Isoparametric hypersurfaces of Riemannian manifolds as initial data for the mean curvature flow

TL;DR

This paper studies how isoparametric hypersurfaces evolve under mean curvature flow in Riemannian manifolds, showing they follow a reparametrized parallel family, and explores singularity types and examples in various ambient spaces.

Contribution

It demonstrates that isoparametric hypersurfaces evolve as reparametrized parallel families under mean curvature flow and identifies conditions for Type I singularities in certain ambient spaces.

Findings

Evolution of isoparametric hypersurfaces follows a reparametrized parallel family.

Existence of hypersurfaces with constant principal curvatures that are not isoparametric.

Development of Type I singularities in specific ambient spaces.

Abstract

We show that the evolution of isoparametric hypersurfaces of Riemannian manifolds by the mean curvature flow is given by a reparametrization of the parallel family in short time, as long as the uniqueness of the mean curvature flow holds for the initial data and the corresponding ambient space. As an application, we provide a class of Riemannian manifolds that admit hypersurfaces with constant principal curvatures, which are not isoparametric hypersurfaces. Furthermore, for a class of ambient spaces, we show that the singularities developed by the mean curvature flow with isoparametric hypersurfaces as the initial data are Type I singularities. We apply our results to describe the evolution of isoparametric hypersurfaces by the mean curvature flow in ambient spaces with nonconstant sectional curvature, such as homogenous 3-manifolds $\mathbb{E}(\kappa, \tau)$ with 4-dimensional isometry groups, and Riemannian products $\mathbb{Q}^2_{c_1} \times \mathbb{Q}^2_{c_2}$ of space forms.