The Mean Curvature Flow of Polar Action Orbits

Mean curvature flow describes the process by which a submanifold is deformed in the direction of its mean curvature vector. A polar action is noteworthy in this context because it admits complete submanifolds called sections that intersect each orbit orthogonally. The mean curvature vector of a polar orbit is tangent to this section, so the problem of solving for the flow of these orbits reduces to solving a system of ordinary dierential equations over a section. For this reason the symmetry properties of a section may be used to construct the possible flow lines and predict which points are invariant under the flow. The corresponding orbits are minimal submanifolds.