Oussama Hijazi

Eigenvalues of the Dirac Operator on Manifolds¶with Boundary

Abstract: Under standard local boundary conditions or certain global APS boundary conditions, we get lower bounds for the eigenvalues of the Dirac operator on compact spin manifolds with boundary. For the local boundary conditions, limiting cases are characterized by the existence of real Killing spinors and the minimality of the boundary.

Dirac Operators on Hypersurfaces of Manifolds with Negative Scalar Curvature

We give a sharp extrinsic lower bound for the first eigenvaluesof the intrinsic Dirac operator of certain hypersurfaces boundinga compact domain in a spin manifold of negative scalar curvature.Limiting-cases are characterized by the existence, on the domain,of imaginary Killing spinors. Some geometrical applications, as anAlexandrov type theorem, are given.

Lower Bounds for the Eigenvalues of the Dirac Operator: Part II. The Submanifold Dirac Operator

In this paper we generalize the results of Part I to the submanifoldDirac operator. In particular, we give optimal lower bounds for thesubmanifold Dirac operator in terms of the mean curvature and othergeometric invariants as the Yamabe number or the energy-momentum tensor.In the limiting case, we prove that the submanifold is Einstein if thenormal bundle is flat.

Lower Bounds for the Eigenvalues of the Dirac Operator: Part I. The Hypersurface Dirac Operator

We give optimal lower bounds for the hypersurface Diracoperator in terms of the Yamabe number, the energy-momentum tensor andthe mean curvature. In the limiting case, we prove that the hypersurfaceis an Einstein manifold with constant mean curvature.

A HOLOGRAPHIC PRINCIPLE FOR THE EXISTENCE OF PARALLEL SPINOR FIELDS AND AN INEQUALITY OF SHI-TAM TYPE

Suppose that

Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant

Sigma=partial M

Branson's Q-curvature in Riemannian and Spin Geometry ?

is the

A holographic principle for the existence of imaginary Killing spinors

n

A positive mass theorem for asymptotically hyperbolic manifolds with inner boundary

-dimensional boundary of a connected compact Riemannian spin manifold

Supersymmetric rigidity of asymptotically locally hyperbolic manifolds

( M,langle;,;rangle)