Lower Bound Eigenvalue Problems of the Compact Riemannian Spin-Submanifold Dirac Operator

Year 2020, Volume: 13 Issue: ÖZEL SAYI I, 56 - 62, 28.02.2020

Serhan Eker

https://doi.org/10.18185/erzifbed.605220

Cited By: 5

Abstract

In
this paper, we construct two modified spinorial LeviCivita connection based on the EnergyMomentum tensor and its trace to bring an optimal
lower bound to the eigenvalues of the compact Riemannian Spinsubmanifold Dirac operator in terms of the the
EnergyMomentum tensor and its trace.  Then, we extend these estimates in terms of
the Yamabe number and the area of the submanifold under the conformal change of
the metric.

Keywords

Spin and 〖Spin〗^c geometry, Dirac operator

Kompakt Riemannian Spin-Alt manifold Dirac Operatörünün Alt Sınır Özdeğer Problemleri

Abstract

Bu makalede, Kompakt Riemannian Spin altmanifold Dirac operatörünün
özdeğerlerine EnergyMomentum tensörü ve onun izine bağlı olarak optimal bir alt sınır getirmek
için spinorial LeviCivita konneksiyonunu EnergyMomentum tensörü ve onun izine bağlı olarak deforme ettik.  Daha sonra bu tahminleri konformal değişim
metriği altında Yamabe sayısı ve altmanifoldun alanına bağlı olarak
genişlettik.

Keywords

Spin and 〖Spin〗^c geometry, Dirac operator, Estimation of eigenvalues

References

• Friedrich T. (2000). “Dirac operators in Riemannian geometry”, American Mathematical Society, 25.
• Friedrich T., Kim E.C. (2001). “Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors”, J. Geom. Phys., 37 (1-2), 1-14.
• Hijazi O. (1986). “A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors”, Comm.Math. Phys., 104, 151–162.
• Hijazi O. (1995). “Lower bounds for the eigenvalues of the Dirac operator”, J. Geom. Phys. 16, 27-38.
• Hijazi O. (1991). “Première valeur propre de l’opérateur de Dirac et nombre de Yamabe”, Comptes rendus de l’Académie, des sciences. Série 1, Mathématique, 313 (12), 865–868.
• Hijazi O., Zhang X. (2001). “ Lower bounds for the eigenvalues of the Dirac operator: part I. The hypersurface Dirac operator”, Ann. Global Anal. Geom., 19 (4), 355–376.
• Hijazi O., Zhang X. (2001). “ Lower bounds for the eigenvalues of the Dirac operator: part II. The Submanifold Dirac operator”, Ann. Global Anal. Geom. 20 (2), 163–181.
• Hijazi O., Montiel S., Zhang X. (2001). “Eigenvalues of the Dirac Operator on Manifolds with Boundary”, Comm. Math. Phys., 221 (2), 255–265.
• Lawson H., Michelsohn M. (1989). “Spin Geometry”, Princeton university press.
• Nakad R., Roth J. (2013). “The Spin ^c Dirac operator on hypersurfaces and applications”, Differential Geom. Appl. 31 (1), 93-103 .
• Lichnerowicz A. (1963). “ “Spineurs harmoniques”, C.R. Acad. Sci. Paris Ser. AB, 257, 1963.
• Naber G.L. (1997). “Topology, geometry, and gauge fields”, Springer, New York.
• Zhang X. (1998). “ Lower bounds for eigenvalues hypersurface Dirac operators, Math. Res. Lett., 5 (2), 199–210. Zhang X. (1998) “ Angular momentum and positive mass theorem”, Comm. Math. Phys., 206 (1), 137–155.