Ancient mean curvature flows out of polytopes

Bourni, Theodora¹ ; Langford, Mat² ; Tinaglia, Giuseppe³

¹ Department of Mathematics, University of Tennessee Knoxville, Knoxville, TN, United States
² School of Mathematical and Physical Sciences, The University of Newcastle, Newcastle, NSW, Australia
³ Department of Mathematics, King’s College London, London, United Kingdom

Geometry & topology, Tome 26 (2022) no. 4, pp. 1849-1905.

Résumé

We develop a theory of convex ancient mean curvature flow in slab regions, with Grim hyperplanes playing a role analogous to that of half-spaces in the theory of convex bodies.

We first construct a large new class of examples. These solutions emerge from circumscribed polytopes at time minus infinity and decompose into corresponding configurations of “asymptotic translators”. This confirms a well-known conjecture attributed to Hamilton; see also Huisken and Sinestrari (2015). We construct examples in all dimensions $n\ge 2$, which include both compact and noncompact examples, and both symmetric and asymmetric examples, as well as a large family of eternal examples that do not evolve by translation. The latter resolve a conjecture of White (2003) in the negative.

We also obtain a partial classification of convex ancient solutions in slab regions via a detailed analysis of their asymptotics. Roughly speaking, we show that such solutions decompose at time minus infinity into a canonical configuration of Grim hyperplanes. An analogous decomposition holds at time plus infinity for eternal solutions. There are many further consequences of this analysis. One is a new rigidity result for translators. Another is that, in dimension two, solutions are necessarily reflection symmetric across the midplane of their slab.

Détail

DOI : 10.2140/gt.2022.26.1849

Keywords: polytopes, mean curvature flow, ancient solutions, translators

Bibliographie

[1] B Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994) 179 | DOI

[2] B Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012) 1413 | DOI

[3] B Andrews, M Langford, J Mccoy, Noncollapsing in fully nonlinear curvature flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013) 23 | DOI

[4] 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

[5] S Angenent, Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow, Netw. Heterog. Media 8 (2013) 1 | DOI

[6] S Angenent, S Brendle, P Daskalopoulos, N Šešum, Unique asymptotics of compact ancient solutions to three-dimensional Ricci flow, Comm. Pure Appl. Math. 75 (2022) 1032 | DOI

[7] S Angenent, P Daskalopoulos, N Sesum, Unique asymptotics of ancient convex mean curvature flow solutions, J. Differential Geom. 111 (2019) 381 | DOI

[8] S Angenent, P Daskalopoulos, N Sesum, Uniqueness of two-convex closed ancient solutions to the mean curvature flow, Ann. of Math. 192 (2020) 353 | DOI

[9] S Angenent, Q You, Ancient solutions to curve shortening with finite total curvature, Trans. Amer. Math. Soc. 374 (2021) 863 | DOI

[10] I Bakas, C Sourdis, Dirichlet sigma models and mean curvature flow, J. High Energy Phys. (2007) | DOI

[11] R H Bamler, B Kleiner, On the rotational symmetry of 3–dimensional κ–solutions, J. Reine Angew. Math. 779 (2021) 37 | DOI

[12] T Bourni, M Langford, Type-II singularities of two-convex immersed mean curvature flow, Geom. Flows 2 (2017) 1 | DOI

[13] T Bourni, M Langford, A Mramor, On the construction of closed nonconvex nonsoliton ancient mean curvature flows, Int. Math. Res. Not. 2021 (2021) 757 | DOI

[14] T Bourni, M Langford, G Tinaglia, Convex ancient solutions to curve shortening flow, Calc. Var. Partial Differential Equations 59 (2020) | DOI

[15] T Bourni, M Langford, G Tinaglia, On the existence of translating solutions of mean curvature flow in slab regions, Anal. PDE 13 (2020) 1051 | DOI

[16] T Bourni, M Langford, G Tinaglia, Collapsing ancient solutions of mean curvature flow, J. Differential Geom. 119 (2021) 187 | DOI

[17] T Bourni, M Langford, G Tinaglia, Convex ancient solutions to mean curvature flow, from: "Differential geometry in the large" (editors O Dearricott, W Tuschmann, Y Nikolayevsky, T Leistner, D Crowley), London Math. Soc. Lecture Note Ser. 463, Cambridge Univ. Press (2021) 50

[18] K A Brakke, The motion of a surface by its mean curvature, 20, Princeton Univ. Press (1978)

[19] S Brendle, Rotational symmetry of Ricci solitons in higher dimensions, J. Differential Geom. 97 (2014) 191

[20] S Brendle, K Choi, Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions, Geom. Topol. 25 (2021) 2195 | DOI

[21] P Bryan, M N Ivaki, Harnack estimate for mean curvature flow on the sphere, Asian J. Math. 24 (2020) 165 | DOI

[22] P Bryan, M N Ivaki, J Scheuer, On the classification of ancient solutions to curvature flows on the sphere, preprint (2016)

[23] P Bryan, J Louie, Classification of convex ancient solutions to curve shortening flow on the sphere, J. Geom. Anal. 26 (2016) 858 | DOI

[24] R Bryant, Ricci flow solitons in dimension three with SO(3)–symmetries, unpublished notes (2005)

[25] F Chini, N M Møller, Ancient mean curvature flows and their spacetime tracks, preprint (2019)

[26] F Chini, N M Møller, Bi-halfspace and convex hull theorems for translating solitons, Int. Math. Res. Not. 2021 (2021) 13011 | DOI

[27] K Choi, C Mantoulidis, Ancient gradient flows of elliptic functionals and Morse index, Amer. J. Math. 144 (2022) 541 | DOI

[28] B Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. Geom. 5 (1997) 389 | DOI

[29] B Chow, R Gulliver, Aleksandrov reflection and geometric evolution of hypersurfaces, Comm. Anal. Geom. 9 (2001) 261 | DOI

[30] P Daskalopoulos, Ancient solutions to geometric flows, from: "Proceedings of the International Congress of Mathematicians, III", Kyung Moon Sa (2014) 773

[31] P Daskalopoulos, R Hamilton, N Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom. 84 (2010) 455

[32] P Daskalopoulos, R Hamilton, N Sesum, Classification of ancient compact solutions to the Ricci flow on surfaces, J. Differential Geom. 91 (2012) 171

[33] P Daskalopoulos, M Del Pino, J King, N Sesum, Type I ancient compact solutions of the Yamabe flow, Nonlinear Anal. 137 (2016) 338 | DOI

[34] P Daskalopoulos, M Del Pino, J King, N Sesum, New type I ancient compact solutions of the Yamabe flow, Math. Res. Lett. 24 (2017) 1667 | DOI

[35] P Daskalopoulos, M Del Pino, N Sesum, Type II ancient compact solutions to the Yamabe flow, J. Reine Angew. Math. 738 (2018) 1 | DOI

[36] P Daskalopoulos, N Sesum, Eternal solutions to the Ricci flow on R2, Int. Math. Res. Not. 2006 (2006) | DOI

[37] K Ecker, G Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991) 547 | DOI

[38] V A Fateev, E Onofri, A B Zamolodchikov, Integrable deformations of the O(3) sigma model : the sausage model, Nuclear Phys. B 406 (1993) 521 | DOI

[39] B Grünbaum, Convex polytopes, 221, Springer (2003) | DOI

[40] R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982) 255

[41] R S Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993) 225

[42] R S Hamilton, Convex hypersurfaces with pinched second fundamental form, Comm. Anal. Geom. 2 (1994) 167 | DOI

[43] R S Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995) 215

[44] R Haslhofer, Uniqueness of the bowl soliton, Geom. Topol. 19 (2015) 2393 | DOI

[45] R Haslhofer, O Hershkovits, Ancient solutions of the mean curvature flow, Comm. Anal. Geom. 24 (2016) 593 | DOI

[46] D Hoffman, T Ilmanen, F Martín, B White, Graphical translators for mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019) | DOI

[47] G Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984) 237

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

[49] G Huisken, C Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999) 1 | DOI

[50] G Huisken, C Sinestrari, Convex ancient solutions of the mean curvature flow, J. Differential Geom. 101 (2015) 267

[51] J Isenberg, H Wu, Mean curvature flow of noncompact hypersurfaces with type-II curvature blow-up, J. Reine Angew. Math. 754 (2019) 225 | DOI

[52] N J Korevaar, R Kusner, B Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989) 465

[53] Y Lai, A family of 3d steady gradient solitons that are flying wings, preprint (2020)

[54] B Lambert, J D Lotay, F Schulze, Ancient solutions in Lagrangian mean curvature flow, Ann. Sc. Norm. Super. Pisa Cl. Sci. 22 (2021) 1169

[55] M Langford, A general pinching principle for mean curvature flow and applications, Calc. Var. Partial Differential Equations 56 (2017) | DOI

[56] M Langford, S Lynch, Sharp one-sided curvature estimates for fully nonlinear curvature flows and applications to ancient solutions, J. Reine Angew. Math. 765 (2020) 1 | DOI

[57] S L Lukyanov, E S Vitchev, A B Zamolodchikov, Integrable model of boundary interaction: the paperclip, Nuclear Phys. B 683 (2004) 423 | DOI

[58] S L Lukyanov, A B Zamolodchikov, Dual form of the paperclip model, Nuclear Phys. B 744 (2006) 295 | DOI

[59] S Lynch, H T Nguyen, Pinched ancient solutions to the high codimension mean curvature flow, Calc. Var. Partial Differential Equations 60 (2021) | DOI

[60] J M S Ma, Ancient solutions to the curve shortening flow spanning the halfplane, Trans. Amer. Math. Soc. 374 (2021) 4207 | DOI

[61] A Mramor, A Payne, Ancient and eternal solutions to mean curvature flow from minimal surfaces, Math. Ann. 380 (2021) 569 | DOI

[62] W W Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956) 900 | DOI

[63] J Von Neumann, Discussion: shape of metal grains, from: "Metal interfaces", American Society for Metals (1952) 108

[64] L Ni, Ancient solutions to Kähler–Ricci flow, Math. Res. Lett. 12 (2005) 633 | DOI

[65] L Ni, Closed type I ancient solutions to Ricci flow, from: "Recent advances in geometric analysis" (editors Y I Lee, C S Lin, M P Tsui), Adv. Lect. Math. 11, International (2010) 147

[66] G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)

[67] S Risa, C Sinestrari, Ancient solutions of geometric flows with curvature pinching, J. Geom. Anal. 29 (2019) 1206 | DOI

[68] S Risa, C Sinestrari, Strong spherical rigidity of ancient solutions of expansive curvature flows, Bull. Lond. Math. Soc. 52 (2020) 94 | DOI

[69] M Sáez, O C Schnürer, Mean curvature flow without singularities, J. Differential Geom. 97 (2014) 545

[70] W Sheng, X J Wang, Singularity profile in the mean curvature flow, Methods Appl. Anal. 16 (2009) 139 | DOI

[71] K Sonnanburg, A Liouville theorem for mean curvature flow, preprint (2017)

[72] J Spruck, L Xiao, Complete translating solitons to the mean curvature flow in R3 with nonnegative mean curvature, Amer. J. Math. 142 (2020) 993 | DOI

[73] X J Wang, Convex solutions to the mean curvature flow, Ann. of Math. 173 (2011) 1185 | DOI

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

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