Advertisement

Inventiones mathematicae

, Volume 69, Issue 2, pp 269–291 | Cite as

A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

  • Peter Li
  • Shing-Tung Yau
Article

Keywords

Compact Surface Willmore Conjecture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berger, M.: Sur les premières valeurs propres des variétés riemanniennes. Compositio Math.26, 129–149 (1973)Google Scholar
  2. 2.
    Berger, M., Gauduchon, P., Mazet, E.: Le spectre d'une variété Riemannienne. Lecture notes in math., vol. 194. Berlin-Heidelberg-New York: Springer 1971Google Scholar
  3. 3.
    Bleecker, D., Weiner, J.: Extrinsic bounds on λ1 of Δ on a compact manifold. Comment. Math. Helv.51, 601–609 (1976)Google Scholar
  4. 4.
    Chen, B.Y.: On the total curvature of immersed manifolds I–V. Amer. J. Math.93, 148–162 (1971); Amer. J. Math.94, 799–809 (1972); Amer. J. Math.95, 636–642 (1973); Bull. Inst. Math. Acad. Sinica7, 301–311 (1979); Bull. Inst. Math. Acad. Sinica9, 509–516 (1981)Google Scholar
  5. 5.
    Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv.51, 43–55 (1976)Google Scholar
  6. 6.
    Cheng, S.Y., Li, P., Yau, S.T.: Heat equations on minimal submanifolds and their applications. Amer. J. Math. in press (1982)Google Scholar
  7. 7.
    Chern, S.S.: La Géométrie des sous-variétés d'une espace Euclidien à plusieurs dimensions. L'Enseigement Math.40, 26–46 (1955)Google Scholar
  8. 8.
    Chern, S.S., Lashof, R.: On the total curvature of immersed manifolds. Amer. J. Math.79, 306–318 (1957)Google Scholar
  9. 9.
    Fary, I.: Sur la courbure totale d'une courbe gauche faisant un noeud. Bull. Soc. Math. France77, 128–138 (1949)Google Scholar
  10. 10.
    Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris270, 1645–1648 (1970)Google Scholar
  11. 11.
    Lawson, H.B.: Lectures on minimal submanifolds. Vol. 1. Math. Lecture Series 9, Publish or Perish, Inc. Berkeley (1980)Google Scholar
  12. 12.
    Milnor, J.W.: On the total curvature of knots. Ann. of Math.52, 248–257 (1950)Google Scholar
  13. 13.
    Reilly, R.C.: On the first eigenvalues of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv.52, 525–533 (1977)Google Scholar
  14. 14.
    Rodriguez, L., Guadalope, I.V.: Normal curvature of surfaces into spaces forms. PreprintGoogle Scholar
  15. 15.
    Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rat. Mech. Anal.3, 343–356 (1954)Google Scholar
  16. 16.
    Willmore, T.J.: Note on embedded surfaces. Anal. Ştüntifice ale Univ., Iasi Sect. I a Mat.11, 493–496 (1965)Google Scholar
  17. 17.
    Wintgen, P.: On the total curvature of surfaces inE 4. Colloq. Math.39, 289–296 (1978)Google Scholar
  18. 18.
    Yang, P., Yau, S.T.: Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Annali della Scuola Sup. di Pisa7, 55–63 (1980)Google Scholar

Copyright information

© Springer-Verlag 1982

Authors and Affiliations

  • Peter Li
    • 1
  • Shing-Tung Yau
    • 2
  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.The Institute for Advanced StudySchool of MathematicsPrincetonUSA

Personalised recommendations