Lower Bounds for the Eigenvalues of the Dirac Operator on Spinc Manifolds

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Lower Bounds for the Eigenvalues of the Dirac Operator on Spinc Manifolds Roger NAKAD May 13, 2010 Institut Elie Cartan, Universite Henri Poincare, Nancy I, B.P 239 54506 Vandœuvre-Les-Nancy Cedex, France. Abstract In this paper, we extend the Hijazi inequality, involving the Energy-Momentum tensor, for the eigenvalues of the Dirac operator on Spinc manifolds without boundary. The limiting case is then studied and an example is given. Key words: Spinc structures, Dirac operator, eigenvalues, Energy-Momentum tensor, perturbed Yamabe operator, conformal geometry. 1 Introduction On a compact Riemannian spin manifold (Mn, g) of dimension n > 2, Th. Friedrich [6] showed that any eigenvalue ? of the Dirac operator satisfies ?2 > ?21 := n 4(n? 1) inf M Sg, (1) where Sg denotes the scalar curvature of M . The limiting case of (1) is characterized by the existence of a special spinor called real Killing spinor. This is a section ? of the spinor bundle satisfying for every X ? ?(TM), ?X? = ? ?1 n X · ?, 1

  • any eigenvalue

  • then any

  • line bundle has

  • killing spinor

  • spinc manifold

  • ?y ?

  • energy- momentum tensor

  • compact riemannian

  • q? defined


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LowerBoundsfortheEigenvaluesoftheDiracOperatoronSpincManifoldsRogerNAKADMay13,2010InstitutE´lieCartan,Universite´HenriPoincare´,NancyI,B.P23954506Vandœuvre-Le`s-NancyCedex,France.nakad@iecn.u-nancy.frAbstractInthispaper,weextendtheHijaziinequality,involvingtheEnergy-Momentumtensor,fortheeigenvaluesoftheDiracoperatoronSpincmanifoldswithoutboundary.Thelimitingcaseisthenstudiedandanexampleisgiven.Keywords:Spincstructures,Diracoperator,eigenvalues,Energy-Momentumtensor,perturbedYamabeoperator,conformalgeometry.1IntroductionOnacompactRiemannianspinmanifold(Mn,g)ofdimensionn>2,Th.Friedrich[6]showedthatanyeigenvalueλoftheDiracoperatorsatisfiesnλ2>λ12:=infSg,(1)4(n1)MwhereSgdenotesthescalarcurvatureofM.Thelimitingcaseof(1)ischaracterizedbytheexistenceofaspecialspinorcalledrealKillingspinor.ThisisasectionψofthespinorbundlesatisfyingforeveryXΓ(TM),λ1rXψ=Xψ,n1