Mean curvature flow of certain kind of isoparametric foliations on non-compact symmetric spaces

TL;DR

This paper studies the long-term behavior of mean curvature flows originating from non-minimal leaves of specific isoparametric foliations on non-compact symmetric spaces, revealing conditions for convergence and asymptotic behavior.

Contribution

It provides new results on the existence, convergence, and asymptotic properties of mean curvature flows for certain isoparametric foliations on non-compact symmetric spaces.

Findings

Flow exists for all time starting from non-minimal leaves.

Flows asymptote to self-similar solutions if no minimal leaf exists.

Flows converge to the minimal leaf in the presence of one.

Abstract

In this paper, we investigate the mean curvature flows starting from all non-minimal leaves of the isoparametric foliation given by a certain kind of solvable group action on a symmetric space of non-compact type. We prove that the mean curvature flow starting from each non-minimal leaf of the foliation exists in infinite time, if the foliation admits no minimal leaf, then the flow asymptotes the self-similar flow starting from another leaf, and if the foliation admits a minimal leaf (in this case, it is shown that there exists the only one minimal leaf), then the flow converges to the minimal leaf of the foliation in $C^{\infty}$-topology.