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Transactions of the American Mathematical Society

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The isoperimetric problem on surfaces of revolution of decreasing Gauss curvature

Authors: Frank Morgan, Michael Hutchings and Hugh Howards
Journal: Trans. Amer. Math. Soc. 352 (2000), 4889-4909
MSC (2000): Primary 53Cxx, 53Axx, 49Qxx
Published electronically: July 12, 2000
MathSciNet review: 1661278
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Abstract | References | Similar Articles | Additional Information


We prove that the least-perimeter way to enclose prescribed area in the plane with smooth, rotationally symmetric, complete metric of nonincreasing Gauss curvature consists of one or two circles, bounding a disc, the complement of a disc, or an annulus. We also provide a new isoperimetric inequality in general surfaces with boundary.

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  • [And] Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Diff. Geom. 43 (1996), 207-230. MR 97m:58045
  • [BaP] Christophe Bavard and Pierre Pansu, Sur le volume minimal de $\mathbb{R}^2$, Ann. Sci. École Norm. Sup. (4) 19 (1986), 479-490. MR 88b:53048
  • [BeC] Itai Benjamini and Jinguo Cao, A new isoperimetric theorem for surfaces of variable curvature, Duke Math. J. 85 (1996), 359-396. MR 97m:58046
  • [Bér] Pierre Bérard, Spectral Geometry: Direct and Inverse Problems, Lecture Notes in Math., No. 1207, Springer-Verlag, New York, 1986. MR 88f:58146
  • [Bow] B. H. Bowditch, The minimal volume of the plane, J. Austral. Math. Soc. 55 (1993), 23-40. MR 94i:53034
  • [Br] Hubert L. Bray, The Penrose inequality in general relativity and volume comparison theorems involving scalar curvature, Ph.D. Dissertation, Stanford Univ., August, 1997.
  • [Ch1] Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, 1984. MR 86g:58140
  • [Ch2] Isaac Chavel, Riemannian Geometry--A Modern Introduction, Cambridge Univ. Press, 1993. MR 95j:53001
  • [Fed] Herbert Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969. MR 41:1976
  • [HM] Joel Hass and Frank Morgan, Geodesics and soap bubbles in surfaces, Math. Z. 223 (1996), 185-196. MR 97j:53009
  • [HH] Erhard Heinz and Stefan Hildebrandt, Some remarks on minimal surfaces in Riemannian manifolds, Comm. Pure Appl. Math. 23 (1970), 371-377. MR 41:4398
  • [HHM] Hugh Howards, Michael Hutchings, and Frank Morgan, The isoperimetric problem on surfaces, Amer. Math. Monthly 106 (1999), 430-439. CMP 99:15
  • [K] Bruce Kleiner, An isoperimetric comparison theorem, Invent. Math. 108 (1992), 37-47. MR 92m:53056
  • [M1] Frank Morgan, Geometric Measure Theory: a Beginner's Guide, 2nd ed., Academic Press, 1995, 3rd ed., 2000. MR 96c:49001
  • [M2] Frank Morgan, Riemannian Geometry: a Beginner's Guide, 2nd ed., A.K. Peters, Ltd., 1998. MR 98i:53001
  • [MJ] Frank Morgan and David L. Johnson, Some sharp isoperimetric theorems for Riemannian manifolds, preprint (1999).
  • [Mo] Charles B. Morrey, Jr., Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. MR 34:2380
  • [Os] Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), 1182-1238. MR 58:18161
  • [Pan] Pierre Pansu, Sur la régularité du profil isopérimétrique des surfaces riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 48 (1998), 247-264. MR 99i:53035
  • [Rit] Manuel Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces.
  • [Rud] Walter Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. MR 35:1420
  • [Sch] Erhard Schmidt, Über eine neue Methode zur Behandlung einer Klasse isoperimetrischer Aufgaben im Grossen, Math. Z. 47 (1942), 489-642. MR7 (1946), p. 527. MR 7:527h
  • [Sp] Michael Spivak, A Comprehensive Introduction to Differential Geometry, Vol. 4, Publish or Perish, Wilmington, 1979. MR 82g:53003d
  • [T1] Peter Topping, The isoperimetric inequality on a surface, Manuscripta Math. 100 (1999), 23-33. CMP 2000:02
  • [T2] Peter Topping, Mean curvature flow and geometric inequalities, Journal für die Reine und Angewandte Math. 503 (1998), 47-61. MR 99m:53080
  • [Wh] Brian White, Complete surfaces of finite total curvature, J. Diff. Geom. 26 (1987), 315-326. MR 88m:53020; correction MR 89j:53009
  • [Ye1] Rugang Ye, Constant mean curvature foliation: singularity structure and curvature estimate, Pacific J. Math. 174 (1996), 569-587. MR 97m:53054
  • [Ye2] Rugang Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147 (1991), 381-396. MR 92f:53030

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Additional Information

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

Michael Hutchings
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305

Hugh Howards
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109

Keywords: Isoperimetric problem
Received by editor(s): July 10, 1998
Received by editor(s) in revised form: November 1, 1998
Published electronically: July 12, 2000
Article copyright: © Copyright 2000 American Mathematical Society

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