Geodesic
On total curvature of immersions and minimal submanifolds of spheres
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 459-463.

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For closed immersed submanifolds of Euclidean spaces, we prove that $\int |\mu |^2\, dV\geq V/R^2$, where $\mu $ is the mean curvature field, $V$ the volume of the given submanifold and $R$ is the radius of the smallest sphere enclosing the submanifold. Moreover, we prove that the equality holds only for minimal submanifolds of this sphere.
Classification : 53A05, 53C40, 53C42, 53C45, 58E12
Keywords: closed submanifold; total mean curvature; minimal submanifold 
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     title = {On total curvature of immersions and minimal submanifolds of spheres},
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Rotondaro, Giovanni. On total curvature of immersions and minimal submanifolds of spheres. Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 459-463. http://geodesic.mathdoc.fr/item/CMUC_1993__34_3_a8/