## On the Riemannian Penrose inequality in dimensions less than eight.(English)Zbl 1168.53016

Authors’ abstract: The positive mass theorem states that a complete asymptotically flat manifold of nonnegative scalar curvature has nonnegative mass and that equality is achieved only for the Euclidean metric. The Riemannian Penrose inequality provides a sharp lower bound for the mass when black holes are present. More precisely, this lower bound is given in terms of the area of an outermost minimal hypersurface, and equality is achieved only for Schwarzschild metrics. The Riemannian Penrose inequality was first proved in three dimensions in 1997 by G. Huisken and T. Ilmanen for the case of a single black hole. In 1999, Bray extended this result to the general case of multiple black holes using a different technique. In this article, we extend the technique of to dimensions less than eight. Part of the argument is contained in a companion article by Lee. The equality case of the theorem requires the added assumption that the manifold be spin.

### MSC:

 53C20 Global Riemannian geometry, including pinching 53C80 Applications of global differential geometry to the sciences 83C25 Approximation procedures, weak fields in general relativity and gravitational theory 83C30 Asymptotic procedures (radiation, news functions, $$\mathcal{H}$$-spaces, etc.) in general relativity and gravitational theory 83C57 Black holes 83E15 Kaluza-Klein and other higher-dimensional theories
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### References:

 [1] R. Arnowitt, S. Deser, and C. W. Misner, Coordinate invariance and energy expressions in general relativity , Phys. Rev. (2) 122 (1961), 997–1006. · Zbl 0094.23003 [2] W. K. Allard, On the first variation of a varifold , Ann. of Math. (2) 95 (1972), 417–491. JSTOR: · Zbl 0252.49028 [3] F. Almgren, Optimal isoperimetric inequalities , Indiana Univ. Math. J. 35 (1986), 451–547. · Zbl 0585.49030 [4] R. Bartnik, The mass of an asymptotically flat manifold , Comm. Pure Appl. Math. 39 (1986), 661–693. · Zbl 0598.53045 [5] H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem , J. Differential Geom. 59 (2001), 177–267. · Zbl 1039.53034 [6] H. L. Bray and K. Iga, Superharmonic functions in $$\mathbfR^ n$$ and the Penrose inequality in general relativity , Comm. Anal. Geom. 10 (2002), 999–1016. · Zbl 1035.31002 [7] G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality , J. Differential Geom. 59 (2001), 353–437. · Zbl 1055.53052 [8] D. A. Lee, On the near-equality case of the positive mass theorem , Duke Math. J. 148 (2009), 63–80. · Zbl 1168.53018 [9] P. Miao, Positive mass theorem on manifolds admitting corners along a hypersurface , Adv. Theor. Math. Phys. 6 (2002), 1163–1182. [10] F. Morgan, Geometric Measure Theory: A Beginner’s Guide , 2nd ed., Academic Press, San Diego, 1995. · Zbl 0819.49024 [11] A. Neves, Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds , preprint,\arxiv0711.4335v1[math.DG] [12] R. Penrose, Naked singularities , Ann. N.Y. Acad. Sci. 224 (1973), 125–134. · Zbl 0925.53023 [13] R. M. Schoen, “Variational theory for the total scalar curvature functional for Riemannian metrics and related topics” in Topics in Calculus of Variations (Montecatini Terme, Italy, 1987) , Lecture Notes in Math. 1365 , Springer, Berlin, 1989, 120–154. · Zbl 0702.49038 [14] R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity , Comm. Math. Phys. 65 (1979), 45–76. · Zbl 0405.53045 [15] Y. Shi and L.-F. Tam, Positive mass theorem and the boundary behaviors of compact manifolds with nonnegative scalar curvature , J. Differential Geom. 62 (2002), 79–125. · Zbl 1071.53018 [16] X. Wang, The mass of asymptotically hyperbolic manifolds , J. Differential Geom. 57 (2001), 273–299. · Zbl 1037.53017 [17] E. Witten, A new proof of the positive energy theorem , Comm. Math. Phys. 80 (1981), 381–402. · Zbl 1051.83532
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