An integral formula for closed surfaces and a generalization of -theorem
On the -theorem for hypersurfaces
On total curvature of immersions and minimal submanifolds of spheres
For closed immersed submanifolds of Euclidean spaces, we prove that , where is the mean curvature field, the volume of the given submanifold and is the radius of the smallest sphere enclosing the submanifold. Moreover, we prove that the equality holds only for minimal submanifolds of this sphere.
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