Polar foliations on symmetric spaces and mean curvature flow

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 in [E. Heintze, X. Liu and C. Olmos, Isoparametric submanifolds and a Chevalley-type restriction theorem, Integrable systems, geometry, and topology, American Mathematical Society, Providence 2006, 151-190]. We will also prove a splitting theorem which, when leaves are compact, 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 in [X. Liu and C.-L. Terng, Ancient solutions to mean curvature flow for isoparametric submanifolds, Math. Ann. 378 2020, 1-2, 289-315] for mean curvature flows of isoparametric submanifolds in spheres.