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

DOI:
https://doi.org/10.1090/S0002-9939-01-06186-X

Published electronically:
December 20, 2001

MathSciNet review:
1879971

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Abstract | References | Similar Articles | Additional Information

Abstract: We give a very general isoperimetric comparison theorem which, as an important special case, gives hypotheses under which the spherically symmetric -spheres of a spherically symmetric -manifold are isoperimetric hypersurfaces, meaning that they minimize -dimensional area among hypersurfaces enclosing the same -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 -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

Email:
bray@math.mit.edu

**Frank Morgan**

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

Email:
Frank.Morgan@williams.edu

DOI:
https://doi.org/10.1090/S0002-9939-01-06186-X

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