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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


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.


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Tobias Holck Colding. William P. Minicozzi. "Wandering singularities." J. Differential Geom. 119 (3) 403 - 420, November 2021.


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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