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

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References

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

    Google Scholar 

  2. Berger, M., Gauduchon, P., Mazet, E.: Le spectre d'une variété Riemannienne. Lecture notes in math., vol. 194. Berlin-Heidelberg-New York: Springer 1971

    Google Scholar 

  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. 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. Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv.51, 43–55 (1976)

    Google Scholar 

  6. Cheng, S.Y., Li, P., Yau, S.T.: Heat equations on minimal submanifolds and their applications. Amer. J. Math. in press (1982)

  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. Chern, S.S., Lashof, R.: On the total curvature of immersed manifolds. Amer. J. Math.79, 306–318 (1957)

    Google Scholar 

  9. Fary, I.: Sur la courbure totale d'une courbe gauche faisant un noeud. Bull. Soc. Math. France77, 128–138 (1949)

    Google Scholar 

  10. Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris270, 1645–1648 (1970)

    Google Scholar 

  11. Lawson, H.B.: Lectures on minimal submanifolds. Vol. 1. Math. Lecture Series 9, Publish or Perish, Inc. Berkeley (1980)

    Google Scholar 

  12. Milnor, J.W.: On the total curvature of knots. Ann. of Math.52, 248–257 (1950)

    Google Scholar 

  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. Rodriguez, L., Guadalope, I.V.: Normal curvature of surfaces into spaces forms. Preprint

  15. Szegö, G.: Inequalities for certain eigenvalues of a membrane of given area. J. Rat. Mech. Anal.3, 343–356 (1954)

    Google Scholar 

  16. Willmore, T.J.: Note on embedded surfaces. Anal. Ştüntifice ale Univ., Iasi Sect. I a Mat.11, 493–496 (1965)

    Google Scholar 

  17. Wintgen, P.: On the total curvature of surfaces inE 4. Colloq. Math.39, 289–296 (1978)

    Google Scholar 

  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 

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, 94305, Stanford, CA, USA

    Peter Li

  2. The Institute for Advanced Study, School of Mathematics, 08540, Princeton, NJ, USA

    Shing-Tung Yau

Authors
  1. Peter Li
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  2. Shing-Tung Yau
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This research is partially supported by NSF grant #MCS81-07911

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Li, P., Yau, ST. A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent Math 69, 269–291 (1982). https://doi.org/10.1007/BF01399507

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