Abstract
In this paper, we obtain lower bounds for the Brown-York quasilocal mass and the Bartnik quasilocal mass for compact three manifolds with smooth boundaries. As a consequence, we derive sufficient conditions for the existence of horizons for a certain class of compact manifolds with boundary and some asymptotically flat complete manifolds. The method is based on analyzing Hawking mass and inverse mean curvature flow.
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References
Bartnik R. (1989). New definition of quasilocal mass. Phys. Rev. Lett. 62: 2346–2348
Bartnik, R.: Energy in general relativity. Tsing Hua lectures on geometry and analysis (Hsinchu, 1990–1991), Cambridge, MA: Internat. Press 1997, pp. 5–27
Bartnik, R.: Mass and 3-metrics of non-negative scalar curvature. In: Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Beijing: Higher Ed. Press 2002, pp. 231–240
Bray H.L. (2001). Proof of the Riemannian Penrose inequality using the positive mass theorem. J. Differ. Geom. 59: 177–267
Beig R.Ó. and Murchadha N. (1991). Trapped surfaces due to concentration of gravitational radiation. Phys. Rev. Lett. 66: 2421–2424
Brown, J.D., York, J.W.: Quasilocal energy in general relativity in Mathematical aspects of classical field theory (Seattle, WA, 1991), Contemp. Math. 132, Providence, RI: Amer. Math. Soc., 1992, pp. 129–142
Brown J.D. and York J.W. (1993). Quasilocal energy and conserved charges derived from the gravitational action. Phys. Rev. D (3) 47(4): 1407–1419
Christodoulou, D., Yau, S.T.: Some remarks on quasi-local mass. In: Contemporary Mathematics 71, Mathematics and General Relativity, J. Isenberg, ed. Providence RI: Amer. Math. Soc., 1988, pp. 9–14
Corvino J. and Schoen R.M. (2006). On the asymptotics for the vacuum Einstein constraint equations. J. Differ. Geom. 73: 185–217
Flanagan E. (1991). Hoop conjecture for black-hole horizon formation. Phys. Rev. D 44: 2409
Galloway G.J. (1993). On the topology of black holes. Commun. Math. Phys. 151: 53–66
Giusti, E.: Minimal surfaces and functions of bounded variation. Notes on pure mathematics 10, Canberra: Department of Pure Mathematics, 1977
Hirsch M.W. (1976). Differential Topology. Springer-Verlag, New York
Huisken G. and Ilmanen T. (2001). The inverse mean curvature flow and the Riemannian Penrose inequality. J. Differ. Geom. 59: 353–437
Klingenberg W. (1978). A course in differential geometry. Graduate Texts in Mathematics, Vol. 51.. Springer-Verlag, New York-Heidelberg
Meeks W.H., Simon L. and Yau S.-T. (1982). Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature. Ann. of Math. 116: 621–659
Meeks W.H. and Yau S.-T. (1980). Topology of three-dimensional manifolds and the embedding problems in minimal surface theory. Ann. of Math. 112: 441–484
Miao P. (2002). Positive mass theorem on manifolds admitting corners along a hypersurface. Adv. Theor. Math. Phys. 6(6): 1163–1182
Miao P. (2004). Asymptotically flat and scalar flat metrics on \({\mathbb{R}}^3\) admitting a horizon. Proc. Amer. Math. Soc. 132: 217–222 (electronic)
Miao P. (2005). A remark on boundary effects in static vacuum initial data sets. Classical Quantum Gravity 22(11): L53–L59
Nirenberg L. (1953). The Weyl and Minkowski problems in differential geometry in the large. Comm. Pure Appl. Math. 6: 337–394
Schoen R. and Yau S.-T. (1993). The existence of a black hole due to condensation of matter. Commun. Math. Phys. 90: 575–579
Shi Y.-G. and Tam L.-F. (2002). Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature. J. Differ. Geom. 62: 79–125
Spivak, M.: A Comprehensive introduction to differential geometry. v. 3, Berkeley, CA: Publish or Perish, 1970–75
Willmore T.J. (1982). Total curvature in Riemannian geometry. E. Horwood, Chichester
Yau, S.-T.: Geometry of three manifolds and existence of black hole due to boundary effect. Adv. Theor. Math. Phys. 5(2001)
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Communicated by G.W. Gibbons
Research Partially Supported by 973 Program (2006CB805905) and Fok YingTong Education Foundation.
Research partially supported by Earmarked Grant of Hong Kong #CUHK403005.
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Shi, Y., Tam, LF. Quasi-Local Mass and the Existence of Horizons. Commun. Math. Phys. 274, 277–295 (2007). https://doi.org/10.1007/s00220-007-0273-8
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DOI: https://doi.org/10.1007/s00220-007-0273-8