On Mean curvature flow of Singular Riemannian foliations: Non compact cases

TL;DR

This paper studies the behavior of mean curvature flow in singular Riemannian foliations, showing finite-time singularities occur at singular leaves under bounded curvature, with implications for convergence and basin structures.

Contribution

It extends previous work by analyzing non-compact leaves and singularities in mean curvature flow within generalized isoparametric foliations.

Findings

Finite-time singularities occur at singular leaves.

Singularities are of type I under bounded curvature.

Cylinder structures influence convergence behavior.

Abstract

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. We also discuss the existence of basin of attractions, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and make a few remarks on MCF of non closed leaves of generalized isoparametric foliation.