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

Roger Nakad

Description

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

Sujets

• Hijazi script
• Poincaré
• Cartan
• Killing spinor

Informations

• Publié par: pefav
• Nombre de lectures: 23
• Langue: English

Extrait

LowerBoundsfortheEigenvaluesof theDiracOperatoronSpin c Manifolds RogerNAKAD May13,2010 InstitutE ´ lieCartan,Universite´HenriPoincare´,NancyI,B.P239 54506Vandœuvre-Le`s-NancyCedex,France. [e-mail] Abstract Inthispaper,weextendtheHijaziinequality,involvingtheEnergy-Momentum tensor,fortheeigenvaluesoftheDiracoperatoronSpin c manifoldswithout boundary.Thelimitingcaseisthenstudiedandanexampleisgiven. Keywords:Spin c structures,Diracoperator,eigenvalues,Energy-Momentum tensor,perturbedYamabeoperator,conformalgeometry. 1Introduction OnacompactRiemannianspinmanifold(M n ,g)ofdimensionn>2,Th. Friedrich[6]showedthatanyeigenvalueλoftheDiracoperatorsatisfies n λ 2 >λ 1 2 :=infS g ,(1) 4(n − 1) M whereS g denotesthescalarcurvatureofM.Thelimitingcaseof(1)is characterizedbytheexistenceofaspecialspinorcalledrealKillingspinor. ThisisasectionψofthespinorbundlesatisfyingforeveryX ∈ Γ(TM), λ 1 r X ψ= − X ∙ ψ, n 1