  A submanifold in space forms is isoparametric if the normal bundle is flat
and principal curvatures along any parallel normal fields are constant. We
study the mean curvature flow with initial data an isoparametric submanifold in
Euclidean space and sphere. We show that the mean curvature flow preserves the
isoparametric condition, develops singularities in finite time, and converges
in finite time to a smooth submanifold of lower dimension. We also give a
precise description of the collapsing.
