The mean curvature flow by parallel hypersurfaces

TL;DR

This paper characterizes isoparametric hypersurfaces in space forms as initial data for mean curvature flow by parallel hypersurfaces, providing explicit solutions and collapsing times based on geometric properties.

Contribution

It establishes a complete characterization of initial hypersurfaces for this flow and derives explicit solutions and collapse times for all isoparametric hypersurfaces in space forms.

Findings

Hypersurfaces are initial data iff they are isoparametric.

Explicit solutions for mean curvature flow are obtained.

Collapse times are explicitly calculated for sphere hypersurfaces.

Abstract

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions are given for all isoparametric hypersurfaces of space forms. In particular, for such hypersurfaces of the sphere, the exact collapsing time into a focal submanifold is given in terms of its dimension, the principal curvatures and their multiplicities.