## Duke Mathematical Journal

### On the Riemannian Penrose inequality in dimensions less than eight

#### 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 (see [HI]). In 1999, Bray extended this result to the general case of multiple black holes using a different technique (see [Br]). In this article, we extend the technique of [Br] to dimensions less than eight. Part of the argument is contained in a companion article by Lee [L]. The equality case of the theorem requires the added assumption that the manifold be spin

#### Article information

Source
Duke Math. J. Volume 148, Number 1 (2009), 81-106.

Dates
First available in Project Euclid: 22 April 2009

http://projecteuclid.org/euclid.dmj/1240432192

Digital Object Identifier
doi:10.1215/00127094-2009-020

Mathematical Reviews number (MathSciNet)
MR2515101

Zentralblatt MATH identifier
05555676

Subjects
Secondary: 83C57: Black holes

#### Citation

Bray, Hubert L.; Lee, Dan A. On the Riemannian Penrose inequality in dimensions less than eight. Duke Math. J. 148 (2009), no. 1, 81--106. doi:10.1215/00127094-2009-020. http://projecteuclid.org/euclid.dmj/1240432192.

#### References

• R. Arnowitt, S. Deser, and C. W. Misner, Coordinate invariance and energy expressions in general relativity, Phys. Rev. (2) 122 (1961), 997--1006.
• W. K. Allard, On the first variation of a varifold, Ann. of Math. (2) 95 (1972), 417--491.
• F. Almgren, Optimal isoperimetric inequalities, Indiana Univ. Math. J. 35 (1986), 451--547.
• R. Bartnik, The mass of an asymptotically flat manifold, Comm. Pure Appl. Math. 39 (1986), 661--693.
• H. L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), 177--267.
• H. L. Bray and K. Iga, Superharmonic functions in $\mathbfR\sp n$ and the Penrose inequality in general relativity, Comm. Anal. Geom. 10 (2002), 999--1016.
• G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), 353--437.
• D. A. Lee, On the near-equality case of the positive mass theorem, Duke Math. J. 148 (2009), 63--80.
• P. Miao, Positive mass theorem on manifolds admitting corners along a hypersurface, Adv. Theor. Math. Phys. 6 (2002), 1163--1182.
• F. Morgan, Geometric Measure Theory: A Beginner's Guide, 2nd ed., Academic Press, San Diego, 1995.
• A. Neves, Insufficient convergence of inverse mean curvature flow on asymptotically hyperbolic manifolds, preprint,\arxiv0711.4335v1[math.DG]
• R. Penrose, Naked singularities, Ann. N.Y. Acad. Sci. 224 (1973), 125--134.
• 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.
• R. Schoen and S. T. Yau, On the proof of the positive mass conjecture in general relativity, Comm. Math. Phys. 65 (1979), 45--76.
• 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.
• X. Wang, The mass of asymptotically hyperbolic manifolds, J. Differential Geom. 57 (2001), 273--299.
• E. Witten, A new proof of the positive energy theorem, Comm. Math. Phys. 80 (1981), 381--402.