The first eigenvalue of the Dirac operator on compact Kähler manifolds
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 \textit{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.
scientific article; zbMATH DE number 778992
scientific article