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On Willmore's Inequality for Submanifolds

Published online by Cambridge University Press:  20 November 2018

Jiazu Zhou
Jiazu Zhou*
Affiliation:
School of Mathematics and Statistics, Southwest University, Chongqing 400715, People's Republic of China e-mail: [email protected]
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Abstract

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Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${{\text{R}}^{n}}$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total squaremean curvature $\int\limits_{M}{{{H}^{2}}d\sigma }$. The low bounds are geometric invariants involving the volume of $M$, the total scalar curvature of $M$, the Euler characteristic and the circumscribed ball of $M$.

Keywords

52A2253C6551C16submanifoldmean curvaturekinematic formulascalar curvature
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 50 , Issue 3 , 01 September 2007 , pp. 474 - 480
Copyright
Copyright © Canadian Mathematical Society 2007

References

[1] Burago, Yu. D. and Zalgaller, V. A., Geometric Inequalities. Grundlehren der Mathematischen Wissenschaften 285, Springer-Verlag, Berlin, 1998.Google Scholar
[2] Chen, B.-Y., Geometry of Submanifolds. Pure and Applied Mathematics 22, Marcel Dekker, New York, 1973.Google Scholar
[3] Chen, B.-Y., Total Mean Curvature and Submanifolds of Finite Type. Series in Pure Mathematics 1, World Scientific, Singapore, 1984.Google Scholar
[4] Chen, C.-S., On the kinematic formula of square of mean curvature vector. Indiana Univ. Math J. 22(1972/73), 1163–1169.Google Scholar
[5] Chern, S.-S., On the kinematic formula in the Euclidean space of n dimensions. Amer. J. Math. 74(1952), 227–236.Google Scholar
[6] Chern, S.-S., On the curvatura integra in a Riemannian manifold. Ann. of Math. 46(1945), 674–684.Google Scholar
[7] Howard, R., The kinematic formula in Riemannian homogenous space. Mem. Amer. Math. Soc. 106(1993), no. 509.Google Scholar
[8] Li, P. and Yau, S. T., A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69(1982), no. 2, 269–291.Google Scholar
[9] Ren, D., Topics in Integral Geometry. Series in Pure Mathematics 19, World Scientific, River Edge, NJ, 1994.Google Scholar
[10] Santaló, L. A., Integral Geometry and Geometric Probability. Encyclopedia of Mathematics and Its Applications, Vol. 1, Addison-Wesley, Reading, MA, 1976.Google Scholar
[11] Schneider, R., Convex bodies: the Brunn-Minkowski theory. Encyclopedia of Mathematics and its Applications, Vol. 44. Cambridge University Press, Cambridge, 1993.Google Scholar
[12] Willmore, T., Total Curvature in Riemannian Geometry. Ellis Horwood, Chichester, 1982.Google Scholar
[13] Zhang, G., A sufficient condition for one convex body containing another. Chinese Ann. Math. Ser. B 9(1988), no. 4, 447–451.Google Scholar
[14] Zhang, G. and Zhou, J., Containment measures in integral geometry. In: Integral Geometry and Convexity, World Scientific, Hackensack, NJ, 2005, pp. 153–168.Google Scholar
[15] Zhou, J., Kinematic formulas for mean curvature powers of hypersurfaces and Hadwiger's theorem in R 2n . Trans. Amer. Math. Soc. 345(1994), no. 1, 243–262.Google Scholar
[16] Zhou, J., On the Willmore deficit of convex surfaces. In: Tomography, Impedance Imaging, and Integral Geometry, Lectures in Appl. Math. 30, American Mathematical Society, Providence, RI, 1994, pp. 279–287.Google Scholar
[17] Zhou, J., Kinematic formula for square mean curvature of hypersurfaces. Bull. Inst. Math. Acad. Sinica 22(1994), no. 1, 31–47.Google Scholar
[18] Zhou, J., On the second fundamental forms of the intersection of submanifolds. Taiwenese J. Math. 11(2007), no. 1, 215–229..Google Scholar
[19] Zhou, J., The Willmore functional and the containment problem in R 4 . Sci. China Ser. A 50(2007), no. 3, 325–333.Google Scholar
[20] Zhou, J., Total square mean curvature of hypersurfaces. Submitted to Bull. Inst. Math. Acad. Sinica.Google Scholar
[21] Zhou, J., On Bonnesen-type isoperimetric inequalities. In: Proceedings of the Tenth International Workshop on Differential Geometry, Kyungpook Nat. Univ., Taegu, 2006, pp. 57–71.Google Scholar