Skip to main content
SpringerLink
Log in

A proof of a general isoperimetric inequality for surfaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Alexandroff, A.D.: Isoperimetric inequalities for curved surfaces. Dokl. Akad. Nauk. SSSR47, 235–238 (1945)

    Google Scholar 

  2. Alexandroff, A.D.: Die innere Geometrie der Konvexen Flächen. Berlin: Akademie-Verlag 1955

    Google Scholar 

  3. Alexandroff, A.D., Streltsov, V.V.: Estimates of the length of a curve on a surface [Russian]. Dokl. Akad. Nauk. SSSR93, 221–224 (1953)

    Google Scholar 

  4. Alexandroff, A.D., Streltsov, V.V.: Isoperimetric problem and estimates of the length of a curve on a surface. Two-dimensional manifolds of bounded curvature. Proc. Steklov Inst. Math.76, 81–99 (1965)

    Google Scholar 

  5. Alexandroff, A.D., Zalgaller, V.A.: Intrinsic Geometry of Surfaces. Translations of Mathematical Monographs15. Providence, R.I.: Amer. Math. Soc. 1967

    Google Scholar 

  6. Bandle, C.: On a differential inequality and its applications to geometry. Math. Z.147, 253–261 (1976)

    Google Scholar 

  7. Beckenbach, E.F., Radó, T.: Subharmonic functions and surfaces of negative curvature. Trans. Amer. Math. Soc.35, 662–674 (1933)

    Google Scholar 

  8. Bernstein, F.: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann.60, 117–136 (1905)

    Google Scholar 

  9. Bol, G.: Isoperimetrische Ungleichung für Bereiche auf Flächen. Jber. Deutsch. Math.-Verein.51, 219–257 (1941)

    Google Scholar 

  10. Burago, Yu.D.: Note on the isoperimetric inequality on two-dimensional surfaces. Siberian Math. J.14, 666–668 (1973)

    Google Scholar 

  11. Carleman, T.: Zur Theorie der Minimal-Flächen. Math. Z.9, 154–160 (1921)

    Google Scholar 

  12. Fiala, F.: Le problème des isoperimètres sur les surfaces ouverts à courbure positive. Comment. Math. Helv.13, 293–396 (1940–41)

    Google Scholar 

  13. Gromov, M.L., Rokhlin, V.A.: Embeddings and immersions in Riemannian Geometry. Russian Math. Surveys25, 5, 1–57 (1970)

    Google Scholar 

  14. Huber, A.: On the isoperimetric inequality on surfaces of variable Gaussian curvature. Ann. of Math60, 237–247 (1954)

    Google Scholar 

  15. Huber, A.: Zum potentialtheoretischen Aspekt der Alexandrowschen Flächentheorie. Comment. Math. Helv.34, 99–126 (1960)

    Google Scholar 

  16. Ionin, V.K.: On isoperimetric and various other inequalities for a manifold of bounded curvature. Siberian Math. J.10, 329–342 (1969)

    Google Scholar 

  17. Osserman, R.: Bonessen-style isoperimetric inequalities. Preprint

  18. Reschetniak, I.G.: Isothermal coordinates in manifolds of bounded curvature. Dokl. Akad. Nauk. SSSR94, 631–633 (1954)

    Google Scholar 

  19. Schmidt, E.: Beweis der Isoperimetrischen Eigenschaft der Kugel im Hyperbolischen und Sphärischen Raum jeder Dimensionenzahl. Math. Z.46, 204–230 (1940)

    Google Scholar 

  20. Toponogov, V.A.: An isoperimetric inequality for surfaces whose Gaussian curvature is bounded above. Siberian Math. J.10, 144–157 (1969)

    Google Scholar 

  21. Whitney, H.: Differentiable manifolds. Ann. of Math.37, 645–680 (1936)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidade Federal Ceara, Campus do Pici, Fortaleza-Ce, Brasil

    João Lucas Barbosa

  2. Instituto de Matematica Pura e Applicada, Rua Luiz de Camões 68, Rio de Janeiro, RJ, Brasil

    Manfredo do Carmo

Authors
  1. João Lucas Barbosa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Manfredo do Carmo
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Barbosa, J.L., Carmo, M.d. A proof of a general isoperimetric inequality for surfaces. Math Z 162, 245–261 (1978). https://doi.org/10.1007/BF01186367

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01186367

Keywords

Search

Navigation