A note on the Yamabe constant of an outermost minimal hypersurface

Schwartz, Fernando

doi:10.1090/S0002-9939-2010-10445-8

A note on the Yamabe constant of an outermost minimal hypersurface
HTML articles powered by AMS MathViewer

by Fernando Schwartz

Proc. Amer. Math. Soc. 138 (2010), 4103-4107

DOI: https://doi.org/10.1090/S0002-9939-2010-10445-8

Published electronically: May 27, 2010

PDF | Request permission

Abstract:

Using an elementary argument we find an upper bound on the Yamabe constant of the outermost minimal hypersurface of an asymptotically flat manifold with nonnegative scalar curvature that satisfies the Riemannian Penrose Inequality. Provided the manifold satisfies the Riemannian Penrose Inequality with rigidity, we show that equality holds in the inequality if and only if the manifold is the Riemannian Schwarzschild manifold.

References

Hubert L. Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Differential Geom. 59 (2001), no. 2, 177–267. MR 1908823
Hubert L. Bray and Dan A. Lee, On the Riemannian Penrose inequality in dimensions less than eight, Duke Math. J. 148 (2009), no. 1, 81–106. MR 2515101, DOI 10.1215/00127094-2009-020
Justin Corvino, A note on asymptotically flat metrics on ${\Bbb R}^3$ which are scalar-flat and admit minimal spheres, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3669–3678. MR 2163606, DOI 10.1090/S0002-9939-05-07926-8
Gregory J. Galloway, Rigidity of marginally trapped surfaces and the topology of black holes, Comm. Anal. Geom. 16 (2008), no. 1, 217–229. MR 2411473
Mingliang Cai and Gregory J. Galloway, On the topology and area of higher-dimensional black holes, Classical Quantum Gravity 18 (2001), no. 14, 2707–2718. MR 1846368, DOI 10.1088/0264-9381/18/14/308
Gregory J. Galloway and Richard Schoen, A generalization of Hawking’s black hole topology theorem to higher dimensions, Comm. Math. Phys. 266 (2006), no. 2, 571–576. MR 2238889, DOI 10.1007/s00220-006-0019-z
S. W. Hawking, Black holes in general relativity, Comm. Math. Phys. 25 (1972), 152–166. MR 293962
Gerhard Huisken and Tom Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Differential Geom. 59 (2001), no. 3, 353–437. MR 1916951
Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479–495. MR 788292