  In this paper we prove that a flat free-boundary minimal $n$-disk, $n\geq3$,
in the unit Euclidean ball $B^{n+1}$ is the unique compact free boundary
minimal hypersurface in the unit Euclidean ball which the squared norm of the
second fundamental form is less than either $\frac{n^2}{4}$ or
$\frac{(n-2)^2}{4|x|^2}$. Moreover, we prove analogous results for compact free
boundary minimal hypersurfaces in annular domains with a conformally Euclidean
metric.
