Opérateurs de Dirac sur les sous-variétés

by Nicolas Ginoux

PhD thesis, Université Henri Poincaré - Nancy 1

Defended on September 10, 2002 - Certificate handed on March 3rd, 2003

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Abstract: In this Ph.D. 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.

Nicolas Ginoux, 17/07/2017