Paper · Sep 10, 2019

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

Marcos M. Alexandrino, Leonardo F. Cavenaghi, Icaro Gonçalves

math.DGWider flowsarXiv:1909.04201v4
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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.

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Submitted
Sep 10, 2019
Last revised
Dec 8, 2019 (v4)
Subjects
math.DG

Comments: In this (last) version, we added in Proposition 5.1 the proof of Type I convergence of MCF of non closed leaves of SRF

Mean curvature flowSingularitiesBoundaries & non-compact

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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