An extension of Liebmann's Theorem to hypersurfaces with boundary

Abstract

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 $\Pi^n\subset \mathbb{R}^{n+1}$ lies in one of the two halfspace determined by $\Pi$ 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.