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Keywords:
closed submanifold; total mean curvature; minimal submanifold
Summary:
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.
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