Mean Curvature Flow of Singular Riemannian Foliations

Given a singular Riemannian foliation on a compact Riemannian manifold, we study the mean curvature flow equation with a regular leaf as initial datum. We prove that if the leaves are compact and the mean curvature vector field is basic, then any finite time singularity is a singular leaf, and the singularity is of type I. This generalizes previous results of Liu–Terng and Koike. In particular, our results can be applied to study the orbits of an isometric action by a compact Lie group.