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On total curvature of immersions and minimal submanifolds of spheres. (English) Zbl 0787.53049
For any smooth immersion of a closed manifold $$M$$ into Euclidean space a lower bound for the total mean curvature is obtained. More precisely, it is shown that $$\int_ M H^ 2 \geq V/R^ 2$$ where $$H$$, $$V$$, $$R$$ denote the norm of the mean curvature vector, the volume, and the circumradius of $$M$$, respectively. This is a straightforward consequence of the Cauchy-Schwarz inequality and the Minkowski formula for the scalar product $$(u,v)\to \int_ M\langle u,v\rangle$$ of vector valued functions. Equality holds if and only if $$M$$ is a minimal immersion in a hypersphere of radius $$R$$.
##### MSC:
 53C40 Global submanifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
##### Keywords:
Willmore conjecture; total mean curvature
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