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 -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.

**[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), no. 5, 430–439. MR**1699261**, 10.2307/2589147**[HH]**Wu-teh Hsiang and Wu-Yi Hsiang,*On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I*, Invent. Math.**98**(1989), no. 1, 39–58. MR**1010154**, 10.1007/BF01388843**[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), no. 1, 37–47. MR**1156385**, 10.1007/BF02100598**[Mon]**Sebastián Montiel,*Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds*, Indiana Univ. Math. J.**48**(1999), no. 2, 711–748. MR**1722814**, 10.1512/iumj.1999.48.1562**[M]**Frank Morgan,*Geometric measure theory*, 2nd ed., Academic Press, Inc., San Diego, CA, 1995. A beginner’s guide. MR**1326605****[MHH]**Frank Morgan, Michael Hutchings, and Hugh Howards,*The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature*, Trans. Amer. Math. Soc.**352**(2000), no. 11, 4889–4909. MR**1661278**, 10.1090/S0002-9947-00-02482-X**[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), 1357-1394. CMP**2000:12****[Pen]**Roger Penrose,*Naked singularities*, Ann. New York Acad. Sci.**224**(1973), 125-134.**[Pet]**Peter Petersen,*Riemannian geometry*, Graduate Texts in Mathematics, vol. 171, Springer-Verlag, New York, 1998. MR**1480173****[R1]**Manuel Ritoré,*Applications of compactness results for harmonic maps to stable constant mean curvature surfaces*, Math. Z.**226**(1997), no. 3, 465–481. MR**1483543**, 10.1007/PL00004351**[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. Amer. Math. Soc.**348**(1996), no. 1, 391–410. MR**1322955**, 10.1090/S0002-9947-96-01496-1**[RR2]**Manuel Ritoré and Antonio Ros,*Stable constant mean curvature tori and the isoperimetric problem in three space forms*, Comment. Math. Helv.**67**(1992), no. 2, 293–305. MR**1161286**, 10.1007/BF02566501

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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/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