Liebmann’s theorem asserts that a compact, connected, convex surface with constant mean curvature (CMC) in the Euclidean space must be a totally umbilical sphere. In this article we extend Liebmann’s result to hypersurfaces with boundary. More precisely, we prove that a locally convex, embedded, compact, connected CMC hypersurface bounded by a closed strictly convex (n − 1) (n-1) -dimensional submanifold in a hyperplane Π n ⊂ R n + 1 ⁿ R^n+1 lies in one of the two halfspaces determined by Π and inherits the symmetries of the boundary. Consequently, spherical caps are the only such hypersurfaces with non-zero constant mean curvature bounded by a (n − 1) (n-1) -sphere.
Cruz et al. (2026) studied this question.