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A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. (English) Zbl 0503.53042


MathOverflow Questions:

Usefulness of Nash embedding theorem

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A30 Conformal differential geometry (MSC2010)

References:

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[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 · Zbl 0223.53034
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[5] Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv.51, 43-55 (1976) · Zbl 0334.35022 · doi:10.1007/BF02568142
[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) · Zbl 0064.17504
[8] Chern, S.S., Lashof, R.: On the total curvature of immersed manifolds. Amer. J. Math.79, 306-318 (1957) · Zbl 0078.13901 · doi:10.2307/2372684
[9] Fary, I.: Sur la courbure totale d’une courbe gauche faisant un noeud. Bull. Soc. Math. France77, 128-138 (1949) · Zbl 0037.23604
[10] Hersch, J.: Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris270, 1645-1648 (1970) · Zbl 0224.73083
[11] Lawson, H.B.: Lectures on minimal submanifolds. Vol. 1. Math. Lecture Series 9, Publish or Perish, Inc. Berkeley (1980) · Zbl 0434.53006
[12] Milnor, J.W.: On the total curvature of knots. Ann. of Math.52, 248-257 (1950) · Zbl 0037.38904 · doi:10.2307/1969467
[13] Reilly, R.C.: On the first eigenvalues of the Laplacian for compact submanifolds of Euclidean space. Comment. Math. Helv.52, 525-533 (1977) · Zbl 0382.53038 · doi:10.1007/BF02567385
[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) · Zbl 0055.08802
[16] Willmore, T.J.: Note on embedded surfaces. Anal. ?tüntifice ale Univ., Iasi Sect. I a Mat.11, 493-496 (1965)
[17] Wintgen, P.: On the total curvature of surfaces inE 4. Colloq. Math.39, 289-296 (1978) · Zbl 0409.53047
[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) · Zbl 0446.58017
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