## Isoperimetric regions in cones

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- by Frank Morgan and Manuel Ritoré PDF
- Trans. Amer. Math. Soc.
**354**(2002), 2327-2339

## Abstract:

We consider cones $C = 0 \smashtimes M^n$ and prove that if the Ricci curvature of $C$ is nonnegative, then geodesic balls about the vertex minimize perimeter for given volume. If strict inequality holds, then they are the only stable regions.## References

- William K. Allard,
*On the first variation of a varifold*, Ann. of Math. (2)**95**(1972), 417–491. MR**307015**, DOI 10.2307/1970868 - Hiroshi Mori,
*On surfaces of right helicoid type in $H^{3}$*, Bol. Soc. Brasil. Mat.**13**(1982), no. 2, 57–62. MR**735120**, DOI 10.1007/BF02584676 - 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**, DOI 10.1007/BF01161634 - Pierre Bérard and Daniel Meyer,
*Inégalités isopérimétriques et applications*, Ann. Sci. École Norm. Sup. (4)**15**(1982), no. 3, 513–541 (French). MR**690651** - H. Bray,
*The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature*, Ph. D. Thesis, Stanford University, 1997. - H. Bray, F. Morgan,
*An isoperimetric comparison theorem for Schwarzchild space and other manifolds*, Proc. Amer. Math. Soc., to appear. - J. Cao and J. F. Escobar,
*A new 3-dimensional curvature integral formula for PL-manifolds of nonpositive curvature*, preprint, 2000. - Isaac Chavel,
*Riemannian geometry—a modern introduction*, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR**1271141** - 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). - Herbert Federer,
*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR**0257325** - A. Gnepp, T. F. Ng, C. Yoder,
*Isoperimetric domains on polyhedra and singular surfaces*, NSF “SMALL” undergraduate research Geometry Group report, Williams College, 1998. - Hugh Howards, Michael Hutchings, and Frank Morgan,
*The isoperimetric problem on surfaces*, Amer. Math. Monthly**106**(1999), no. 5, 430–439. MR**1699261**, DOI 10.2307/2589147 - Sebastián Montiel,
*Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds*, Indiana Univ. Math. J.**48**(1999), no. 2, 711–748. MR**1722814**, DOI 10.1512/iumj.1999.48.1562 - Frank Morgan,
*Geometric measure theory*, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR**1775760** - F. Morgan,
*Area-minimizing surfaces in cones*, Comm. Anal. Geom., to appear. - F. Morgan, D. Johnson,
*Some sharp isoperimetric theorems for Riemannian manifolds*, Indiana Univ. Math. J.,**49**(2000) 1017–1041. - 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** - R. Pedrosa,
*The isoperimetric problem in spherical cylinders*, preprint, 2002. - 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**, DOI 10.1512/iumj.1999.48.1614 - 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**, DOI 10.1007/PL00004351 - 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**, DOI 10.1007/BF02566501 - Richard Schoen and Leon Simon,
*Regularity of stable minimal hypersurfaces*, Comm. Pure Appl. Math.**34**(1981), no. 6, 741–797. MR**634285**, DOI 10.1002/cpa.3160340603 - 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** - 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**, DOI 10.4310/CAG.1999.v7.n1.a7 - Yoshihiro Tashiro,
*Complete Riemannian manifolds and some vector fields*, Trans. Amer. Math. Soc.**117**(1965), 251–275. MR**174022**, DOI 10.1090/S0002-9947-1965-0174022-6

## 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
- Received by editor(s): May 23, 2001
- Received by editor(s) in revised form: November 1, 2001
- Published electronically: February 12, 2002
- © Copyright 2002 by the authors
- Journal: Trans. Amer. Math. Soc.
**354**(2002), 2327-2339 - MSC (2000): Primary 53C42; Secondary 49Q20
- DOI: https://doi.org/10.1090/S0002-9947-02-02983-5
- MathSciNet review: 1885654