zbl

[14] O. Hijazi, Eigenvalues of the Dirac operator on compact Kähler manifolds, Commun. Math. Phys., 160 (1994), 563-579. | MR | Zbl

[15] N. Hitchin, Harmonic Spinors, Adv. in Math., 14 (1974), 1-55. | MR | Zbl

[16] 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., 3 (1986), 291-325. | MR | Zbl

[17] K.-D. Kirchberg, The first Eigenvalue of the Dirac Operator on Kähler Manifolds, J. Geom. Phys., 7 (1990), 449-468. | MR | Zbl

[18] S. Kobayashi, On compact Kähler Manifolds with Positive Definite Ricci Tensor, Ann. of Math., 74 (1961), 570-574. | MR | Zbl

[19] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris, 257 (1963), 7-9. | MR | Zbl

[20] A. Lichnerowicz, La première valeur propre de l'opérateur de Dirac pour une variété kählérienne et son cas limite, C. R. Acad. Sci. Paris, 311, Série I (1990), 717-722. | MR | Zbl

[21] A. Moroianu, La première valeur propre de l'opérateur de Dirac sur les variétés kählériennes compactes, Commun. Math. Phys., 169 (1995), 373-384. | MR | Zbl

[22] A. Moroianu, On Kirchberg inequality for compact Kähler manifolds of even complex dimension, Ann. Global Anal. Geom., 15 (1997), 235-242. | MR | Zbl

[23] A. Moroianu, Parallel and Killing Spinors on Spinc Manifolds, Commun. Math. Phys., 187 (1997), 417-428. | MR | Zbl

[24] A. Moroianu, Spinc Manifolds and Complex Contact Structures, Commun. Math. Phys., 193 (1998), 661-673. | MR | Zbl

[25] B. O'Neill, The fundamental equations of a submersion, Michigan Math J., 13 (1966), 459-469. | MR | Zbl

Cité par Sources :