Dynamics of closed singularities
Annales de l'Institut Fourier to appear, , 44 p.

Parabolic geometric flows have the property of 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 this paper is that, by bringing in the dynamical properties of the flow, we obtain also smoothing for long time for generic initial conditions. When combined with one our earlier paper, this allows us to show that, in an important special case, the singularities are the simplest possible.

We take here the first steps towards understanding the dynamics of the flow. 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 of the flow of things close to the singularity.

That is, if the flow leaves a neighborhood of a singularity, is it possible for it to re-enter the same neighborhood at a much later time? The first part is addressed in this paper, while the second will be addressed in a forthcoming paper.

Les flots géométriques paraboliques ont la propriété de régulariser en temps court, néanmoins, en temps long, l’apparition de singularités est inévitable et elle peuvent être compliquées et impossibles à classifier. L’idée qui sous-tend cet article est que l’utilisation des propriétés dynamiques du flot permet d’obtenir une régularisation en temps long aussi, pour des conditions initiales génériques. Combiné avec un article récent, il nous permet de montrer, dans un cas particulier important, que les singularités sont les plus simples possibles.

Il s’agit du premier pas vers une compréhension de la dynamique du flot. La question de la dynamique d’une singularité revêt deux aspects. Le premier est : quelles sont les dynamiques proches d’une singularité. Le second est : quel est le comportement en temps long des éléments proches de la singularité. Plus précisément, si le flot sort du voisinage d’une singularité, est-il possible qu’il revienne dans ce voisinage ultérieurement ? La première question est discutée dans cet article, la seconde dans un article à venir.

Classification:  58J35,  35K10
Keywords: Parabolic geometric flows, singularities, dynamics of parabolic flows
     author = {Colding, Tobias Holck and Minicozzi II, William P.},
     title = {Dynamics of closed singularities},
     note = {to appear in \emph{Annales de l'Institut Fourier}},
Colding, Tobias Holck; Minicozzi II, William P. Dynamics of closed singularities. Annales de l'Institut Fourier, to appear, 44 p.

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