An isoperimetric comparison theorem for Schwarzschild space and other manifolds

Bray, Hubert; Morgan, Frank

doi:10.1090/S0002-9939-01-06186-X

An isoperimetric comparison theorem for Schwarzschild space and other manifolds
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by Hubert Bray and Frank Morgan

Proc. Amer. Math. Soc. 130 (2002), 1467-1472

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

Published electronically: December 20, 2001

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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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