The mean curvature flow for isoparametric submanifolds

TL;DR

This paper investigates the evolution of isoparametric submanifolds under mean curvature flow, showing preservation of properties, finite-time singularities, and convergence to lower-dimensional submanifolds.

Contribution

It demonstrates that the isoparametric condition is preserved under mean curvature flow and provides a detailed description of the collapsing process.

Findings

Flow preserves isoparametric condition

Finite-time development of singularities

Convergence to lower-dimensional submanifolds

Abstract

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.