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An isoperimetric comparison theorem for Schwarzschild space and other manifolds

Authors: Hubert Bray and Frank Morgan
Journal: Proc. Amer. Math. Soc. 130 (2002), 1467-1472
MSC (1991): Primary 53C42, 53A10, 49Q20, 83C57
Published electronically: December 20, 2001
MathSciNet review: 1879971
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Abstract: We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric $(n-1)$-spheres of a spherically symmetric $n$-manifold are isoperimetric hypersurfaces, meaning that they minimize $(n-1)$-dimensional area among hypersurfaces enclosing the same $n$-volume. This result greatly generalizes the result of Bray (Ph.D. thesis, 1997), which proved that the spherically symmetric 2-spheres of 3-dimensional Schwarzschild space (which is defined to be a totally geodesic, space-like slice of the usual $(3+1)$-dimensional Schwarzschild metric) are isoperimetric. We also note that this Schwarzschild result has applications to the Penrose inequality in general relativity, as described by Bray.

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Additional Information

Hubert Bray
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Frank Morgan
Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267

Keywords: Isoperimetric problem, Penrose inequality, Schwarzschild space
Received by editor(s): August 18, 2000
Received by editor(s) in revised form: November 14, 2000
Published electronically: December 20, 2001
Communicated by: Bennett Chow
Article copyright: © Copyright 2001 Hubert Bray and Frank Morgan