An extension of Liebmann’s theorem to hypersurfaces with boundary

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)-dimensional submanifold in a hyperplane lies in one of the two halfspaces determined by the hyperplane 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)-sphere.