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On total curvature of immersions and minimal submanifolds of spheres

Giovanni Rotondaro — 1993

Commentationes Mathematicae Universitatis Carolinae

For closed immersed submanifolds of Euclidean spaces, we prove that ${\int | μ |}^{2} d V \geq V / R^{2}$ , where $μ$ 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.