Polar foliations on symmetric spaces and the mean curvature flow

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.

20 pages. Major changes throughout, following suggestions and remarks. Appendix removed, the result proved there was recently proved by Silva and Speranca. Added assumption of compact leaves on theorem 1.2. Generalized old theorem 1.3 to more general ambient spaces