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Transactions of the American Mathematical Society

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Regularity of isoperimetric hypersurfaces in Riemannian manifolds

Author: Frank Morgan
Journal: Trans. Amer. Math. Soc. 355 (2003), 5041-5052
MSC (2000): Primary 49Q20
Published electronically: July 28, 2003
MathSciNet review: 1997594
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Abstract: We add to the literature the well-known fact that an isoperimetric hypersurface $S$ of dimension at most six in a smooth Riemannian manifold $M$ is a smooth submanifold. If the metric is merely Lipschitz, then $S$ is still Hölder differentiable.

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  • [A] F. J. Almgren Jr., Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints, Mem. Amer. Math. Soc. 4 (1976), no. 165, viii+199. MR 0420406
  • [B] Enrico Bombieri, Regularity theory for almost minimal currents, Arch. Rational Mech. Anal. 78 (1982), no. 2, 99–130. MR 648941, 10.1007/BF00250836
  • [CW] Ya-Zhe Chen and Lan-Cheng Wu, Second order elliptic equations and elliptic systems, Translations of Mathematical Monographs, vol. 174, American Mathematical Society, Providence, RI, 1998. Translated from the 1991 Chinese original by Bei Hu. MR 1616087
  • [Co] Andrew Cotton, David Freeman, Andrei Gnepp, Ting Ng, John Spivack, and Cara Yoder, The isoperimetric problem on some singular surfaces, J. Austral. Math. Soc., to appear.
  • [E] Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
  • [F1] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [F2] 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, 10.1090/S0002-9904-1970-12542-3
  • [G] Mariano Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Annals of Mathematics Studies, vol. 105, Princeton University Press, Princeton, NJ, 1983. MR 717034
  • [GT] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190
  • [Gi] Enrico Giusti, Minimal surfaces and functions of bounded variation, Monographs in Mathematics, vol. 80, Birkhäuser Verlag, Basel, 1984. MR 775682
  • [GMT] E. Gonzalez, U. Massari, and I. Tamanini, On the regularity of boundaries of sets minimizing perimeter with a volume constraint, Indiana Univ. Math. J. 32 (1983), no. 1, 25–37. MR 684753, 10.1512/iumj.1983.32.32003
  • [Gr] Misha Gromov, Isoperimetry of waists and concentration of maps, IHES, 2002,$\sim$gromov/topics/topic11.html.
  • [H] E. Hopf, Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus, Sitzungberichte der Preussischen Akademie der Wissenshaften zu Berlin, Physikalischen-Mathematische Klasse 19 (1927), 147-152.
  • [LU] Olga A. Ladyzhenskaya and Nina N. Ural′tseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. MR 0244627
  • [LM] Gary Lawlor and Frank Morgan, Curvy slicing proves that triple junctions locally minimize area, J. Differential Geom. 44 (1996), no. 3, 514–528. MR 1431003
  • [M] Frank Morgan, Geometric measure theory, 3rd ed., Academic Press, Inc., San Diego, CA, 2000. A beginner’s guide. MR 1775760
  • [My1] Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
  • [My2] C. B. Morrey Jr., Second-order elliptic systems of differential equations, Contributions to the theory of partial differential equations, Annals of Mathematics Studies, no. 33, Princeton University Press, Princeton, N. J., 1954, pp. 101–159. MR 0068091
  • [R] Antonio Ros, The isoperimetric problem, Proc. Clay Research Institution Summer School, 2001, to appear.
  • [SS] Richard Schoen and Leon Simon, A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals, Indiana Univ. Math. J. 31 (1982), no. 3, 415–434. MR 652826, 10.1512/iumj.1982.31.31035
  • [Se] James Serrin, On the strong maximum principle for quasilinear second order differential inequalities, J. Functional Analysis 5 (1970), 184–193. MR 0259328
  • [Si] 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
  • [Sis] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 0233295
  • [T1] Jean E. Taylor, The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces, Ann. of Math. (2) 103 (1976), no. 3, 489–539. MR 0428181
  • [T2] Jean E. Taylor, The structure of singularities in solutions to ellipsoidal variational problems with constraints in 𝑅³, Ann. of Math. (2) 103 (1976), no. 3, 541–546. MR 0428182
  • [W] Brian White, Existence of smooth embedded surfaces of prescribed genus that minimize parametric even elliptic functionals on 3-manifolds, J. Differential Geom. 33 (1991), no. 2, 413–443. MR 1094464

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Additional Information

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

Keywords: Isoperimetric hypersurface, area-minimizing, fixed volume, regularity, Lipschitz metric, constant mean curvature
Received by editor(s): December 12, 2001
Received by editor(s) in revised form: March 27, 2002, and October 18, 2002
Published electronically: July 28, 2003
Article copyright: © Copyright 2003 by the author