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 ) (n-1) -dimensional submanifold in a hyperplane Π n ⊂ R n + 1 \Pi ^n\subset \mathbb {R}^{n+1} lies in one of the two halfspaces determined by Π \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 ) (n-1) -sphere.
Fields of Study & Subjects
Geometric Analysis and Curvature Flows
Advanced Differential Geometry Research
Point processes and geometric inequalities
Publication details
- Journal / source
- Proceedings of the American Mathematical Society
- Volume
- 154
- Issue
- 3
- Pages
- 1269–1278
- Publication date
- 16 January 2026
- Language
- EN