Uniqueness Results for Free Boundary Minimal Hypersurfaces in Conformally Euclidean Balls and Annular Domains

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.