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 ) (n-1) -dimensional submanifold in a hyperplane Π n ⊂ R n + 1 Π ^n⊂ Rⁿ⁺¹ lies in one of the two halfspaces determined by Π 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.