Area Rigidity for the Equatorial Disk in the Ball

It is proved by Brendle (Geom Funct Anal 22:621–626, 2012) that the equatorial disk \(D^k\) $D^k$ has least area among k-dimensional free boundary minimal submanifolds in the Euclidean ball \(B^n\) $B^n$ . By comparing the excess of free boundary minimal submanifolds with the excess of the associated cones over their boundary, we prove the existence of a gap for the area.