2006/01/29 by Yanyan Li, YanYan Li, Louis Nirenberg +2
Mathematics · #35J60 #53A05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities #math.AP #math.DG #msc:35J60 #msc:53A05
paper · pdf · doi:10.48550/arxiv.math/0601704
corrected 3 typo
openalex publication_date 2006/01/29 · arxiv created 2006/02/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A classical result of A.D. Alexandrov states that a connected compact smooth n-dimensional manifold without boundary, embedded in \Bbb Rn+1, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of M in a hyperplane Xn+1=constant in case M satisfies: for any two points (X', Xn+1), (X', Xn+1) on M, with Xn+1> Xn+1, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional conditions. Some variations of the Hopf Lemma are also presented. Several open problems are described. Part I dealt with corresponding one dimensional problems.