2024/12/04 by Flávio França Cruz, Cruz, Flávio França, Nelli, Barbara
Computer Science · Engineering · Mathematics · #35J60 #53C42 #Advanced Numerical Analysis Techniques #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2412.03368
openalex publication_date 2024/12/04 · openalex created_date 2024/12/06 · openalex updated_date 2026/07/28
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 Πn⊂ ℝn+1 lies in one of the two halfspace determined by 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)-sphere.