Mean curvatures and symmetry of convex hypersurfaces

Abstract

Let $M^n$ be a $C^2$ closed convex hypersurface in Euclidean space, and ${\sigma }_m$ be its $m$th mean curvature. We show that $M$ is symmetric with respect to a hyperplane orthogonal to a given direction $e$, if ${\sigma }_m(p)\le {\sigma }_m(q)$ whenever $p-q$ is parallel to $e$. For convex hypersurfaces, this settles a conjecture of Li, extends the mean-curvature theorem of Li-Yan-Yao to all ${\sigma }_m$, and strengthens some earlier results of Li-Nirenberg by removing nondegeneracy assumptions. The proof is based on the theory of mixed volumes, specifically the rigidity of quermassintegrals under Steiner symmetrization.

arXiv comment14 pages