Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A note on the Yamabe constant of an outermost minimal hypersurface


Author: Fernando Schwartz
Journal: Proc. Amer. Math. Soc. 138 (2010), 4103-4107
MSC (2010): Primary 53C21, 83C99
DOI: https://doi.org/10.1090/S0002-9939-2010-10445-8
Published electronically: May 27, 2010
MathSciNet review: 2679631
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] 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 (2004j:53046)
  • [2] 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
  • [3] Justin Corvino, A note on asymptotically flat metrics on $ {\mathbb{R}}^3$ which are scalar-flat and admit minimal spheres, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3669-3678 (electronic), 10.1090/S0002-9939-05-07926-8. MR 2163606 (2007a:53077)
  • [4] 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 (2009e:53087)
  • [5] 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, 10.1088/0264-9381/18/14/308. MR 1846368 (2002k:83051)
  • [6] 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 (2007i:53078)
  • [7] S. W. Hawking, Black holes in general relativity, Comm. Math. Phys. 25 (1972), 152-166. MR 0293962 (45:3037)
  • [8] 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 (2003h:53091)
  • [9] Richard Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, J. Differential Geom. 20 (1984), no. 2, 479-495. MR 788292 (86i:58137)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C21, 83C99

Retrieve articles in all journals with MSC (2010): 53C21, 83C99


Additional Information

Fernando Schwartz
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-0614
Email: fernando@math.utk.edu

DOI: https://doi.org/10.1090/S0002-9939-2010-10445-8
Received by editor(s): November 5, 2009
Received by editor(s) in revised form: February 8, 2010
Published electronically: May 27, 2010
Additional Notes: This work was partially supported by The Leverhulme Trust
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society