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An estimation for the first eigenvalue of the Dirac operator on closed Kähler manifolds of positive scalar curvature. (English) Zbl 0629.53058
The main result of the paper is the following Theorem: Let \(\lambda\) be an eigenvalue of the Dirac operator D on a closed Kähler spin manifold \(M^ n\) of real dimension n with positive scalar curvature R, then \[ (*)\quad \lambda^ 2\geq ((n+2)/n)R_ 0/4 \] where \(R_ 0\) denotes the minimum of R on \(M^ n\). If \(\lambda_ 0:=\sqrt{(n+2)R_ 0/4n}\) itself is an eigenvalue of D, then \(M^ n\) is an Einstein space of odd complex dimension \(m=n/2.\)
The estimate (*) is sharp in the sense that there exist Kähler manifolds for which \(\lambda_ 0\) itself is an eigenvalue of D. In dimension \(n=6\) the complex projective space \(P^ 3({\mathbb{C}})\) and the flag manifold \(F({\mathbb{C}}^ 3)\) with respect to their natural Kähler structures are examples of such spaces. Later we have shown in another paper that for \(n=6\) there are no other examples.

53C55 Global differential geometry of Hermitian and Kählerian manifolds
58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.)
Full Text: DOI
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