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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?)

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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