Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



In orbifolds, small isoperimetric regions are small balls

Author: Frank Morgan
Journal: Proc. Amer. Math. Soc. 137 (2009), 1997-2004
MSC (2000): Primary 49Q20
Published electronically: January 21, 2009
MathSciNet review: 2480281
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In a compact orbifold, for a small prescribed volume, an isoperimetric region is close to a small metric ball; in a Euclidean orbifold, it is a small metric ball.

References [Enhancements On Off] (What's this?)

  • [A] William K. Allard, The first variation of a varifold, Ann. of Math. (2) 95 (1972) 417-491. MR 0307015 (46:6136)
  • [Alm] Frederick J. Almgren, Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Memoirs Amer. Math. Soc. 4, No. 165, 1976. MR 0420406 (54:8420)
  • [Dr] Olivier Druet, Sharp local isoperimetric inequalities involving the scalar curvature, Proc. Amer. Math. Soc. 130 (2002) 2351-2361. MR 1897460 (2003b:53036)
  • [F] Herbert Federer, The singular sets of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension, Bull. Amer. Math. Soc. 76 (1970) 767-771. MR 0260981 (41:5601)
  • [HHM] Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999) 430-439. MR 1699261 (2000i:52027)
  • [K] B. Kleiner, cited by Per Tomter, Constant mean curvature surfaces in the Heisenberg group, Proc. Symp. Pure Math. 54, Part 1, Amer. Math. Soc., Providence, RI, 1993, 485-495. MR 1216601 (94a:53098)
  • [M1] Frank Morgan, Geometric Measure Theory: A Beginner's Guide, Academic Press, 4th ed., San Diego, CA, 2008. MR 1775760 (2001j:49001)
  • [M2] Frank Morgan, In polytopes, small balls about some vertex minimize perimeter, J. Geom. Anal. 17 (2007) 97-106. MR 2302876 (2007k:49090)
  • [M3] Frank Morgan, Regularity of area-minimizing surfaces in $ 3$D polytopes and of invariant surfaces in $ \rn$, J. Geom. Anal. 15 (2005) 321-341. MR 2152485 (2006b:53008)
  • [M4] Frank Morgan, Regularity of isoperimetric hypersurfaces in Riemannian manifolds, Trans. Amer. Math. Soc. 355 (2003) 5041-5052. MR 1997594 (2004j:49066)
  • [MJ] Frank Morgan and David L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, Indiana U. Math J. 49 (2000) 1017-1041. MR 1803220 (2002e:53043)
  • [MR] Frank Morgan and Manuel Ritoré, Isoperimetric regions in cones, Trans. Amer. Math. Soc. 354 (2002) 2327-2339. MR 1885654 (2003a:53089)
  • [Ros] Antonio Ros, The isoperimetric problem, Global Theory of Minimal Surfaces (Proc. Clay Research Institution Summer School, 2001, David Hoffman, editor), Amer. Math. Soc., Providence, RI, 2005. MR 2167260 (2006e:53023)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 49Q20

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

Additional Information

Frank Morgan
Affiliation: Department of Mathematics and Statistics, Williams College, Williamstown, Massachusetts 01267

Keywords: Isoperimetric, perimeter minimizing, orbifold
Received by editor(s): March 19, 2008
Published electronically: January 21, 2009
Communicated by: Richard A. Wentworth
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society