1999, English, Article edition: A Short Proof Of Eigenvalue Estimates For The Dirac Operator On Riemannian And Kähler Manifolds Uwe Semmelmann

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  • 1999-11-04
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  • English

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Title
  • A Short Proof Of Eigenvalue Estimates For The Dirac Operator On Riemannian And Kähler Manifolds
Author
  • Uwe Semmelmann
Other Authors
  • The Pennsylvania State University CiteSeer Archives
Published
  • unknown
  • 1999-11-04
Physical Description
  • ps
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Notes
  • . Aim of this note is to present short and natural proofs for the lower bounds of the spectrum of the Dirac operator on compact Riemannian and Kahler manifolds with positive scalar curvature. Keywords: Dirac operator, eigenvalue, lower bound 1991 MSC: 53A50, 53C25, 58G25, 58G30 1. Introduction For Riemannian manifolds a lower bound of the spectrum of the Dirac operator was first given by Th. Friedrich (c.f.: [Fri80]). The corresponding result in the Kahler case is due to K.-D. Kirchberg (c.f.: [Kir86] and [Kir90] ). A modified proof was given by O. Hijazi (c.f. [Hij94]). The idea of the proof in this note is to estimate the norm of the covariant derivative of an eigenspinor using the direct sum decomposition of the tensor product of spinor bundle and cotangent bundle. To achieve this we use the explicit formula for the embedding of the spinor bundle S into the tensor product SOmega TM (with modifications for Kahler manifolds). 2. Riemannian manifolds Let (M n ; g) be a compact R...
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