Transactions of the American Mathematical Society

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

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

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.


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

  • [And] Ben Andrews, Contraction of convex hypersurfaces by their affine normal, J. Differential Geom. 43 (1996), no. 2, 207–230. MR 1424425
  • [BaP] Christophe Bavard and Pierre Pansu, Sur le volume minimal de 𝑅², Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 4, 479–490 (French). MR 875084
  • [BeC] Itai Benjamini and Jianguo Cao, A new isoperimetric comparison theorem for surfaces of variable curvature, Duke Math. J. 85 (1996), no. 2, 359–396. MR 1417620, 10.1215/S0012-7094-96-08515-4
  • [Bér] Pierre H. Bérard, Spectral geometry: direct and inverse problems, Lecture Notes in Mathematics, vol. 1207, Springer-Verlag, Berlin, 1986. With appendixes by Gérard Besson, and by Bérard and Marcel Berger. MR 861271
  • [Bow] B. H. Bowditch, The minimal volume of the plane, J. Austral. Math. Soc. Ser. A 55 (1993), no. 1, 23–40. MR 1231692
  • [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, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
  • [Ch2] Isaac Chavel, Riemannian geometry—a modern introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. MR 1271141
  • [Fed] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR 0257325
  • [HM] Joel Hass and Frank Morgan, Geodesics and soap bubbles in surfaces, Math. Z. 223 (1996), no. 2, 185–196. MR 1417428, 10.1007/PL00004560
  • [HH] Erhard Heinz and Stefan Hildebrandt, Some remarks on minimal surfaces in Riemannian manifolds, Comm. Pure Appl. Math. 23 (1970), 371–377. MR 0259765
  • [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), no. 1, 37–47. MR 1156385, 10.1007/BF02100598
  • [M1] Frank Morgan, Geometric measure theory, 2nd ed., Academic Press, Inc., San Diego, CA, 1995. A beginner’s guide. MR 1326605
  • [M2] Frank Morgan, Riemannian geometry, 2nd ed., A K Peters, Ltd., Wellesley, MA, 1998. A beginner’s guide. MR 1600519
  • [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, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511
  • [Os] Robert Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc. 84 (1978), no. 6, 1182–1238. MR 0500557, 10.1090/S0002-9904-1978-14553-4
  • [Pan] Pierre Pansu, Sur la régularité du profil isopérimétrique des surfaces riemanniennes compactes, Ann. Inst. Fourier (Grenoble) 48 (1998), no. 1, 247–264 (French, with English and French summaries). MR 1614957
  • [Rit] Manuel Ritoré, Constant geodesic curvature curves and isoperimetric domains in rotationally symmetric surfaces.
  • [Rud] Walter Rudin, Real and complex analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210528
  • [Sch] Erhard Schmidt, Über eine neue Methode zur Behandlung einer Klasse isoperimetrischer Aufgaben im Grossen, Math. Z. 47 (1942), 489–642 (German). MR 0016219
  • [Sp] Michael Spivak, A comprehensive introduction to differential geometry. Vol. IV, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532833
  • [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, J. Reine Angew. Math. 503 (1998), 47–61. MR 1650335, 10.1515/crll.1998.099
  • [Wh] Brian White, Complete surfaces of finite total curvature, J. Differential Geom. 26 (1987), no. 2, 315–326. MR 906393
    Brian White, Correction to: “Complete surfaces of finite total curvature” [J. Differential Geom. 26 (1987), no. 2, 315–326; MR0906393 (88m:53020)], J. Differential Geom. 28 (1988), no. 2, 359–360. MR 961520
  • [Ye1] Rugang Ye, Constant mean curvature foliation: singularity structure and curvature estimate, Pacific J. Math. 174 (1996), no. 2, 569–587. MR 1405602
  • [Ye2] Rugang Ye, Foliation by constant mean curvature spheres, Pacific J. Math. 147 (1991), no. 2, 381–396. MR 1084717

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 53Cxx, 53Axx, 49Qxx

Retrieve articles in all journals with MSC (2000): 53Cxx, 53Axx, 49Qxx


Additional Information

Frank Morgan
Affiliation: Department of Mathematics, Williams College, Williamstown, Massachusetts 01267
Email: Frank.Morgan@williams.edu

Michael Hutchings
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: hutching@math.stanford.edu

Hugh Howards
Affiliation: Department of Mathematics, Wake Forest University, Winston-Salem, North Carolina 27109
Email: howards@wfu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02482-X
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