Abstract
In this paper, we prove that a flat free boundary minimal n-disk, $$n\ge 3$$ , in the unit Euclidean ball $${\mathbb {B}}^{n+1}$$ is the unique compact free boundary minimal hypersurface whose 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 or balls with a conformally Euclidean metric.
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