Minimal hypersurfaces in the ball with free boundary

In this note we use the strong maximum principle and integral estimates prove two results on minimal hypersurfaces F:Mn→Rn+1 with free boundary on the standard unit sphere. First we show that if F is graphical with respect to any Killing field, then F(Mn) is a flat disk. This result is independent of the topology or number or boundaries. Second, if Mn=Dn is a disk, we show the supremum of the curvature squared on the interior is bounded below by n times the infimum of the curvature on the boundary. These may be combined the give an impression of the curvature of non-flat minimal hyperdisks with free boundary.