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
@unpublished{AIF_0__0_0_A22_0,
     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.

[1] Angenent, Sigurd B. Shrinking doughnuts, Nonlinear diffusion equations and their equilibrium states, 3 (Gregynog, 1989) (Progress in Nonlinear Differential Equations and their Applications) Tome 7, Birkhäuser, 1992, pp. 21-38 | MR 1167827

[2] Chopp, David L. Computation of self-similar solutions for mean curvature flow, Exp. Math., Tome 3 (1994) no. 1, pp. 1-15 | MR 1302814

[3] Colding, Tobias H.; Minicozzi, William P. II Generic mean curvature flow I: generic singularities, Ann. Math., Tome 175 (2012) no. 2, pp. 755-833 | Article | MR 2993752

[4] Colding, Tobias H.; Minicozzi, William P. II Uniqueness of blowups and Łojasiewicz inequalities, Ann. Math., Tome 182 (2015) no. 1, pp. 221-285 | Article | MR 3374960

[5] Colding, Tobias H.; Minicozzi, William P. II Level set method for motion by mean curvature, Notices Am. Math. Soc., Tome 63 (2016) no. 10, pp. 1148-1153 | Article | MR 3559343

[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.)

[8] Colding, Tobias H.; Minicozzi, William P. II; Pedersen, Erik Kjær Mean curvature flow, Bull. Am. Math. Soc., Tome 52 (2015) no. 2, pp. 297-333 | Article | MR 3312634

[9] Ecker, Klaus; Huisken, Gerhard Mean curvature evolution of entire graphs, Ann. Math., Tome 130 (1989) no. 3, pp. 453-471 | Article | MR 1025164

[10] Ecker, Klaus; Huisken, Gerhard Interior estimates for hypersurfaces moving by mean curvature, Invent. Math., Tome 105 (1991) no. 3, pp. 547-569 | Article | MR 1117150

[11] Epstein, Charles L.; Weinstein, Michael I. A stable manifold theorem for the curve shortening equation, Commun. Pure Appl. Math., Tome 40 (1987) no. 1, pp. 119-139 | Article | MR 865360

[12] Gage, M.; Hamilton, Richard S. The heat equation shrinking convex plane curves, J. Differ. Geom., Tome 23 (1986) no. 1, pp. 69-96 | MR 840401 | Zbl 0621.53001

[13] Gilbarg, David; Trudinger, Neil S. Elliptic partial differential equations of second order, Classics in Mathematics, Springer, 2001, xiv+517 pages | MR 1814364

[14] Hirsch, Morris W.; Pugh, Charles C. Stable manifolds and hyperbolic sets, Global Analysis (Berkeley, 1968) (Proceedings of Symposia in Pure Mathematics) Tome 16 (1970), pp. 133-163 | MR 0271991 | Zbl 0215.53001

[15] Huisken, Gerhard Flow by mean curvature of convex surfaces into spheres, J. Differ. Geom., Tome 20 (1984) no. 1, pp. 237-266 | MR 772132

[16] Huisken, Gerhard Nonparametric mean curvature evolution with boundary conditions, J. Differ. Equations, Tome 77 (1989) no. 2, pp. 369-378 | Article | MR 983300

[17] Huisken, Gerhard Asymptotic behavior for singularities of the mean curvature flow, J. Differ. Geom., Tome 31 (1990) no. 1, pp. 285-299 | MR 1030675

[18] Huisken, Gerhard; Polden, Alexander Geometric evolution equations for hypersurfaces, Calculus of variations and geometric evolution problems (Cetraro, 1996) (Lecture Notes in Mathematics) Tome 1713, Springer, 1999, pp. 45-84 | Article | MR 1731639

[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., Tome 108 (1994) no. 520, x+90 pages | Article | MR 1196160

[21] Kapouleas, Nikolaos; Kleene, Stephen James; Møller, Niels Martin Mean curvature self-shrinkers of high genus: non-compact examples, J. Reine Angew. Math., Tome 739 (2018), pp. 1-39 | Article | MR 3808256

[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, Tome 23, American Mathematical Society, 1968, xi+648 pages | MR 0241822

[24] Lieberman, Gary M. The first initial-boundary value problem for quasilinear second order parabolic equations, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Tome 13 (1986) no. 3, pp. 347-387 | MR 881097

[25] Lieberman, Gary M. Second order parabolic differential equations, World Scientific, 1996, xii+439 pages | Article | MR 1465184 | Zbl 0884.35001

[26] Nguyen, Xuan Hien Construction of complete embedded self-similar surfaces under mean curvature flow, Part III, Duke Math. J., Tome 163 (2014) no. 11, pp. 2023-2056 | Article | MR 3263027

[27] Sesum, Natasa Rate of convergence of the mean curvature flow, Commun. Pure Appl. Math., Tome 61 (2008) no. 4, pp. 464-485 | Article | MR 2383930

[28] White, Brian Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math., Tome 488 (1997), pp. 1-35 | Article | MR 1465365