Inverse curvature flow for hypersurfaces in Riemannian manifold and its application

Abstract

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 discussits 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.