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Extrinsic estimates for eigenvalues of the Dirac operator

“…We first compare the Dirac operator on M with the one on M. We will use notations in [3]. We also refer the reader to [6,15,17,18,19,20] and the references therein. Basic facts concerning Clifford algebras and spinor representations can be found in classical books [4,23].…”

“…For lower bounds estimates of submanifold Dirac operators, Hijazi and Zhang in [19,20] proved that for D H ϕ = λ H ϕ, ϕ ∈ Γ(ΣM)| M , it holds:…”

“…Under some extra conditions on the extrinsic curvature, they also obtained some lower bound in terms of the Yamabe constant, the curvature and volume of M (see [20] for details).…”

The Dirac operator on locally reducible Riemannian manifolds

“…Using the hypersurface Dirac operator, Witten [47] proved the positive energy conjecture for arbitrary demension. Motivated by Witten's work, Zhang [49,50] considered the lower bounds for the hypersurface Dirac operators and some related results later are generalized to submanifold Dirac operators by Hijazi and Zhang [34,35], where they gave optimal lower bounds for the submanifold Dirac operator with respect to the mean curvature and other geometric invaraints, for example, the Yamabe number or the energy momentum tensor under certain extra conditions. Ginoux and Morel [30] also investigated the eigenvalue problem for submanifold Dirac operators.…”

Eigenvalue Estimates of the spincDirac Operator and Harmonic Forms on Kähler-Einstein Manifolds

“…Wang [38]). Useful geometric information has been also obtained by restricting parallel and Killing spinors to hypersurfaces [3,13,14,15,16,17]. O. Hijazi proved that the Clifford multiplication between a harmonic k-form β (k = 0, n) and a Killing spinor vanishes.…”