Work

An extension of Liebmann’s theorem to hypersurfaces with boundary

Year: 2026

Type: article

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.

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17553

Authors: Flávio Cruz, Barbara Nelli

Institutions: Universidade Regional do Cariri, University of L'Aquila

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Fundação Cearense de Apoio ao Desenvolvimento Científico e Tecnológico

Award: G7219734420