2022/02/20 by Shibing Chen, Xiang Ma, Chen, Shibing +3
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2202.09824
openalex publication_date 2022/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We prove that a hemisphere in the Euclidean space Rn+1, viewed as the graph of a function, admits no smooth perturbations as graphs with mean curvature H≥ 1 whose boundary equator is fixed up to C2. This is an extension of the Mean Curvature Rigidity phenomenon discovered by Gromov and Souam on non-compact totally umbilic hypersurfaces in space forms. The proof uses a Tangency Principle. On the other hand, we show that there exist nontrivial smooth perturbations with H≥ 1 on a great spherical cap whose boundary is fixed up to C2. Similar results hold true for perturbations decreasing H, and for the r mean curvature function Hr. This contrast between rigidity and non-rigidity is even true in the 1-dimensional case for circles and for discrete objects (polygons inscribed in a circle).