On an inequality of T. J. Willmore
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- by Bang-yen Chen PDF
- Proc. Amer. Math. Soc. 26 (1970), 473-479 Request permission
Erratum: Proc. Amer. Math. Soc. 29 (1971), 627.
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
- Bang-yen Chen, Some integral formulas of the Gauss-Kronecker curvature, K\B{o}dai Math. Sem. Rep. 20 (1968), 410–413. MR 234479 —, Surfaces of curvature ${\lambda _N} = 0in{E^{2 + N}}$, Kōdai Math. Sem. Rep. 20(1969), 331-334.
- Shiing-shen Chern and Richard K. Lashof, On the total curvature of immersed manifolds. II, Michigan Math. J. 5 (1958), 5–12. MR 97834 G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, New York, 1934.
- Tominosuke Ôtsuki, On the total curvature of surfaces in Euclidean spaces, Jpn. J. Math. 35 (1966), 61–71. MR 200806, DOI 10.4099/jjm1924.35.0_{6}1
- T. J. Willmore, Note on embedded surfaces, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 11B (1965), 493–496 (English, with Romanian and Russian summaries). MR 202066
- T. J. Willmore, Mean curvature of immersed surfaces, An. Şti. Univ. “All. I. Cuza” Iaşi Secţ. I a Mat. (N.S.) 14 (1968), 99–103 (English, with Romanian summary). MR 0238220
Additional Information
- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 26 (1970), 473-479
- MSC: Primary 53.75
- DOI: https://doi.org/10.1090/S0002-9939-1970-0266113-3
- MathSciNet review: 0266113