Lower bounds for the eigenvalues of the Spin<sup><i>c</i></sup> Dirac operator on manifolds with boundary

Roger Nakad; Julien Roth

doi:10.1016/j.crma.2015.12.017

Differential geometry

Lower bounds for the eigenvalues of the Spin^c Dirac operator on manifolds with boundary
[Minorations des valeurs propres de l'opérateur de Dirac sur les variétés Spin^c à bord]

Présenté par : the Editorial Board

Roger Nakad ¹ ; Julien Roth ²

¹ Notre Dame University-Louaize, Faculty of Natural and Applied Sciences, Department of Mathematics and Statistics, P. O. Box 72, Zouk Mikael, Lebanon
² LAMA, Université Paris-Est Marne-la-Vallée, Cité Descartes, Champs-sur-Marne, 77454 Marne-la-Vallée cedex 2, France

Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 425-431.

Résumé
Abstract

Nous étendons l'inégalité de Friedrich pour les valeurs propres de l'opérateur de Dirac sur les variétés ${Spin}^{c}$ à bord pour différentes conditions à bord. Le cas limite est étudié et des exemples sont donnés.

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

Reçu le : 2014-10-30
Accepté le : 2015-12-16
Publié le : 2016-02-11

DOI : 10.1016/j.crma.2015.12.017

Affiliations des auteurs :

Roger Nakad ¹ ; Julien Roth ²

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     author = {Roger Nakad and Julien Roth},
     title = {Lower bounds for the eigenvalues of the {Spin\protect\textsuperscript{\protect\emph{c}}} {Dirac} operator on manifolds with boundary},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {425--431},
     publisher = {Elsevier},
     volume = {354},
     number = {4},
     year = {2016},
     doi = {10.1016/j.crma.2015.12.017},
     language = {en},
}

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%A Julien Roth
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Roger Nakad; Julien Roth. Lower bounds for the eigenvalues of the Spin^c Dirac operator on manifolds with boundary. Comptes Rendus. Mathématique, Volume 354 (2016) no. 4, pp. 425-431. doi : 10.1016/j.crma.2015.12.017. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.12.017/

Bibliographie
Cité par

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[2] D. Chen Eigenvalue estimates for the Dirac operator with generalized APS boundary condition, J. Geom. Phys., Volume 57 (2007), pp. 379-386

[3] T. Friedrich Der erste Eigenwert des Dirac-operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung, Math. Nachr., Volume 97 (1980), pp. 117-146

[4] T. Friedrich Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, vol. 25, American Mathematical Society, 2000

[5] N. Grosse; R. Nakad Complex generalized Killing spinors on Riemannian ${Spin}^{c}$ manifolds, Results Math., Volume 67 (2015) no. 1, pp. 177-195

[6] M. Herzlich; Moroianu Generalized Killing spinors and conformal eigenvalue estimates for ${Spin}^{c}$ manifold, Ann. Global Anal. Geom., Volume 17 (1999), pp. 341-370

[7] O. Hijazi; S. Montiel; X. Zhang Eigenvalues of the Dirac operator on manifolds with boundary, Comm. Math. Phys., Volume 221 (2001), pp. 255-265

[8] O. Hijazi; S. Montiel; S. Roldán Eigenvalue boundary problems for the Dirac operator, Comm. Math. Phys., Volume 231 (2002), pp. 375-390

[9] S. Montiel Unicity of constant mean curvature hypersurface in some Riemannian manifolds, Indiana Univ. Math. J., Volume 48 (1999) no. 2, pp. 711-748

[10] R. Nakad; J. Roth Hypersurfaces of ${Spin}^{c}$ manifolds and Lawson type correspondence, Ann. Global Anal. Geom., Volume 42 (2012) no. 3, pp. 421-442

[11] R. Nakad; J. Roth The ${Spin}^{c}$ Dirac operator on hypersurfaces and applications, Differ. Geom. Appl., Volume 31 (2013) no. 1, pp. 93-103

[12] S. Raulot Optimal eigenvalues estimate for the Dirac operator on domains with boundary, Lett. Math. Phys., Volume 73 (2005) no. 2, pp. 135-145

[13] F. Torralbo Compact minimal surfaces in the Berger spheres, Ann. Global Anal. Geom., Volume 4 (2012) no. 41, pp. 391-405

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