Estimation on the Spin${}^c$ twisted Dirac operators

Year 2022, Volume: 51 Issue: 6, 1621 - 1629, 01.12.2022

Serhan Eker

https://doi.org/10.15672/hujms.1054157

Abstract

We generalize the lower bound estimates for eigenvalues of the twisted Dirac operator on compact Riemannian Spin${}^c-$submanifold obtained by Roger Nakad and Julien Roth in (Archiv der Mathematik 104(5), 453-461, 2015).

Keywords

Spin geometry, Dirac operator, estimation of eigenvalues

References

• [1] C. Bär, Extrinsic Bounds for Eigenvalues of the Dirac Operator, Ann. Glob. Anal. Geom. 16 (6), 573–596, 1998.
• [2] S. Eker, Seiberg−Witten-like equations on the strictly pseudoconvex CR−3 manifolds, Miskolc Math. Notes, 20 (1), 233–43, 2019.
• [3] S. Eker, Lower Bound Eigenvalue Problems of the Compact Riemannian Spin- Submanifold Dirac Operator, Erzincan Üniversitesi Fen Bilimleri Enstitüsü Dergisi, (special issue I), 13, 56–62, 2020.
• [4] S. Eker, Lower Bounds for the Eigenvalues of the Dirac Operator on Spinc Manifolds, Iran. J. Sci. Technol. Trans. A Sci., 44, 251–257, 2020.
• [5] T. Friedrich, Dirac operators in Riemannian geometry, Graduate Studies in Mathematics, 25, Amer. Math. Soc., Providence, RI, 2000.
• [6] T. Friedrich and E.C. Kim, Some remarks on the Hijazi inequality and generalizations of the Killing equation for spinors, J. Geom. Phys. 37 (1-2), 1–14, 2001.
• [7] N. Ginoux and B. Morel, On eigenvalue estimates for the submanifold Dirac operator, Internat. J. Math. 13 (5), 553–548, 2002.
• [8] G. Habib, Energy-Momentum tensor on foliations, J. Geom. Phys. 57, 2234–2248, 2007.
• [9] M. Herzlich and A. Moroianu, Generalized Killing spinors and conformal eigenvalue estimates for Spinc manifolds, Ann. Global Anal. Geom. 17, 341–370, 1999.
• [10] O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104, 151–162, 1986.
• [11] O. Hijazi, Lower bounds for the eigenvalues of the Dirac operator, J. Geom. Phys, 16, 27–38, 1995.
• [12] O. Hijazi and X. Zhang, Lower bounds for the eigenvalues of the Dirac operator: part I. The hypersurface Dirac operator, Ann. Global Anal. Geom. 19 (4), 355–376, 2001.
• [13] O. Hijazi and X. Zhang, Lower Bounds for the Eigenvalues of the Dirac Operator: Part II. The Submanifold Dirac Operator, Ann. Global Anal. Geom. 20 (2), 163–181, 2001.
• [14] O. Hijazi, S. Montiel and X. Zhang, Eigenvalues of the Dirac Operator on Manifolds with Boundary, Comm. Math. Phys. 221 (2), 255–265, 2001.
• [15] H. Lawson and M. Michelsohn, Spin geometry, Princeton university press, 1989.
• [16] A. Lichnerowicz, Spineurs harmoniques, C.R. Acad. Sci. Paris Ser. AB, 257, 1963.
• [17] R. Nakad, Lower bounds for the eigenvalues of the Dirac operator on Spinc manifolds, J. Geom. Phys. 60 (10), 1634–1642, 2010.
• [18] R. Nakad and J. Roth, Lower bounds for the eigenvalues of the Spinc Dirac operator on submanifolds, Arch. Math. 104 (5), 453–461, 2015.
• [19] D. Salamon, Spin geometry adn Seiberg−Witten invariants, Citeseer, 1996.
• [20] E. Witten, Monopoles and four-manifolds, Math. Ress. Lett. 1 (6), 769–796, 1994.
• [21] X.Zhang, Lower bounds for eigenvalues of hypersurface Dirac operators, Math. Res. Lett. 5 (2), 199–210, 1998.