凸超曲面上的曲率积分
Curvature integral on convex hypersurfaces
- Shantou University（汕头大学）
- Tsinghua University（清华大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究证明凸超曲面无穷远处归一化标量曲率积分的不等式上界，并刻画取等条件，推广至任意余维数。
AI中文摘要：
设 $M^n$，$n\ge3$，为 $\mathbb{R}^{n+1}$ 中具有非空内部的闭凸集的平滑、连通、完备、非紧边界。我们证明，其无穷远处的归一化标量曲率积分至多为 $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$，除非在整体刚性运动下，$M$ 是 $\mathbb{R}^3$ 中光滑闭凸曲面与 $\mathbb{R}^{n-2}$ 的乘积。这些例外乘积具有零渐近体积比，且极限为 $8\pi\omega_{n-2}$。我们还刻画了 $4\pi\omega_{n-2}(1-\operatorname{AVR}(M))$ 界取等号的情形。证明使用了外部管体积以及锥 Steiner 和 Gauss--Bonnet 公式。对内蕴距离与欧氏径向距离的基本比较证明了内蕴极限的存在性，并从锥系数中确定了其精确值。相同的结论推广到任意余维数的可定向凸子流形。
英文摘要：
Let $M^n$, $n\ge3$, be the smooth, connected, complete, noncompact boundary of a closed convex set with nonempty interior in $\mathbb{R}^{n+1}$. We prove that its normalized scalar-curvature integral at infinity is at most $4πω_{n-2}(1-\operatorname{AVR}(M))$, except when, up to an ambient rigid motion, $M$ is the product of a smooth closed convex surface in $\mathbb{R}^3$ and $\mathbb{R}^{n-2}$. These exceptional products have zero asymptotic volume ratio and limit $8πω_{n-2}$. We also characterize the equality case for the $4πω_{n-2}(1-\operatorname{AVR}(M))$ bound. The proof uses exterior tube volumes together with the cone Steiner and Gauss--Bonnet formulas. An elementary comparison of intrinsic and Euclidean radial distances proves existence of the intrinsic limit and identifies its exact value from the cone coefficients. The same conclusions extend to orientable convex submanifolds of arbitrary codimension
