×

zbMATH — the first resource for mathematics

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

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A30 Conformal differential geometry (MSC2010)
MathOverflow Questions:
Usefulness of Nash embedding theorem
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Berger, M.: Sur les premières valeurs propres des variétés riemanniennes. Compositio Math.26, 129-149 (1973) · Zbl 0257.53048
[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
[3] Bleecker, D., Weiner, J.: Extrinsic bounds on ?1 of ? on a compact manifold. Comment. Math. Helv.51, 601-609 (1976) · Zbl 0341.53034
[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) · Zbl 0209.52803
[5] Cheng, S.Y.: Eigenfunctions and nodal sets. Comment. Math. Helv.51, 43-55 (1976) · Zbl 0334.35022
[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
[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
[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
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.