Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

   
 
 

 

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
DOI: https://doi.org/10.1090/S0002-9947-02-02983-5
Published electronically: February 12, 2002
MathSciNet review: 1885654
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider cones $C = 0\, \times{\kern-10.5pt}\times \,M^n$ 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 [Enhancements On Off] (What's this?)

  • [A] W. K. Allard, On the first variation of a varifold, Ann. of Math., 95 (1972) 417-491. MR 46:6136
  • [BdC] J. L. Barbosa and M. do Carmo, Stability of hypersurfaces with constant mean curvature, Math. Z. 185 (1984) 339-353. 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) 123-138. MR 88m:53109
  • [BM] P. Bérard, D. Meyer, Inégalités isopérimétriques et applications, Ann. Scient. Éc. Norm. Sup. (4), 15 (1982) 513-542. 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 3-dimensional curvature integral formula for PL-manifolds 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, Springer-Verlag, 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) 430-439. MR 2000i:52027
  • [Mo] S. Montiel, Unicity of constant mean curvature hypersurfaces in foliated Riemannian manifolds, Indiana Univ. Math. J., 48, no. 2, (1999) 711-748. MR 2001f:53131
  • [M1] F. Morgan, Geometric measure theory: a beginner's guide. Third edition, Academic Press, 2000. MR 2001j:49001
  • [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] B. O'Neill, Semi-Riemmanian 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) 1357-1394. MR 2001k:53120
  • [R1] M. Ritoré, Applications of compactness results for harmonic maps to stable constant mean curvature surfaces, Math. Z., 226 (1997) 465-481. 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) 293-305. MR 93a:53055
  • [SS] R. Schoen, L. Simon, Regularity of stable minimal hypersurfaces, Comm. Pure App. Math., 34 (1981) 741-797. 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) 199-220. MR 2000d:49062
  • [T] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., 117 (1965) 251-275. MR 30:4229

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53C42, 49Q20

Retrieve articles in all journals with MSC (2000): 53C42, 49Q20


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

American Mathematical Society