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November 2021 Wandering singularities
Tobias Holck Colding, William P. Minicozzi
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J. Differential Geom. 119(3): 403-420 (November 2021). DOI: 10.4310/jdg/1635368532

Abstract

Parabolic geometric flows are smoothing for short time however, over long time, singularities are typically unavoidable, can be very nasty and may be impossible to classify. The idea of [CM6] and here is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for large time for generic initial conditions. When combined with [CM1], this shows, in an important special case, the singularities are the simplest possible.

The question of the dynamics of a singularity has two parts. One is: What are the dynamics near a singularity? The second is: What is the long time behavior? That is, if the flow leaves a neighborhood of a singularity, can it return at a much later time? The first question was addressed in [CM6] and the second here.

Combined with [CM1], [CM6], we show that all other closed singularities than the (round) sphere have a neighborhood where “nearly every” closed hypersurface leaves under the flow and never returns, even to a dilated, rotated or translated copy of the singularity. In other words, it wanders off. In contrast, by Huisken, any closed hypersurface near a sphere remains close to a dilated or translated copy of the sphere at each time.

Funding Statement

The authors were partially supported by NSF Grants DMS 1812142 and DMS 1707270.

Citation

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Tobias Holck Colding. William P. Minicozzi. "Wandering singularities." J. Differential Geom. 119 (3) 403 - 420, November 2021. https://doi.org/10.4310/jdg/1635368532

Information

Received: 11 September 2018; Accepted: 3 July 2019; Published: November 2021
First available in Project Euclid: 1 November 2021

Digital Object Identifier: 10.4310/jdg/1635368532

Rights: Copyright © 2021 Lehigh University

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Vol.119 • No. 3 • November 2021
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