Kähler manifolds with small eigenvalues of the Dirac operator and a conjecture of Lichnerowicz
Moroianu, Andrei
Annales de l'Institut Fourier, Tome 49 (1999), p. 1637-1659 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

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},
     volume = {49},
     year = {1999},
     pages = {1637-1659},
     doi = {10.5802/aif.1732},
     mrnumber = {2001i:58062},
     zbl = {0946.53040},
     language = {en},
     url = {http://dml.mathdoc.fr/item/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) pp. 1637-1659. doi : 10.5802/aif.1732. http://gdmltest.u-ga.fr/item/AIF_1999__49_5_1637_0/

[1] C. Bär, Real Killing spinors and holonomy, Commun. Math. Phys., 154 (1993), 509-521. | MR 94i:53042 | Zbl 0778.53037

[2] H. Baum, Th. Friedrich, R. Grunewald, I. Kath, Twistors and Killing Spinors on Riemannian Manifolds, Teubner-Verlag Stuttgart/Leipzig (1991). | Zbl 0734.53003

[3] Th. Friedrich, Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., 97 (1980), 117-146. | Zbl 0462.53027

[4] Th. Friedrich, A remark on the first eigenvalue of the Dirac operator on 4-dimensional manifolds, Math. Nachr., 102 (1981), 53-56. | MR 83d:58069 | Zbl 0481.53039

[5] Th. Friedrich, The Classification of 4-dimensional Kähler Manifolds with Small Eigenvalue of the Dirac Operator, Math. Ann., 295 (1993), 565-574. | Zbl 0798.53065

[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 87a:58156 | Zbl 0577.58034

[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 90e:58158 | Zbl 0633.53069

[8] Th. Friedrich, I. Kath, Compact Seven-dimensional Manifolds with Killing Spinors, Commun. Math. Phys., 133 (1990), 543-561. | Zbl 0722.53038

[9] S. Gallot, Équations différentielles caractéristiques de la sphère, Ann. Sci. Ec. Norm. Sup. Paris, 12 (1979), 235-267. | Numdam | MR 80h:58051 | Zbl 0412.58009

[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 88b:58142 | Zbl 0606.53024

[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 87j:58096 | Zbl 0593.58040

[14] O. Hijazi, Eigenvalues of the Dirac operator on compact Kähler manifolds, Commun. Math. Phys., 160 (1994), 563-579. | MR 95b:58156 | Zbl 0794.53042

[15] N. Hitchin, Harmonic Spinors, Adv. in Math., 14 (1974), 1-55. | MR 50 #11332 | Zbl 0284.58016

[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 89b:58221 | Zbl 0629.53058

[17] K.-D. Kirchberg, The first Eigenvalue of the Dirac Operator on Kähler Manifolds, J. Geom. Phys., 7 (1990), 449-468. | MR 92h:58199 | Zbl 0734.53050

[18] S. Kobayashi, On compact Kähler Manifolds with Positive Definite Ricci Tensor, Ann. of Math., 74 (1961), 570-574. | MR 24 #A2922 | Zbl 0107.16002

[19] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris, 257 (1963), 7-9. | MR 27 #6218 | Zbl 0136.18401

[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 92a:58147 | Zbl 0713.53040

[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 96g:58198 | Zbl 0832.53054

[22] A. Moroianu, On Kirchberg inequality for compact Kähler manifolds of even complex dimension, Ann. Global Anal. Geom., 15 (1997), 235-242. | MR 98d:58192 | Zbl 0890.53058

[23] A. Moroianu, Parallel and Killing Spinors on Spinc Manifolds, Commun. Math. Phys., 187 (1997), 417-428. | MR 98i:58245 | Zbl 0888.53035

[24] A. Moroianu, Spinc Manifolds and Complex Contact Structures, Commun. Math. Phys., 193 (1998), 661-673. | MR 99c:53048 | Zbl 0908.53024

[25] B. O'Neill, The fundamental equations of a submersion, Michigan Math J., 13 (1966), 459-469. | MR 34 #751 | Zbl 0145.18602