Some Estimates for the Spin−Submanifold Twisted Dirac Operators

Year 2020, Volume: 5 Issue: 1, 8 - 15, 30.03.2020

Mehmet Ergen

Abstract

In this paper, we generalize lower bound estimates for the eigenvalue estimates of the submanifold twisted Dirac operator on a compact Riemannian Spin−submanifold proved by N. Ginoux and B. Morel in 2002.

Keywords

Spin and Spinc geometry, geometry, Dirac operator, Estimation of eigenvalues

References

• Bar, C. (1998). Extrinsic Bounds for Eigenvalues of the Dirac Operator. Ann. Glob. Anal. Geom., 16(6), 573–596.
• Eker, S., Degirmenci, N. (2018). Seiberg −Witten−like equations without Self−Duality on odd dimensional manifolds. Journal of Partial Differential Equations, 31(4), 291–303.
• Eker, S. (2019). The Bochner vanishing theorems on the conformal killing vector fields, TWMS Journal of Applied and Engineering Mathematics, 9(1) Special Issue, 114–120.
• Eker, S . (2020). Lower Bound Eigenvalue Problems of the Compact Riemannian Spin-Submanifold Dirac Operator. Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 13 (OZEL SAYI I) , 56–62.
• Eker, S. (2020). Lower Bounds for the Eigenvalues of the Dirac Operator on Spinc Manifolds. Iranian Journal of Science and Technology, Transactions A: Science, 44(1), 251—257.
• Friedrich, T. (1980). Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Mathematische Nachrichten, 97(1), 117–146.
• Friedrich, T. (2001). Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors. Journal of Geometry and Physics, 37(1-2), 1–14.
• Ginoux, N., Bertrand, M. (2002). On eigenvalue estimates for the submanifold Dirac operator. International Journal of Mathematics, 13(05), 533-548.
• Hijazi, O. (1986). A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Communications in Mathematical Physics, 104(1), 151–162.
• Hijazi, O. (1991). Premiere 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. (1995). Lower bounds for the eigenvalues of the Dirac operator. Journal of Geometry and Physics, 16(1), 27–38.
• Hijazi, O., Xiao, Z. (2001). Lower Bounds for the Eigenvalues of the Dirac Operator: Part II. The Submanifold Dirac Operator. Annals of Global Analysis and Geometry, 20(2), 163–181.
• Hijazi, O., Xiao, Z. (2001). Lower Bounds for the Eigenvalues of the Dirac Operator: Part II. The Submanifold Dirac Operator. Annals of Global Analysis and Geometry, 20(2), 163–181.
• Hijazi, O., Montiel, S., Xiao, Z. (2001). Eigenvalues of the Dirac Operator on Manifolds with Boundary. Communications in Mathematical Physics, 221(2), 255–265.
• Lawson, H.B., Michelsohn, M.L. (2016). Spin geometry. Princeton university press, 38.
• Lichnerowicz, O. (1963). Spineurs harmoniques. C.R. Acad. Sci. Paris Ser. A–B, 257.
• Morel, B. (2001). Eigenvalue estimates for the Dirac–Schrödinger operators. Journal of Geometry and Physics, 38(1), 1–18.
• Adriana, T. (2009). On the twisted Dirac operators. Bull. Math. Soc. Sci. Math. Roumanie Tome, 52(3), 383–386.
• Xiao, Z. (1998). Lower bounds for eigenvalues of hypersurface Dirac operators. Mathematical Research Letters, 5(2), 199–210.
• Witten, E. (1981). A new proof of the positive energy theorem. Communications in Mathematical Physics, 80(3), 381–402.