Sharp entropy bounds for self-shrinkers in mean curvature flow

Hershkovits, Or¹ ; White, Brian¹

¹ Department of Mathematics, Stanford University, Stanford, CA, United States

Geometry & topology, Tome 23 (2019) no. 3, pp. 1611-1619.

Résumé

Let $M\subset {ℝ}^{m+1}$ be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial $k^{\text{ th}}$ homology. We show that the entropy of $M$ is greater than or equal to the entropy of a round $k$-sphere, and that if equality holds, then $M$ is a round $k$-sphere in ${ℝ}^{k+1}$.

Détail

DOI : 10.2140/gt.2019.23.1611

Classification : 53C44, 49Q20
Keywords: mean curvature flow, entropy, shrinker

Bibliographie

[1] S B Angenent, Shrinking doughnuts, from: "Nonlinear diffusion equations and their equilibrium states, III" (editors N G Lloyd, W M Ni, L A Peletier, J Serrin), Progr. Nonlinear Differential Equations Appl. 7, Birkhäuser (1992) 21 | DOI

[2] J Bernstein, L Wang, A topological property of asymptotically conical self-shrinkers of small entropy, Duke Math. J. 166 (2017) 403 | DOI

[3] J Bernstein, L Wang, Topology of closed hypersurfaces of small entropy, Geom. Topol. 22 (2018) 1109 | DOI

[4] T H Colding, T Ilmanen, W P Minicozzi Ii, B White, The round sphere minimizes entropy among closed self-shrinkers, J. Differential Geom. 95 (2013) 53 | DOI

[5] T H Colding, W P Minicozzi Ii, Generic mean curvature flow, I : Generic singularities, Ann. of Math. 175 (2012) 755 | DOI

[6] R Haslhofer, O Hershkovits, Singularities of mean convex level set flow in general ambient manifolds, Adv. Math. 329 (2018) 1137 | DOI

[7] O Hershkovits, B White, The regularity and structure theory of advancing fronts motion by mean curvature plus a forcing term, in preparation

[8] O Hershkovits, B White, Avoidance for set-theoretic solutions of mean-curvature-type flows, preprint (2018)

[9] O Hershkovits, B White, A Morse-theoretic approach to topological changes in mean curvature flow, in preparation (2018)

[10] G Huisken, Asymptotic behavior for singularities of the mean curvature flow, J. Differential Geom. 31 (1990) 285 | DOI

[11] G Huisken, C Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999) 45 | DOI

[12] T Ilmanen, The level-set flow on a manifold, from: "Differential geometry: partial differential equations on manifolds" (editor R Greene), Proc. Sympos. Pure Math. 54, Amer. Math. Soc. (1993) 193 | DOI

[13] T Ilmanen, Singularities of mean curvature flow of surfaces, preprint (1995)

[14] T Ilmanen, Problems in mean curvature flow, preprint (2003)

[15] N Kapouleas, S J Kleene, N M Møller, Mean curvature self-shrinkers of high genus: non-compact examples, J. Reine Angew. Math. 739 (2018) 1 | DOI

[16] D Ketover, Self-shrinking platonic solids, preprint (2016)

[17] A Stone, A density function and the structure of singularities of the mean curvature flow, Calc. Var. Partial Differential Equations 2 (1994) 443 | DOI

[18] B White, The topology of hypersurfaces moving by mean curvature, Comm. Anal. Geom. 3 (1995) 317 | DOI

[19] B White, Stratification of minimal surfaces, mean curvature flows, and harmonic maps, J. Reine Angew. Math. 488 (1997) 1 | DOI

[20] B White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000) 665 | DOI

[21] B White, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 16 (2003) 123 | DOI

[22] B White, Topological change in mean convex mean curvature flow, Invent. Math. 191 (2013) 501 | DOI

[23] B White, Subsequent singularities in mean-convex mean curvature flow, Calc. Var. Partial Differential Equations 54 (2015) 1457 | DOI