On the isoperimetric rigidity of extrinsic minimal balls

We consider an m-dimensional minimal submanifold P and a metric R-sphere in the Euclidean space R-n. If the sphere has its center p on P, then it will cut out a well defined connected component of P which contains this center point. We call this connected component an extrinsic minimal R-ball of P. The quotient of the volume of the extrinsic ball and the volume of its boundary is not larger than the corresponding quotient obtained in the space form standard situation, where the minimal submanifold is the totally geodesic linear subspace R-m. Here we show that if the minimal submanifold has dimension larger than 3, if P is not too curved along the boundary of an extrinsic minimal R-ball, and if the inequality alluded to above is an equality for the extrinsic minimal ball, then the minimal submanifold is totally geodesic.