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On an inequality of T. J. Willmore


Author: Bang-yen Chen
Journal: Proc. Amer. Math. Soc. 26 (1970), 473-479
MSC: Primary 53.75
DOI: https://doi.org/10.1090/S0002-9939-1970-0266113-3
Erratum: Proc. Amer. Math. Soc. 29 (1971), 627.
MathSciNet review: 0266113
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Abstract: Willmore proved that the integral of the square of mean curvature $ H$ over a closed surface $ {M^2}$ in $ {E^3},{\smallint _{{M^2}}}{H^2}dV$, is $ \geqq 4\pi $, and equal to $ 4\pi $ when and only when $ {M^2}$ is a sphere in $ {E^3}$. In this paper we give some generalizations of Willmore's result.


References [Enhancements On Off] (What's this?)

  • [1] B-y. Chen, Some integral formulas of the Gauss-Kronecker curvature, Kōdai Math. Sem. Rep. 20 (1968), 410-413. MR 38 #2796. MR 0234479 (38:2796)
  • [2] -, Surfaces of curvature $ {\lambda _N} = 0in{E^{2 + N}}$, Kōdai Math. Sem. Rep. 20(1969), 331-334.
  • [3] S. S. Chern and R. K. Lashof, On the total curvature of immersed manifolds. II, Michigan Math. J. 5 (1958), 5-12. MR 20 #4301. MR 0097834 (20:4301)
  • [4] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
  • [5] T. Ōtsuki, On the total curvature of surfaces in Euclidean spaces, Japan. J. Math. 35 (1966), 61-71. MR 34 #692. MR 0200806 (34:692)
  • [6] T. J. Willmore, Note on embedded surfaces, An. Sti. Univ. ``Al. I. Cuza'' Iaşi. Sect. Ia Mat. 11B (1965), 493-496. MR 34 #1940. MR 0202066 (34:1940)
  • [7] -, Mean curvature of immersed surface, An Sti. Univ. ``Al. I. Cuza'' Iaşi. Sect. Ia Mat. 14 (1968), 99-103. MR 38 #6496. MR 0238220 (38:6496)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1970-0266113-3
Keywords: $ i$th mean curvature, Lipschitz-Killing curvature, $ \alpha $th curvatures of first or second kinds, Willmore's inequality, betti number
Article copyright: © Copyright 1970 American Mathematical Society

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