Geodesic
On total curvature of immersions and minimal submanifolds of spheres
Commentationes Mathematicae Universitatis Carolinae, Tome 34 (1993) no. 3, pp. 459-463 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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     author = {Rotondaro, Giovanni},
     title = {On total curvature of immersions and minimal submanifolds of spheres},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {459--463},
     year = {1993},
     volume = {34},
     number = {3},
     mrnumber = {1243078},
     zbl = {0787.53049},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/CMUC_1993_34_3_a8/}
}
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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/