Riemann流形中超曲面的逆曲率流及其几何应用

Inverse curvature flow for hypersurfaces in Riemannian manifold and its application

本文是关于Riemann流形中超曲面逆曲率流的综述文章.首先介绍Euclid空间超曲面的逆曲率流的收敛性,以及其在证明Alexandrov-Fenchel不等式中的应用.其次,介绍在双曲空间以及球面中类似的结论.接着讨论Kottler空间的逆平均曲率流.Kottler空间是一类扭曲乘积空间,它满足物理中的稳态方程且在无穷远处渐近于局部双曲空间.本文将介绍此类空间中的逆平均曲率流的收敛性并用来对星形平均凸超曲面证明Minkowski型不等式.逆曲率流是近几年比较热门的一个研究领域,然而,由于篇幅有限,本文不能一一全部介绍.因此,本文最后列举一些相关的文献供感兴趣的读者参考.

This is a survey paper on the inverse curvature flow for hypersurfaces in Riemannian manifold. We first discuss the long time behavior of the inverse curvature flow in Euclidean space, and its application in proving the Alexandrov-Fenchel inequalities for star-shaped hypersurfaces. Then we discuss the related results in hyperbolic space and in sphere. Finally, we discuss the inverse mean curvature flow in Kottler space. Kottler space is an important example of warped product space, and is aymptotically locally hyperbolic at the infinity and satisfies the static equation. We will consider the convergence result of inverse mean curvature flow in such space and also discuss its application in proving the Minkowski-type inequality for star-shaped and mean convex hypersurfaces. Inverse curvature flow is an active research area in recent years. We cannot include all results in this short article. For the convenience of the interested readers, we list a few related references on other topics that we do not mention.