A free boundary CMC surface in the 3-ball with traceless second fundamental form pinched by (2+H<x,N>)^2/2 is a spherical cap or a Delaunay annulus.
Uniqueness results for free-boundary minimal hypersurfaces in conformally Euclidean balls and annular domains
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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.
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Gap phenomena for constant mean curvature surfaces
A free boundary CMC surface in the 3-ball with traceless second fundamental form pinched by (2+H<x,N>)^2/2 is a spherical cap or a Delaunay annulus.