Collapse of the mean curvature flow for equifocal submanifolds

TL;DR

This paper studies the behavior of mean curvature flows starting from equifocal submanifolds in symmetric spaces, revealing how these flows evolve and collapse, using a novel approach involving lifts to infinite-dimensional Hilbert spaces.

Contribution

It introduces a new method to analyze mean curvature flows of equifocal submanifolds via lifts to Hilbert spaces, independent of their characterization as Hermann action orbits.

Findings

Analysis of flow collapse behavior

Application of Hilbert space lifts for flow investigation

Insights into focal submanifold evolution

Abstract

In this paper, we investigate the mean curvature flows for an equifocal submanifold in a symmetric space of compact type and its focal submanifolds as initial data. It is known that equifocal submanifolds of codimension greater than one in irreducible symmetric spaces of compact type occur as principal orbits of Hermann actions. However, we investigate the flows conceptionally without use of this fact. Concretely the investigation is performed by investigating the mean curvature flows having the lifts of the submanifolds to an (infinite deiminsional separable) Hilbert space through a Riemannian submersion as initial data.