Skip to main content
SpringerLink
Log in

Two geometric inequalities for the torus

  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

Given an arbitrary Riemannian metric on the torus, a sharp lower bound for the area and a sharp upper bound for the first eigenvalue of the Laplacian is given in terms of the primitive length spectrum.

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

Similar content being viewed by others

References

  1. Accola, R., ‘Differential and extremal lengths on Riemann surfaces’, Proc. Nat. Acad. Sci., U.S.A. 46 (1960), 540–543.

    Google Scholar 

  2. Berger, M., Lectures on Geodesics in Riemannian Geometry, Tata Institute, Bombay, 1965.

    Google Scholar 

  3. Berger, M., ‘Du côté de chez Pu’, Ann. Scient. Ec. Norm. Sup. 5 (1972), 1–44.

    Google Scholar 

  4. Berger, M., A l'Ombre de Loewner, Ann. Scient. Ec. Norm. Sup. 5 (1972), 241–260.

    Google Scholar 

  5. Berger, M., Sur les premières valeurs propres des variétés Riemanniennes, Compositio Math. 26 (1973), 129–149.

    Google Scholar 

  6. Berger, M., ‘Filling Riemannian manifolds or Isosystolic inequalities’, in Global Riemannian Geometry (T. J. Willmore and N. J. Hitchin, eds), Wiley, New York, 1984, pp. 75–84.

    Google Scholar 

  7. Blatter, C., ‘Über Extremallänge auf gesschlossenen Flächen’, Comment. Math. Helv. 35 (1961), 153–168.

    Google Scholar 

  8. Epstein, D., ‘Curves on 2-manifolds and isotopies’, Acta Math. 115 (1966), 83–107.

    Google Scholar 

  9. Gromov, M., structures métriques pour les variétés riemanniennes, Cedic/Fernand Nathan, 1981.

  10. Gromov, M., ‘Filling Riemannian manifolds’, J. Diff. Geom. 18 (1983), 1–148.

    Google Scholar 

  11. Hebda, J., ‘Some lower bounds for the area of surfaces’, Invent. Math. 65 (1982), 485–490.

    Google Scholar 

  12. Hebda, J., ‘An inequality between the volume and the convexity radius of a Riemannian manifold’, Illinois J. Math. 26 (1982), 642–649.

    Google Scholar 

  13. Hebda, J., ‘The collars of a Riemannian manifold and stable isosystolic inequalities’, Pacific J. Math. 121 (1986), 339–356.

    Google Scholar 

  14. Pu, P. M., ‘Some inequalities in certain non-orientable manifolds’, Pacific J. Math. 11 (1952), 55–71.

    Google Scholar 

  15. Swokowski, E., Fundamentals of Trigonometry (7th edn), PWS-Kent, Boston, 1989.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Saint Louis University, 63103, St Louis, MO, USA

    James J. Hebda

Authors
  1. James J. Hebda
    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

Hebda, J.J. Two geometric inequalities for the torus. Geom Dedicata 38, 101–106 (1991). https://doi.org/10.1007/BF00147738

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF00147738

Keywords

Search

Navigation