Giovanni Rotondaro
On total curvature of immersions and minimal submanifolds of spheres

Comment.Math.Univ.Carolinae 34,3 (1993) 459-463.

Abstract: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.

Keywords: closed submanifold, total mean curvature, minimal submanifold
AMS Subject Classification: Primary 53A05; Secondary 53C45