×

Lower bounds for the eigenvalues of the \(\operatorname{Spin}^c\) Dirac operator on manifolds with boundary. (Minorations des valeurs propres de l’opérateur de Dirac sur les variétés \(\operatorname{Spin}^c\) à bord.) (English. Abridged French version) Zbl 1387.58010

Summary: We extend the Friedrich inequality for the eigenvalues of the Dirac operator on \(\operatorname{Spin}^c\) manifolds with boundary under different boundary conditions. The limiting case is then studied and examples are given.

MSC:

58C40 Spectral theory; eigenvalue problems on manifolds
53C27 Spin and Spin\({}^c\) geometry
Full Text: DOI

References:

[1] Atiyah, M. F.; Patodi, V. K.; Singer, I. M., Spectral asymmetry and Riemannian geometry III, Math. Proc. Camb. Philos. Soc., 79, 71-99 (1976) · Zbl 0325.58015
[2] Chen, D., Eigenvalue estimates for the Dirac operator with generalized APS boundary condition, J. Geom. Phys., 57, 379-386 (2007) · Zbl 1108.58027
[3] Friedrich, T., Der erste Eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., 97, 117-146 (1980) · Zbl 0462.53027
[4] Friedrich, T., Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, vol. 25 (2000), American Mathematical Society · Zbl 0949.58032
[5] Grosse, N.; Nakad, R., Complex generalized Killing spinors on Riemannian \(Spin^c\) manifolds, Results Math., 67, 1, 177-195 (2015) · Zbl 1318.53046
[6] Herzlich, M.; Moroianu, Generalized Killing spinors and conformal eigenvalue estimates for \(Spin^c\) manifold, Ann. Global Anal. Geom., 17, 341-370 (1999) · Zbl 0988.53020
[7] Hijazi, O.; Montiel, S.; Zhang, X., Eigenvalues of the Dirac operator on manifolds with boundary, Comm. Math. Phys., 221, 255-265 (2001) · Zbl 0997.58015
[8] Hijazi, O.; Montiel, S.; Roldán, S., Eigenvalue boundary problems for the Dirac operator, Comm. Math. Phys., 231, 375-390 (2002) · Zbl 1018.58020
[9] Montiel, S., Unicity of constant mean curvature hypersurface in some Riemannian manifolds, Indiana Univ. Math. J., 48, 2, 711-748 (1999) · Zbl 0973.53048
[10] Nakad, R.; Roth, J., Hypersurfaces of \(Spin^c\) manifolds and Lawson type correspondence, Ann. Global Anal. Geom., 42, 3, 421-442 (2012) · Zbl 1258.53048
[11] Nakad, R.; Roth, J., The \(Spin^c\) Dirac operator on hypersurfaces and applications, Differ. Geom. Appl., 31, 1, 93-103 (2013) · Zbl 1262.58008
[12] Raulot, S., Optimal eigenvalues estimate for the Dirac operator on domains with boundary, Lett. Math. Phys., 73, 2, 135-145 (2005) · Zbl 1085.53040
[13] Torralbo, F., Compact minimal surfaces in the Berger spheres, Ann. Global Anal. Geom., 4, 41, 391-405 (2012) · Zbl 1242.53076
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.