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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: zhoujz@swu.edu.cn
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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

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