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Opérateurs de Dirac sur les sous-variétés

Abstract : In this thesis, we study the spectrum of two Dirac operators defined on a submanifold. First, we prove a lower bound for an operator which is canonically associated with the Dirac-Witten's operator. We then show that equality holds in these inequalities only if the submanifold admits a "twisted Killing" spinor. On the other hand, we give extrinsic upper bounds for the smallest eigenvalues of the Dirac operator on the submanifold twisted with its normal bundle. Completing C. Bär's work for hypersurfaces of the hyperbolic space, we obtain new estimates for hypersurfaces of manifolds admitting twistor-spinors. We finally extend these results to submanifolds of some particular Kählerian manifolds. The existence of Kählerian Killing spinors on such manifolds yields new eigenvalue estimates for CR-submanifolds. As a consequence, we obtain a comparison theorem for the eigenvalues of Dirac operators between Kählerian submanifolds of the complex projective space.
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Contributor : Nicolas Ginoux <>
Submitted on : Friday, October 10, 2003 - 9:26:12 AM
Last modification on : Friday, February 26, 2021 - 3:21:58 AM
Long-term archiving on: : Monday, September 20, 2010 - 11:31:16 AM


  • HAL Id : tel-01746710, version 3



Nicolas Ginoux. Opérateurs de Dirac sur les sous-variétés. Mathématiques [math]. Université Henri Poincaré - Nancy 1, 2002. Français. ⟨NNT : 2002NAN10047⟩. ⟨tel-01746710v3⟩



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