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.

中文翻译：

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保形欧几里德球和环形域中自由边界最小超曲面的唯一性结果

在本文中，我们证明了以欧几里得球\（{\ mathbb {B}} ^ {n + 1} \）为单位的自由边界最小n圆盘\（n \ ge 3 \）是唯一的紧致的自由边界极小超曲面，其第二基本形式的平方范数小于\（\ frac {n ^ 2} {4} \）或\（\ frac {（n-2）^ 2} {4 | x | ^ 2} \）。此外，我们证明了具有保形欧几里德度量的环形域或球中的紧致自由边界极小超曲面的相似结果。