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On Willmore's Inequality for Submanifolds
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- Journal:
- Canadian Mathematical Bulletin / Volume 50 / Issue 3 / 01 September 2007
- Published online by Cambridge University Press:
- 20 November 2018, pp. 474-480
- Print publication:
- 01 September 2007
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- Article
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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 $M$.
