Mean curvatures and symmetry of convex hypersurfaces

Mohammad Ghomi

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arXiv · 2608.28410

Mean curvatures and symmetry of convex hypersurfaces

Mohammad Ghomi

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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 $\sigma_m(p)\leq\sigma_m(q)$ 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.

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Mohammad Ghomi. 2026-08-28. Mean curvatures and symmetry of convex hypersurfaces. https://arxiv.org/abs/2608.28410

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