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Eigenvalues of the Dirac operator. (English) Zbl 0568.53022
Arbeitstag. Bonn 1984, Proc. Meet. Max-Planck-Inst. Math., Bonn 1984, Lect. Notes Math. 1111, 251-260 (1985).
[For the entire collection see Zbl 0547.00007.]
This article presents results of C. Vaffa and E. Witten [Commun. Math. Phys. 95, 257-276 (1984)]. Let (M,g) be a compact Riemannian spin manifold of dimension d and D be the Dirac operator of M acting on the spin bundle S. Given a Hermitian vector bundle V with a connection A, we denote by \(D_ A: S\otimes V\to S\otimes V\) the extended Dirac operator. \(D_ A\) is self-adjoint, has discrete eigenvalues \(\lambda_ j\) indexed by increasing absolute value. The following uniform upper bounds are obtained. There exist constants C and C’, depending on (M,g) but not on V, A or n, such that, for any n, \(| \lambda_ n| \leq C_ n^{1/d}\) and, if d is odd, every interval of length \(C'_ n{}^{1/d}\) contains n eigenvalues. The proof uses index theorem, deformation arguments relating the connection A to another one \(A_ 0\) and, when d is odd, the spectral flow of a periodic family of self-adjoint elliptic operators.
Reviewer: P.Cherrier

53C20 Global Riemannian geometry, including pinching
58J99 Partial differential equations on manifolds; differential operators
35J99 Elliptic equations and elliptic systems
Zbl 0547.00007