Dynamics of closed singularities
Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 2973-3016.

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.

Published online:
DOI: 10.5802/aif.3343
Classification: 58J35, 35K10
Keywords: Parabolic geometric flows, singularities, dynamics of parabolic flows
Mot clés : Flots géométiques paraboliques, singularités, dynamique des flots paraboliques
Colding, Tobias Holck 1; Minicozzi II, William P. 1

1 MIT, Dept. of Math. 77 Massachusetts Avenue Cambridge, MA 02139-4307 (USA)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     author = {Colding, Tobias Holck and Minicozzi II, William P.},
     title = {Dynamics of closed singularities},
     journal = {Annales de l'Institut Fourier},
     pages = {2973--3016},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {7},
     year = {2019},
     doi = {10.5802/aif.3343},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3343/}
AU  - Colding, Tobias Holck
AU  - Minicozzi II, William P.
TI  - Dynamics of closed singularities
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 2973
EP  - 3016
VL  - 69
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3343/
DO  - 10.5802/aif.3343
LA  - en
ID  - AIF_2019__69_7_2973_0
ER  - 
%0 Journal Article
%A Colding, Tobias Holck
%A Minicozzi II, William P.
%T Dynamics of closed singularities
%J Annales de l'Institut Fourier
%D 2019
%P 2973-3016
%V 69
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3343/
%R 10.5802/aif.3343
%G en
%F AIF_2019__69_7_2973_0
Colding, Tobias Holck; Minicozzi II, William P. Dynamics of closed singularities. Annales de l'Institut Fourier, Volume 69 (2019) no. 7, pp. 2973-3016. doi : 10.5802/aif.3343. https://aif.centre-mersenne.org/articles/10.5802/aif.3343/

[1] Angenent, Sigurd B. Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) (Progress in Nonlinear Differential Equations and their Applications), Volume 7, Birkhäuser, 1992, pp. 21-38 | DOI | MR | Zbl

[2] Chopp, David L. Computation of self-similar solutions for mean curvature flow, Exp. Math., Volume 3 (1994) no. 1, pp. 1-15 | DOI | MR | Zbl

[3] Colding, Tobias H.; Minicozzi, William P. II Generic mean curvature flow I: generic singularities, Ann. Math., Volume 175 (2012) no. 2, pp. 755-833 | DOI | MR | Zbl

[4] Colding, Tobias H.; Minicozzi, William P. II Uniqueness of blowups and Łojasiewicz inequalities, Ann. Math., Volume 182 (2015) no. 1, pp. 221-285 | DOI | MR | Zbl

[5] Colding, Tobias H.; Minicozzi, William P. II Level set method for motion by mean curvature, Notices Am. Math. Soc., Volume 63 (2016) no. 10, pp. 1148-1153 | DOI | MR | Zbl

[6] Colding, Tobias H.; Minicozzi, William P. II Complexity of parabolic systems (2019) (https://arxiv.org/abs/1903.03499)

[7] Colding, Tobias H.; Minicozzi, William P. II In search of stable geometric structures (2019) (https://arxiv.org/abs/1907.03672, to appear in Notices Am. Math. Soc.) | Zbl

[8] Colding, Tobias H.; Minicozzi, William P. II; Pedersen, Erik Kjær Mean curvature flow, Bull. Am. Math. Soc., Volume 52 (2015) no. 2, pp. 297-333 | DOI | MR | Zbl

[9] Ecker, Klaus; Huisken, Gerhard Mean curvature evolution of entire graphs, Ann. Math., Volume 130 (1989) no. 3, pp. 453-471 | DOI | MR | Zbl

[10] Ecker, Klaus; Huisken, Gerhard Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., Volume 105 (1991) no. 3, pp. 547-569 | DOI | MR | Zbl

[11] Epstein, Charles L.; Weinstein, Michael I. A stable manifold theorem for the curve shortening equation, Commun. Pure Appl. Math., Volume 40 (1987) no. 1, pp. 119-139 | DOI | MR | Zbl

[12] Gage, M.; Hamilton, Richard S. The heat equation shrinking convex plane curves, J. Differ. Geom., Volume 23 (1986) no. 1, pp. 69-96 | DOI | MR | Zbl

[13] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, xiv+517 pages | MR | Zbl

[14] Hirsch, Morris W.; Pugh, Charles C. Stable manifolds and hyperbolic sets, Global Analysis (Berkeley, 1968) (Proceedings of Symposia in Pure Mathematics), Volume 16 (1970), pp. 133-163 | MR | Zbl

[15] Huisken, Gerhard Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., Volume 20 (1984) no. 1, pp. 237-266 | DOI | MR | Zbl

[16] Huisken, Gerhard Nonparametric mean curvature evolution with boundary conditions, J. Differ. Equations, Volume 77 (1989) no. 2, pp. 369-378 | DOI | MR | Zbl

[17] Huisken, Gerhard Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom., Volume 31 (1990) no. 1, pp. 285-299 | DOI | MR | Zbl

[18] Huisken, Gerhard; Polden, Alexander Geometric evolution equations for hypersurfaces, Calculus of variations and geometric evolution problems (Cetraro, 1996) (Lecture Notes in Mathematics), Volume 1713, Springer, 1999, pp. 45-84 | DOI | MR | Zbl

[19] Ilmanen, Tom Singularities of Mean Curvature Flow of Surfaces (https://people.math.ethz.ch/~ilmanen/papers/sing.ps)

[20] Ilmanen, Tom Elliptic regularization and partial regularity for motion by mean curvature, Mem. Am. Math. Soc., Volume 108 (1994) no. 520, x+90 pages | DOI | MR | Zbl

[21] Kapouleas, Nikolaos; Kleene, Stephen James; Møller, Niels Martin Mean curvature self-shrinkers of high genus: non-compact examples, J. Reine Angew. Math., Volume 739 (2018), pp. 1-39 | DOI | MR | Zbl

[22] Ketover, Daniel Self-shrinking Platonic solids (2016) (https://arxiv.org/abs/1602.07271)

[23] Ladyženskaja, Olga A.; Solonnikov, Vsevolod A.; Uralʼceva, Nina N. Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, 23, American Mathematical Society, 1968, xi+648 pages | MR

[24] Lieberman, Gary M. The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 13 (1986) no. 3, pp. 347-387 | Numdam | MR | Zbl

[25] Lieberman, Gary M. Second order parabolic differential equations, World Scientific, 1996, xii+439 pages | DOI | MR | Zbl

[26] Nguyen, Xuan Hien Construction of complete embedded self-similar surfaces under mean curvature flow, Part III, Duke Math. J., Volume 163 (2014) no. 11, pp. 2023-2056 | DOI | MR | Zbl

[27] Sesum, Natasa Rate of convergence of the mean curvature flow, Commun. Pure Appl. Math., Volume 61 (2008) no. 4, pp. 464-485 | DOI | MR | Zbl

[28] White, Brian Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math., Volume 488 (1997), pp. 1-35 | DOI | MR | Zbl

Cited by Sources: