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The first eigenvalue of the Dirac operator on compact Kähler manifolds. (La première valeur propre de l’opérateur de Dirac sur les variétés kählériennes compactes.) (French) Zbl 0832.53054

The first eigenvalue \(\lambda_1\) of the Dirac operator on a compact Kähler spin manifold of odd complex dimension \(m\) satisfies \(\lambda^2_1 \geq m^{-1} (m + 1){s\over 4}\). Here \(s\) is the lower bound of the scalar curvature. This estimate is due to K.-D. Kirchberg [Ann. Global Anal. Geom. 4, 291-325 (1986; Zbl 0629.53058)]. The author describes the manifolds for which equality holds in this estimate. For \(m = 4l+ 1\) this is the complex projective space and for \(m = 4l + 3\) these are twistor spaces of quaternionic Kähler manifolds of positive scalar curvature.

MSC:

53C55 Global differential geometry of Hermitian and Kählerian manifolds
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 0629.53058
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References:

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