  We show that a minimal disk satisfying the free boundary condition in a
constant curvature ball of any dimension is totally geodesic. We weaken the
condition to parallel mean curvature vector in which case we show that the disk
lies in a three dimensional constant curvature submanifold and is totally
umbilic. These results extend to higher dimensions earlier three dimensional
work of J. C. C. Nitsche and R. Souam.
