Mass endomorphism, surgery and perturbations
[Endomorphisme de masse, chirurgie et perturbations]
Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 467-487.

Nous montrons que l’endomorphisme de masse associé à l’opérateur de Dirac sur une variété riemannienne est non nul pour une métrique générique. La preuve s’appuie sur l’étude du comportement par chirurgie de l’endomorphisme de masse, de son comportement au voisinage d’une métrique possédant des spineurs harmoniques et par des arguments de perturbations analytiques.

We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.

DOI : 10.5802/aif.2855
Classification : 53C27, 57R65, 58J05, 58J60
Keywords: Dirac operator, mass endomorphism, surgery.
Mot clés : Opérateur de Dirac, endomorphisme de masse, chirurgie.
Ammann, Bernd 1 ; Dahl, Mattias 2 ; Hermann, Andreas 3 ; Humbert, Emmanuel 3

1 Fakultät für Mathematik Universität Regensburg 93040 Regensburg Germany
2 Institutionen för Matematik Kungliga Tekniska Högskolan 100 44 Stockholm Sweden
3 Laboratoire de Mathématiques et Physique Théorique, Université de Tours, Parc de Grandmont, 37200 Tours, France
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     title = {Mass endomorphism, surgery and~perturbations},
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Ammann, Bernd; Dahl, Mattias; Hermann, Andreas; Humbert, Emmanuel. Mass endomorphism, surgery and perturbations. Annales de l'Institut Fourier, Tome 64 (2014) no. 2, pp. 467-487. doi : 10.5802/aif.2855. http://www.numdam.org/articles/10.5802/aif.2855/

[1] Ammann, B. A spin-conformal lower bound of the first positive Dirac eigenvalue, Diff. Geom. Appl., Volume 18 (2003), pp. 21-32 | DOI | MR | Zbl

[2] Ammann, B. The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions, Comm. Anal. Geom., Volume 17 (2009), pp. 429-479 | DOI | MR | Zbl

[3] Ammann, B. A variational problem in conformal spin geometry, Habilitationsschrift, Universität Hamburg, 2003

[4] Ammann, B.; Dahl, M.; Humbert, E. Surgery and harmonic spinors, Adv. Math., Volume 220 (2009), pp. 523-539 | DOI | MR | Zbl

[5] Ammann, B.; Dahl, M.; Humbert, E. Harmonic spinors and local deformations of the metric, Comm. Anal. Geom., Volume 18 (2011), pp. 927-936 | MR | Zbl

[6] Ammann, B.; Grosjean, J.-F.; Humbert, E.; Morel, B. A spinorial analogue of Aubin’s inequality, Math. Z., Volume 260 (2008), pp. 127-151 | DOI | MR | Zbl

[7] Ammann, B.; Humbert, E.; Morel, B. Mass endomorphism and spinorial Yamabe type problems, Comm. Anal. Geom., Volume 14 (2006), pp. 163-182 | DOI | MR | Zbl

[8] Bär, C.; Dahl, M. Surgery and the Spectrum of the Dirac Operator, J. reine angew. Math., Volume 552 (2002), pp. 53-76 | MR | Zbl

[9] Beig, R.; Murchadha, N. Ó Trapped surfaces due to concentration of gravitational radiation, Phys. Rev. Lett., Volume 66 (1991) no. 19, pp. 2421-2424 | DOI | MR | Zbl

[10] Bourguignon, J.-P.; Gauduchon, P. Spineurs, opérateurs de Dirac et variations de métriques, Comm. Math. Phys., Volume 144 (1992), pp. 581-599 | DOI | MR | Zbl

[11] Friedrich, T. Dirac Operators in Riemannian Geometry, Graduate Studies in Mathematics, 25, AMS, Providence, Rhode Island, 2000 | MR | Zbl

[12] Hermann, A. Generic metrics and the mass endomorphism on spin 3-manifolds, Ann. Glob. Anal. Geom., Volume 37 (2010), pp. 163-171 | DOI | MR | Zbl

[13] Hermann, A. Dirac eigenspinors for generic metrics, Universität Regensburg (2012) (Ph. D. Thesis arXiv:1201.5771)

[14] Hijazi, O Première valeur propre de l’opérateur de Dirac et nombre de Yamabe, C. R. Acad. Sci. Paris, Série I, Volume 313 (1991), pp. 865-868 | MR | Zbl

[15] Kato, T. Perturbation theory for linear operators, Grundlehren der mathematischen Wissenschaften, 132, Springer-Verlag, 1966 | MR | Zbl

[16] Lawson, H. B.; Michelsohn, M.-L. Spin geometry, Princeton University Press, Princeton, 1989 | MR | Zbl

[17] Lee, J. M.; Parker, T. H. The Yamabe problem, Bull. Am. Math. Soc., New Ser., Volume 17 (1987), pp. 37-91 | DOI | MR | Zbl

[18] Maier, S. Generic metrics and connections on spin- and spin c -manifolds, Comm. Math. Phys., Volume 188 (1997), pp. 407-437 | DOI | MR | Zbl

[19] Schoen, R. Conformal deformation of a Riemannian metric to constant scalar curvature, J. Diff. Geom., Volume 20 (1984), pp. 479-495 | MR | Zbl

[20] Stolz, S. Simply connected manifolds of positive scalar curvature, Ann. of Math. (2), Volume 136 (1992) no. 3, pp. 511-540 | DOI | MR | Zbl

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