Isoperimetric regions in cones

Authors:
Frank Morgan and Manuel Ritoré

Journal:
Trans. Amer. Math. Soc. **354** (2002), 2327-2339

MSC (2000):
Primary 53C42; Secondary 49Q20

Published electronically:
February 12, 2002

MathSciNet review:
1885654

Full-text 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.

**[A]**William K. Allard,*On the first variation of a varifold*, Ann. of Math. (2)**95**(1972), 417–491. MR**0307015****[BdC]**João Lucas Barbosa and Manfredo do Carmo,*Stability of hypersurfaces with constant mean curvature*, Math. Z.**185**(1984), no. 3, 339–353. MR**731682**, 10.1007/BF01215045**[BdCE]**J. Lucas Barbosa, Manfredo do Carmo, and Jost Eschenburg,*Stability of hypersurfaces of constant mean curvature in Riemannian manifolds*, Math. Z.**197**(1988), no. 1, 123–138. MR**917854**, 10.1007/BF01161634**[BM]**Pierre Bérard and Daniel Meyer,*Inégalités isopérimétriques et applications*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 3, 513–541 (French). MR**690651****[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 3-dimensional curvature integral formula for PL-manifolds of nonpositive curvature*, preprint, 2000.**[C]**Isaac Chavel,*Riemannian geometry—a modern introduction*, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR**1271141****[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]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[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]**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**[Mo]**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**[M1]**Frank Morgan,*Geometric measure theory*, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR**1775760****[M2]**F. Morgan,*Area-minimizing 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) 1017-1041.**[ON]**Barrett O’Neill,*Semi-Riemannian geometry*, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR**719023****[P]**R. Pedrosa,*The isoperimetric problem in spherical cylinders*, preprint, 2002.**[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), no. 4, 1357–1394. MR**1757077**, 10.1512/iumj.1999.48.1614**[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**[RR]**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**[SS]**Richard Schoen and Leon Simon,*Regularity of stable minimal hypersurfaces*, Comm. Pure Appl. Math.**34**(1981), no. 6, 741–797. MR**634285**, 10.1002/cpa.3160340603**[S]**Leon Simon,*Lectures on geometric measure theory*, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR**756417****[SZ]**Peter Sternberg and Kevin Zumbrun,*On the connectivity of boundaries of sets minimizing perimeter subject to a volume constraint*, Comm. Anal. Geom.**7**(1999), no. 1, 199–220. MR**1674097**, 10.4310/CAG.1999.v7.n1.a7**[T]**Yoshihiro Tashiro,*Complete Riemannian manifolds and some vector fields*, Trans. Amer. Math. Soc.**117**(1965), 251–275. MR**0174022**, 10.1090/S0002-9947-1965-0174022-6

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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:
https://doi.org/10.1090/S0002-9947-02-02983-5

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