Polar foliations on symmetric spaces and the mean curvature flow

TL;DR

This paper investigates polar foliations on symmetric spaces with non-negative curvature, proving they are isoparametric, and explores their behavior under mean curvature flow, showing solutions are always ancient.

Contribution

It establishes that all polar foliations on certain symmetric spaces are isoparametric and demonstrates that mean curvature flow solutions are always ancient, generalizing previous results.

Findings

All polar foliations are isoparametric.

Solutions to mean curvature flow are always ancient.

Reduces study to compact symmetric spaces.

Abstract

In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined by Heintze, Liu and Olmos. We will also prove a splitting theorem which reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results of Liu and Terng for the mean curvature flow of isoparametric submanifolds in spheres.