On the isoperimetric rigidity of extrinsic minimal balls. (Q1405247)

Let \(P\) be an \(m\)-dimensional minimal submanifold in the Euclidean space \(\mathbb R^n\). If a metric \(R\)-sphere has its center on \(P\), then it cuts out a well defined connected component of \(P\), called an extrinsic minimal \(R\)-ball of \(P\), which contains the center point. Conditions for \(P\) to be totally geodesic are described by using the dimension \(m\), the quotient of the volume of an extrinsic ball and the volume of its boundary and by using the second fundamental form of \(P\).