相对Alexandrov-Fenchel型不等式  

The relative Alexandrov-Fenchel type inequalities

作　　者：王国芳 翁良俊 夏超 Guofang Wang;Liangjun Weng;Chao Xia

机构地区：[1]Mathematisches Institut,Albert-Ludwigs-Universitat Freiburg,Freiburg D-79104,Germany [2]安徽大学数学科学学院,合肥230601 [3]厦门大学数学科学学院,厦门361005

出　　处：《中国科学：数学》2024年第10期1671-1684,共14页Scientia Sinica：Mathematica

基　　金：中国博士后科学基金会(批准号:2021M702143);国家自然科学基金(批准号:11871406和12171260)资助项目。

摘　　要：Alexandrov-Fenchel不等式是凸几何中的重要不等式之一,它的特殊形式可以看作是高阶均质积分之间的等周型不等式.近年来,带自由边界或毛细边界超曲面获得了很多关注.本文从微分几何角度研究球体内或半空间中带有自由边界或毛细边界凸超曲面上的一类相对Alexandrov-Fenchel型不等式,并综述它的最新进展.首先,介绍单位球体或半空间中毛细边界超曲面的高阶Minkowski型积分公式.从第一变分角度给出单位球体或半空间中毛细边界超曲面的均质积分的定义.然后,本文利用Minkowski型公式构造局部限制型的超曲面曲率流,通过分析流的长时间存在性和收敛性,证明单位球体和半空间中的一类相对Alexandrov-Fenchel型不等式.The Alexandrov-Fenchel inequality is one of the important inequalities in convex geometry,and its special form can be regarded as an isoperimetric inequality between higher-order quermassintegrals.In recent years,hypersurfaces with free or capillary boundaries have received much attention.In this paper,we will study a class of relative Alexandrov-Fenchel type inequalities on convex hypersurfaces with free or capillary boundaries in the unit ball or half-space from the perspective of differential geometry,and review its recent progress.First,we introduce the higher-order Minkowski-type integral formula for capillary hypersurfaces in the unit ball or half-space.From the first variational point of view,we give the definition of the quermassintegrals of the capillary hypersurfaces in the unit ball or half-space.After that,we use the Minkowski-type formula to construct a locally constrained curvature fow.By analyzing the long-time existence and convergence of the flow,we prove a class of relative Alexandrov-Fenchel type inequalities in the unit ball and half-space.

关 键 词：相对均质积分 相对Alexandrov-Fenchel型不等式 限制型超曲面曲率流 毛细边界 

分 类 号：O186.5[理学—数学]