Collapse of the mean curvature flow for isoparametric submanifolds in non-compact symmetric spaces

TL;DR

This paper studies the behavior of mean curvature flow for specific isoparametric submanifolds in non-compact symmetric spaces, revealing conditions under which the flow collapses.

Contribution

It introduces a method to analyze mean curvature flow for curvature-adapted isoparametric submanifolds via their lift to an infinite-dimensional pseudo-Hilbert space.

Findings

Mean curvature flow collapses for the studied submanifolds.

The analysis uses pseudo-Riemannian submersion techniques.

Results extend understanding of curvature flow in non-compact symmetric spaces.

Abstract

It is known that principal orbits of Hermann actions on a symmetric space of non-compact type are curvature-adapted isoparametric submanifolds having no focal point of non-Euclidean type on the ideal boundary of the ambient symmetric space. In this paper, we investigate the mean curvature flows for such a curvature-adapted isoparametric submanifold and its focal submanifold. Concretely the investigation is performed by investigating the mean curvature flows for the lift of the submanifold to an infinite dimensional pseudo-Hilbert space through a pseudo-Riemannian submersion.