Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-26T01:08:30.501Z Has data issue: false hasContentIssue false
  • English
  • Français

A Weighted ${{L}^{2}}$-Estimate of the Witten Spinor in Asymptotically Schwarzschild Manifolds

Published online by Cambridge University Press:  20 November 2018

Felix Finster  and
Margarita Kraus
Felix Finster
Affiliation:
NWF I–Mathematik, Universität Regensburg, 93040 Regensburg, Germany email: [email protected], [email protected]
Margarita Kraus
Affiliation:
NWF I–Mathematik, Universität Regensburg, 93040 Regensburg, Germany email: [email protected], [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We derive a weighted ${{L}^{2}}$-estimate of the Witten spinor in a complete Riemannian spin manifold $({{M}^{n}},\,g)$ of non-negative scalar curvature which is asymptotically Schwarzschild. The interior geometry of $M$ enters this estimate only via the lowest eigenvalue of the square of the Dirac operator on a conformal compactification of $M$.

Keywords

83C6035Q7535J4558J05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 59 , Issue 5 , 01 October 2007 , pp. 943 - 965
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Ammann, B., The smallest Dirac eigenvalue in a spin-conformal class and cmc-immersions. http://arxiv.org/abs/math/0309061, 2003.Google Scholar
[2] Bär, C., Lower eigenvalue estimates for Dirac operators. Math. Ann. 293(1992), no. 1, 39–46.Google Scholar
[3] Bartnik, R., The mass of an asymptotically flat manifold. Commun. Pure Appl. Math. 39(1986), no. 5, 661–693.Google Scholar
[4] Bray, H. and Finster, F., Curvature estimates and the positive mass theorem. Comm. Anal. Geom. 10(2002), no. 2, 291–306.Google Scholar
[5] Finster, F. and Kath, I., Curvature estimates in asymptotically flat manifolds of positive scalar curvature. Comm. Anal. Geom. 10(2002), no. 5, 1017–1031.Google Scholar
[6] Finster, F. and Kraus, M., Curvature estimates in asymptotically flat Lorentzian manifolds. Canad. J. Math. 57(2005), no. 4, 708–723..Google Scholar
[7] Friedrich, T., Der erste Eigenwert des Dirac-Operators einer kompakten Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachrichten 97(1980), 117–146.Google Scholar
[8] Gibbons, G. W., Hawking, S. W., Horowitz, G. T., and Perry, M. J., Positive mass theorems for black holes. Comm. Math. Phys. 88(1983), no. 3, 295–308.Google Scholar
[9] Hebey, E., Sobolev Spaces on Riemannian Manifolds. Lecture Notes in Mathematics 1635, Springer-Verlag, Berlin, 1996.Google Scholar
[10] Herzlich, M., A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds. Comm. Math. Phys. 188(1997), no. 1, 121–133.Google Scholar
[11] Hijazi, O., A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors. Comm. Math. Phys. 104(1986), no. 1, 151–162.Google Scholar
[12] Herzlich, M., Twistor operators and eigenvalues of the Dirac operator. In: Quaternionic Structures in Mathematics and Physics. Int. Sch. Adv. Stud. (SISSA), Trieste, 1998, pp. 151–174.Google Scholar
[13] Hitchin, N., Harmonic spinors. Adv. in Math. 14(1974), 1–55.Google Scholar
[14] Kraus, M. and Tretter, C., A new method for eigenvalue estimates for Dirac operators on certain manifolds with Sk-symmetry. Differential Geom. Appl. 19(2003), no. 1, 1–14.Google Scholar
[15] Lawson, H.-B. and Michelsohn, M.-L., Spin Geometry. Princeton Mathematical Series 38, Princeton University Press, Princeton, NJ, 1989.Google Scholar
[16] Lott, J., Eigenvalue bounds for the Dirac operator. Pacific J. Math. 125(1986), no. 1, 117–126.Google Scholar
[17] Parker, T. and Taubes, C. H., On Witten's proof of the positive energy theorem. Comm. Math. Phys. 84(1982), 223–238.Google Scholar
[18] Schoen, R. and Yau, S.-T., On the proof of the positive mass conjecture in general relativity. Comm. Math. Phys. 65(1979), no. 1, 45–76.Google Scholar
[19] Schoen, R. and Yau, S.-T., Proof of the positive mass theorem. I. Comm. Math. Phys. 79(1981), no. 2, 231–360.Google Scholar
[20] Witten, E., A new proof of the positive energy theorem. Comm.Math. Phys. 80(1981), no. 3, 381–402.Google Scholar