发表机构
University of Science and Technology of China（中国科学技术大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明球面中弱凸域的Alexandrov--Fenchel不等式，通过构造全局约束曲率流并利用曲率估计与对偶性，得到等号仅对测地球成立，并将结果推广至弱凸情形。
AI 中文摘要
我们证明了在开半球内包含的光滑弱凸域的任何两个球面quermassintegral之间的Alexandrov--Fenchel不等式，当且仅当该域为测地球时取等号。对于严格凸超曲面，我们引入了一个全局约束曲率流，该流保持一个quermassintegral并减小下一个。我们通过结合收缩估计、支撑函数论证和球面对偶性建立了均匀曲率估计。因此，该流对所有时间存在，并光滑且指数地收敛到一个测地球面。quermassintegral的单调性给出了完整的Alexandrov--Fenchel不等式族。一个短时平均曲率流逼近和局部刚性论证将结果（包括等号刻画）推广到弱凸域。
英文摘要
We prove the Alexandrov--Fenchel inequalities between any two spherical quermassintegrals for smooth weakly convex domains contained in an open hemisphere, with equality if and only if the domain is a geodesic ball. For strictly convex hypersurfaces, we introduce a globally constrained curvature flow which preserves one quermassintegral and decreases the next one. We establish uniform curvature estimates by combining a pinching estimate, a support function argument, and spherical polarity. As a consequence, the flow exists for all time and converges smoothly and exponentially to a geodesic sphere. The monotonicity of the quermassintegrals gives the full family of Alexandrov--Fenchel inequalities. A short-time mean curvature flow approximation and a localized rigidity argument extend the result, including the equality characterization, to weakly convex domains.
Comments33 pages. Comments are welcome!
