No Access

A positive mass theorem for asymptotically hyperbolic manifolds with inner boundary

Oussama Hijazi

Institut Élie Cartan de Lorraine, Université de Lorraine, Nancy, B.P. 239, 54506 Vandœuvre-Lès-Nancy Cedex, France

E-mail Address: Oussama.Hijazi@univ-lorraine.fr

Search for more papers by this author

Sebastián Montiel

Departamento de Geometría y Topología Universidad de Granada, 18071 Granada, Spain

E-mail Address: smontiel@ugr.es

Search for more papers by this author

, and

Simon Raulot

Laboratoire de Mathématiques R. Salem, UMR 6085, CNRS-Université de Rouen Avenue de l’Université, B.P. 12, Technopôle du Madrillet 76801 Saint-Étienne-du-Rouvray, France

E-mail Address: simon.raulot@univ-rouen.fr

Search for more papers by this author

https://doi.org/10.1142/S0129167X15501013Cited by:4 (Source: Crossref)

Abstract

In this paper, we prove an optimal Positive Mass theorem for Asymptotically Hyperbolic spin manifolds with compact inner boundary. This improves a previous result of Chruściel and Herzlich [The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math.212(2) (2003) 231–264].

Keywords:

AMSC: 53C27, 53C40, 53C80, 58G25

References

1. L. Andersson, M. Cai and C. J. Galloway, Rigidity and positivity of mass for asymptotically hyperbolic manifolds, Ann. Henri Poincaré 9 (2009) 1–33. Crossref, Web of Science, Google Scholar
2. L. Andersson and M. Dahl, Scalar curvature rigidity for asymptotically locally hyperbolic manifolds, Ann. Global Anal. Geom. 16 (1998) 1–27. Crossref, Web of Science, Google Scholar
3. M. F. Atiyah, V. K. Patodi and I. M. Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43–69. Crossref, Web of Science, Google Scholar
4. C. Bär, Lower eigenvalues estimates for Dirac operators, Math. Ann. 293 (1992) 39–46. Crossref, Web of Science, Google Scholar
5. R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986) 661–693. Crossref, Web of Science, Google Scholar
6. R. Bartnik and P. Chruściel, Boundary value problems for Dirac-type equations, J. Reine Angew. Math. 579 (2005) 13–73. Crossref, Web of Science, Google Scholar
7. R. Bartnik and J. Isenberg, The constraint equations, preprint (2004), arXiv:gr-qc/0405092v1. Google Scholar
8. H. Baum, Complete Riemannian manifolds with imaginary Killing spinors, Ann. Global Anal. Geom. 7 (1989) 205–226. Crossref, Google Scholar
9. J.-P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu and S. Moroianu, A Spinorial Approach to Riemannian and Conformal Geometry, Monographs in Mathematics (European Mathematical Society, 2015), 462 pp. Crossref, Google Scholar
10. H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59(2) (2001) 177–267. Crossref, Web of Science, Google Scholar
11. H. L. Bray and D. A. Lee, On the Riemannian Penrose inequality in dimension less than eight, Duke Math. J. 148(1) (2009) 81–106. Crossref, Web of Science, Google Scholar
12. P. Chruściel and M. Herzlich, The mass of asymptotically hyperbolic Riemannian manifolds, Pacific J. Math 212(2) (2003) 231–264. Crossref, Web of Science, Google Scholar
13. J. Cortier, A family of asymptotically hyperbolic manifolds with arbitrary energy-momentum vector, J. Math. Phys. 53 (2012) 102504. Crossref, Web of Science, Google Scholar
14. M. Dahl, R. Gicquaud and A. Sakovich, Penrose type inequalities for asymptotically hyperbolic graphs, Ann. Henri Poincaré 14(5) (2013) 1135–1168. Crossref, Web of Science, Google Scholar
15. L. L. de Lima and F. Gir ao, An Alexandrov–Frenchel-type inequality in hyperbolic space with an application to a Penrose inequality, preprint (2012), arXiv:1209.0438v3. Google Scholar
16. G. W. Gibbons, S. W. Hawking, G. T. Horowitz and M. J. Perry, Positive mass theorems for black holes, Comm. Math. Phys. 88 (1983) 295–308. Crossref, Web of Science, Google Scholar
17. N. Ginoux, The Dirac Spectrum, Lecture Notes in Mathematics, Vol. 1976 (Springer, 2009). Crossref, Google Scholar
18. M. Herzlich, A Penrose-like inequality for the mass of Riemannian asymptotically flat manifolds, Comm. Math. Phys. 188(1) (1997) 121–133. Crossref, Web of Science, Google Scholar
19. M. Herzlich, Minimal surfaces, the Dirac operator and the Penrose inequality, Sémin. Théor. Spectr. Géom. 20 (2002) 9–16. Google Scholar
20. O. Hijazi, A conformal lower bound for the smallest eigenvalue of the Dirac operator and Killing spinors, Comm. Math. Phys. 104 (1986) 121–162. Crossref, Web of Science, Google Scholar
21. O. Hijazi, Première valeur propre de l’opérateur de Dirac et nombre de Yamabe, C. R. Acad. Sci. Paris 313 (1991) 865–868. Google Scholar
22. O. Hijazi, S. Montiel and A. Roldán, Dirac operator on hypersurfaces of manifolds with negative scalar curvature, Ann. Global Anal. Geom. 23 (2003) 247–264. Crossref, Web of Science, Google Scholar
23. G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59(3) (2001) 353–467. Crossref, Web of Science, Google Scholar
24. M. A. Khuri, A Penrose-like inequality with charge, Gen. Relativ. Gravit. 45(11) (2013) 2341–2361. Crossref, Web of Science, Google Scholar
25. M.-K. G. Lam, The graph cases of the Riemannian positive mass and Penrose inequalities in all dimension, preprint (2010), arXiv:1010.4256. Google Scholar
26. H. B. Lawson and M. L. Michelsohn, Spin Geometry, Princeton Mathematical Series, Vol. 38 (Princeton University Press, 1989). Google Scholar
27. M. Mars, Present status of the Penrose inequality, Class. Quantum Grav. 26(1) (2009) 193001. Crossref, Web of Science, Google Scholar
28. R. Penrose, Naked singularities, Ann. New York Acad. Sci. 224 (1973) 125–134. Crossref, Web of Science, Google Scholar
29. R. Schoen and S.-T. Yau, On the proof of the positive mass conjecture in General Relativity, Comm. Math. Phys. 65(1) (1979) 45–76. Crossref, Web of Science, Google Scholar
30. R. Schoen and S.-T. Yau, Proof of the positive mass theorem II, Comm. Math. Phys. 79(2) (1981) 231–260. Crossref, Web of Science, Google Scholar
31. X. Wang, The mass of asymptotically hyperbolic manifold, J. Differential Geom. 57(2) (2001) 273–199. Crossref, Web of Science, Google Scholar
32. E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981) 381–402. Crossref, Web of Science, Google Scholar