Geometry & Topology Volume 26, issue 4 (2022)

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Page

Policies for Authors

Ethics Statement

ISSN 1364-0380 (online)

ISSN 1465-3060 (print)

Author Index

To Appear

Other MSP Journals

Ancient mean curvature flows out of polytopes

Theodora Bourni, Mat Langford and Giuseppe Tinaglia

Geometry & Topology 26 (2022) 1849–1905

DOI: 10.2140/gt.2022.26.1849

Bibliography

1	B Andrews, Harnack inequalities for evolving hypersurfaces, Math. Z. 217 (1994) 179 MR1296393
2	B Andrews, Noncollapsing in mean-convex mean curvature flow, Geom. Topol. 16 (2012) 1413 MR2967056
3	B Andrews, M Langford, J McCoy, Noncollapsing in fully nonlinear curvature flows, Ann. Inst. H. Poincaré C Anal. Non Linéaire 30 (2013) 23 MR3011290
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 MR1167827
5	S Angenent, Formal asymptotic expansions for symmetric ancient ovals in mean curvature flow, Netw. Heterog. Media 8 (2013) 1 MR3043925
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 MR4400906
7	S Angenent, P Daskalopoulos, N Sesum, Unique asymptotics of ancient convex mean curvature flow solutions, J. Differential Geom. 111 (2019) 381 MR3934596
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 MR4151080
9	S Angenent, Q You, Ancient solutions to curve shortening with finite total curvature, Trans. Amer. Math. Soc. 374 (2021) 863 MR4196380
10	I Bakas, C Sourdis, Dirichlet sigma models and mean curvature flow, J. High Energy Phys. (2007) MR2326594
11	R H Bamler, B Kleiner, On the rotational symmetry of 3–dimensional κ–solutions, J. Reine Angew. Math. 779 (2021) 37 MR4319063
12	T Bourni, M Langford, Type-II singularities of two-convex immersed mean curvature flow, Geom. Flows 2 (2017) 1 MR3562950
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 MR4199238
14	T Bourni, M Langford, G Tinaglia, Convex ancient solutions to curve shortening flow, Calc. Var. Partial Differential Equations 59 (2020) MR4127403
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 MR4109899
16	T Bourni, M Langford, G Tinaglia, Collapsing ancient solutions of mean curvature flow, J. Differential Geom. 119 (2021) 187 MR4318294
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 MR4420785
18	K A Brakke, The motion of a surface by its mean curvature, 20, Princeton Univ. Press (1978) MR0485012
19	S Brendle, Rotational symmetry of Ricci solitons in higher dimensions, J. Differential Geom. 97 (2014) 191 MR3231974
20	S Brendle, K Choi, Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions, Geom. Topol. 25 (2021) 2195 MR4310889
21	P Bryan, M N Ivaki, Harnack estimate for mean curvature flow on the sphere, Asian J. Math. 24 (2020) 165 MR4143987
22	P Bryan, M N Ivaki, J Scheuer, On the classification of ancient solutions to curvature flows on the sphere, preprint (2016) arXiv:1604.01694
23	P Bryan, J Louie, Classification of convex ancient solutions to curve shortening flow on the sphere, J. Geom. Anal. 26 (2016) 858 MR3472819
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) arXiv:1901.05481
26	F Chini, N M Møller, Bi-halfspace and convex hull theorems for translating solitons, Int. Math. Res. Not. 2021 (2021) 13011 MR4307681
27	K Choi, C Mantoulidis, Ancient gradient flows of elliptic functionals and Morse index, Amer. J. Math. 144 (2022) 541 MR4401511
28	B Chow, Geometric aspects of Aleksandrov reflection and gradient estimates for parabolic equations, Comm. Anal. Geom. 5 (1997) 389 MR1483984
29	B Chow, R Gulliver, Aleksandrov reflection and geometric evolution of hypersurfaces, Comm. Anal. Geom. 9 (2001) 261 MR1846204
30	P Daskalopoulos, Ancient solutions to geometric flows, from: "Proceedings of the International Congress of Mathematicians, III", Kyung Moon Sa (2014) 773 MR3729051
31	P Daskalopoulos, R Hamilton, N Sesum, Classification of compact ancient solutions to the curve shortening flow, J. Differential Geom. 84 (2010) 455 MR2669361
32	P Daskalopoulos, R Hamilton, N Sesum, Classification of ancient compact solutions to the Ricci flow on surfaces, J. Differential Geom. 91 (2012) 171 MR2971286
33	P Daskalopoulos, M del Pino, J King, N Sesum, Type I ancient compact solutions of the Yamabe flow, Nonlinear Anal. 137 (2016) 338 MR3485129
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 MR3762689
35	P Daskalopoulos, M del Pino, N Sesum, Type II ancient compact solutions to the Yamabe flow, J. Reine Angew. Math. 738 (2018) 1 MR3794888
36	P Daskalopoulos, N Sesum, Eternal solutions to the Ricci flow on ℝ², Int. Math. Res. Not. 2006 (2006) MR2264733
37	K Ecker, G Huisken, Interior estimates for hypersurfaces moving by mean curvature, Invent. Math. 105 (1991) 547 MR1117150
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 MR1239623
39	B Grünbaum, Convex polytopes, 221, Springer (2003) MR1976856
40	R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982) 255 MR664497
41	R S Hamilton, The Harnack estimate for the Ricci flow, J. Differential Geom. 37 (1993) 225 MR1198607
42	R S Hamilton, Convex hypersurfaces with pinched second fundamental form, Comm. Anal. Geom. 2 (1994) 167 MR1312684
43	R S Hamilton, Harnack estimate for the mean curvature flow, J. Differential Geom. 41 (1995) 215 MR1316556
44	R Haslhofer, Uniqueness of the bowl soliton, Geom. Topol. 19 (2015) 2393 MR3375531
45	R Haslhofer, O Hershkovits, Ancient solutions of the mean curvature flow, Comm. Anal. Geom. 24 (2016) 593 MR3521319
46	D Hoffman, T Ilmanen, F Martín, B White, Graphical translators for mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019) MR3962912
47	G Huisken, Flow by mean curvature of convex surfaces into spheres, J. Differential Geom. 20 (1984) 237 MR772132
48	G Huisken, C Sinestrari, Convexity estimates for mean curvature flow and singularities of mean convex surfaces, Acta Math. 183 (1999) 45 MR1719551
49	G Huisken, C Sinestrari, Mean curvature flow singularities for mean convex surfaces, Calc. Var. Partial Differential Equations 8 (1999) 1 MR1666878
50	G Huisken, C Sinestrari, Convex ancient solutions of the mean curvature flow, J. Differential Geom. 101 (2015) 267 MR3399098
51	J Isenberg, H Wu, Mean curvature flow of noncompact hypersurfaces with type-II curvature blow-up, J. Reine Angew. Math. 754 (2019) 225 MR4000574
52	N J Korevaar, R Kusner, B Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989) 465 MR1010168
53	Y Lai, A family of 3d steady gradient solitons that are flying wings, preprint (2020) arXiv:2010.07272
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 MR4334316
55	M Langford, A general pinching principle for mean curvature flow and applications, Calc. Var. Partial Differential Equations 56 (2017) MR3669776
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 MR4129354
57	S L Lukyanov, E S Vitchev, A B Zamolodchikov, Integrable model of boundary interaction: the paperclip, Nuclear Phys. B 683 (2004) 423 MR2057110
58	S L Lukyanov, A B Zamolodchikov, Dual form of the paperclip model, Nuclear Phys. B 744 (2006) 295 MR2229176
59	S Lynch, H T Nguyen, Pinched ancient solutions to the high codimension mean curvature flow, Calc. Var. Partial Differential Equations 60 (2021) MR4201652
60	J M S Ma, Ancient solutions to the curve shortening flow spanning the halfplane, Trans. Amer. Math. Soc. 374 (2021) 4207 MR4251227
61	A Mramor, A Payne, Ancient and eternal solutions to mean curvature flow from minimal surfaces, Math. Ann. 380 (2021) 569 MR4263692
62	W W Mullins, Two-dimensional motion of idealized grain boundaries, J. Appl. Phys. 27 (1956) 900 MR78836
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 MR2189227
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 MR2648942
66	G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002) arXiv:math/0211159
67	S Risa, C Sinestrari, Ancient solutions of geometric flows with curvature pinching, J. Geom. Anal. 29 (2019) 1206 MR3935256
68	S Risa, C Sinestrari, Strong spherical rigidity of ancient solutions of expansive curvature flows, Bull. Lond. Math. Soc. 52 (2020) 94 MR4072034
69	M Sáez, O C Schnürer, Mean curvature flow without singularities, J. Differential Geom. 97 (2014) 545 MR3263514
70	W Sheng, X J Wang, Singularity profile in the mean curvature flow, Methods Appl. Anal. 16 (2009) 139 MR2563745
71	K Sonnanburg, A Liouville theorem for mean curvature flow, preprint (2017) arXiv:1711.02261
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 MR4101337
73	X J Wang, Convex solutions to the mean curvature flow, Ann. of Math. 173 (2011) 1185 MR2800714
74	B White, The size of the singular set in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 13 (2000) 665 MR1758759
75	B White, The nature of singularities in mean curvature flow of mean-convex sets, J. Amer. Math. Soc. 16 (2003) 123 MR1937202