An isoperimetric comparison theorem for Schwarzschild space and other manifolds
Authors:
Hubert Bray and Frank Morgan
Journal:
Proc. Amer. Math. Soc. 130 (2002), 14671472
MSC (1991):
Primary 53C42, 53A10, 49Q20, 83C57
Published electronically:
December 20, 2001
MathSciNet review:
1879971
Fulltext PDF Free Access
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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 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 2spheres of 3dimensional Schwarzschild space (which is defined to be a totally geodesic, spacelike 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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Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), 13571394. CMP 2000:12
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 [B1]
 Hubert Bray, The Penrose conjecture in general relativity and volume comparison theorems involving scalar curvature, Ph.D. dissertation, Stanford Univ., 1997.
 [B2]
 Hubert Bray, Proof of the Riemannian Penrose inequality using the positive mass theorem, J. Diff. Geom. (to appear).
 [HHM]
 Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), 430439. MR 2000i:52027
 [HH]
 W.T. Hsiang and W.Y. Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces, Inv. Math. 85 (1989), 3958. MR 90h:53078
 [HI]
 G. Huisken and T. Ilmanen, The inverse mean curvature flow and the Riemannian Penrose inequality, J. Diff. Geom. (to appear).
 [K]
 Bruce Kleiner, An isoperimetric comparison theorem, Invent. Math. 108 (1992), 3747. MR 92m:53056
 [Mon]
 Sebastián Montiel, Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds, Indiana U. Math. J. 48 (1999), 711748. MR 2001f:53131
 [M]
 Frank Morgan, Geometric Measure Theory: a Beginner's Guide, Academic Press, second edition, 1995, third edition, 2000. MR 96c:49001 (review of second edition)
 [MHH]
 Frank Morgan, Michael Hutchings, and Hugh Howards, The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature, Trans. AMS 352 (2000), 48894909. MR 2001b:58024
 [MR]
 Frank Morgan and Manuel Ritoré, Isoperimetric regions in cones, Trans. AMS (to appear).
 [P]
 Renato H. L. Pedrosa, The isoperimetric problem in spherical cylinders, preprint (1998).
 [PR]
 Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), 13571394. CMP 2000:12
 [Pen]
 Roger Penrose, Naked singularities, Ann. New York Acad. Sci. 224 (1973), 125134.
 [Pet]
 Peter Petersen, Riemannian Geometry, Springer, 1998. MR 98m:53001
 [R1]
 Manuel Ritoré, Applications of compactness results for harmonic maps to stable constant mean curvature surfaces, Math. Z. 226 (1997), 465481. MR 98m:53082
 [R2]
 Manuel Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces, Comm. Anal. Geom. (to appear).
 [RR1]
 Manuel Ritoré and Antonio Ros, The spaces of index one minimal surfaces and stable constant mean curvature surfaces embedded in flat three manifolds, Trans. AMS 258 (1996), 391410. MR 96f:58038
 [RR2]
 Manuel Ritoré and Antonio Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comm. Math. Helv. 67 (1992), 293305. MR 93a:53055
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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:
http://dx.doi.org/10.1090/S000299390106186X
PII:
S 00029939(01)06186X
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
