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# On Willmore's Inequality for Submanifolds

Published:2007-09-01
Printed: Sep 2007
• Jiazu Zhou
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## Abstract

Let $M$ be an $m$ dimensional submanifold in the Euclidean space ${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain some low geometric estimates of the total square mean curvature $\int_M H^2 d\sigma$. 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: submanifold, mean curvature, kinematic formul, scalar curvature
 MSC Classifications: 52A22 - Random convex sets and integral geometry [See also 53C65, 60D05] 53C65 - Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx] 51C16 - unknown classification 51C16