凸超曲面的平均曲率与对称性
Mean curvatures and symmetry of convex hypersurfaces
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中文总结 AI 辅助
本文研究凸超曲面的平均曲率与对称性问题，利用混合体积理论及Steiner对称化下quermassintegral的刚性，证明了满足特定方向平均曲率单调条件的凸超曲面对称，解决了相关猜想并推广强化了已有结果。
中文摘要 AI 辅助
设$M^n$为欧氏空间中的$C^2$闭凸超曲面，$σ_m$为其第$m$平均曲率。我们证明：若对任意满足$p-q$平行于给定方向$e$的点$p,q$，都有$σ_m(p)\leqσ_m(q)$，则$M$关于垂直于$e$的超平面对称。对凸超曲面而言，该结论解决了Li的一个猜想，将Li-Yan-Yao的平均曲率定理推广到所有$σ_m$，并通过去除非退化假设强化了Li-Nirenberg的部分早期结果。证明基于混合体积理论，具体是Steiner对称化下quermassintegral（平均曲率积分）的刚性性质。
英文摘要
Let $M^n$ be a $C^2$ closed convex hypersurface in Euclidean space, and $σ_m$ be its $m$th mean curvature. We show that $M$ is symmetric with respect to a hyperplane orthogonal to a given direction $e$, if $σ_m(p)\leqσ_m(q)$ 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 $σ_m$, 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.
