Isoperimetric regions in cones
Authors:
Frank Morgan and Manuel Ritoré
Journal:
Trans. Amer. Math. Soc. 354 (2002), 23272339
MSC (2000):
Primary 53C42; Secondary 49Q20
Published electronically:
February 12, 2002
MathSciNet review:
1885654
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: We consider cones and prove that if the Ricci curvature of is nonnegative, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions.
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 [A]
 W. K. Allard, On the first variation of a varifold, Ann. of Math., 95 (1972) 417491. MR 46:6136
 [BdC]
 J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984) 339353. MR 85k:58021c
 [BdCE]
 J. L. Barbosa, M. do Carmo and J. Eschenburg, Stability of hypersurfaces with constant mean curvature in Riemannian manifolds, Math. Z., 197 (1988) 123138. MR 88m:53109
 [BM]
 P. Bérard, D. Meyer, Inégalités isopérimétriques et applications, Ann. Scient. Éc. Norm. Sup. (4), 15 (1982) 513542. MR 84h:58147
 [Br]
 H. Bray, The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, Ph. D. Thesis, Stanford University, 1997.
 [BrM]
 H. Bray, F. Morgan, An isoperimetric comparison theorem for Schwarzchild space and other manifolds, Proc. Amer. Math. Soc., to appear.
 [CE]
 J. Cao and J. F. Escobar, A new 3dimensional curvature integral formula for PLmanifolds of nonpositive curvature, preprint, 2000.
 [C]
 I. Chavel, Riemannian geometry: a modern introduction, Cambridge Tracts in Mathematics, no. 108, Cambridge University Press, 1993. MR 95j:53001
 [CFG]
 A. Cotton, D. Freeman, A. Gnepp, T. Ng, J. Spivack, C. Yoder (Williams College NSF ``SMALL'' undergraduate research Geometry Groups 1998, 2000), The isoperimetric problem on singular surfaces, preprint (2000).
 [F]
 H. Federer, Geometric measure theory, Grundlehren Math. Wissen. 153, SpringerVerlag, New York, 1969. MR 41:1976
 [GNY]
 A. Gnepp, T. F. Ng, C. Yoder, Isoperimetric domains on polyhedra and singular surfaces, NSF ``SMALL'' undergraduate research Geometry Group report, Williams College, 1998.
 [HHM]
 H. Howards, M. Hutchings, F. Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly, 106, no. 5, (1999) 430439. MR 2000i:52027
 [Mo]
 S. Montiel, Unicity of constant mean curvature hypersurfaces in foliated Riemannian manifolds, Indiana Univ. Math. J., 48, no. 2, (1999) 711748. MR 2001f:53131
 [M1]
 F. Morgan, Geometric measure theory: a beginner's guide. Third edition, Academic Press, 2000. MR 2001j:49001
 [M2]
 F. Morgan, Areaminimizing surfaces in cones, Comm. Anal. Geom., to appear.
 [MJ]
 F. Morgan, D. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana Univ. Math. J., 49 (2000) 10171041.
 [ON]
 B. O'Neill, SemiRiemmanian geometry, Academic Press, New York, 1983. MR 85f:53002
 [P]
 R. Pedrosa, The isoperimetric problem in spherical cylinders, preprint, 2002.
 [PR]
 R. Pedrosa, M. 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. MR 2001k:53120
 [R1]
 M. Ritoré, Applications of compactness results for harmonic maps to stable constant mean curvature surfaces, Math. Z., 226 (1997) 465481. MR 98m:53082
 [RR]
 M. Ritoré, A. Ros, Stable constant mean curvature tori and the isoperimetric problem in three space forms, Comment. Math. Helv., 67 (1992) 293305. MR 93a:53055
 [SS]
 R. Schoen, L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure App. Math., 34 (1981) 741797. MR 82k:49054
 [S]
 L. Simon, Lectures on geometric measure theory, Proc. Centre Math. Anal. 3, Australian National University, 1983. MR 87a:49001
 [SZ]
 P. Sternberg, K. Zumbrun, On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint, Comm. Anal. Geom., 7, no. 1, (1999) 199220. MR 2000d:49062
 [T]
 Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., 117 (1965) 251275. MR 30:4229
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Additional Information
Frank Morgan
Affiliation:
Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email:
Frank.Morgan@williams.edu
Manuel Ritoré
Affiliation:
Departamento de Geometría y Topología, Universidad de Granada, E–18071 Granada, España
Email:
ritore@ugr.es
DOI:
http://dx.doi.org/10.1090/S0002994702029835
PII:
S 00029947(02)029835
Received by editor(s):
May 23, 2001
Received by editor(s) in revised form:
November 1, 2001
Published electronically:
February 12, 2002
Article copyright:
© Copyright 2002
by the authors
