Geodesic
Classification of bubble-sheet ovals in ℝ4
Choi, Beomjun1 ; Daskalopoulos, Panagiota2 ; Du, Wenkui3 ; Haslhofer, Robert4 ; Šešum, Nataša5
1Department of Mathematics, POSTECH, Gyeongbuk, South Korea
2Department of Mathematics, Columbia University, New York, NY, United States
3Department of Mathematics, Massachusetts Insitute of Technology, Cambridge, MA, United States
4Department of Mathematics, University of Toronto, Toronto, ON, Canada
5Department of Mathematics, Rutgers University, Piscataway, NJ, United States
Geometry & topology, Tome 29 (2025) no. 2, pp. 931-1016
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We prove that any bubble-sheet oval for the mean curvature flow in ℝ4, up to scaling and rigid motion, either is the O(2)×O(2)-symmetric ancient oval constructed by Haslhofer and Hershkovits, or belongs to the one-parameter family of ℤ22×O(2)-symmetric ancient ovals constructed by Du and Haslhofer. In particular, this seems to be the first instance of a classification result for geometric flows that are neither cohomogeneity-one nor selfsimilar.

DOI : 10.2140/gt.2025.29.931
Keywords: mean curvature flow, singularities, ancient solutions

Choi, Beomjun  1   ; Daskalopoulos, Panagiota  2   ; Du, Wenkui  3   ; Haslhofer, Robert  4   ; Šešum, Nataša  5

1 Department of Mathematics, POSTECH, Gyeongbuk, South Korea
2 Department of Mathematics, Columbia University, New York, NY, United States
3 Department of Mathematics, Massachusetts Insitute of Technology, Cambridge, MA, United States
4 Department of Mathematics, University of Toronto, Toronto, ON, Canada
5 Department of Mathematics, Rutgers University, Piscataway, NJ, United States 
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Choi, Beomjun; Daskalopoulos, Panagiota; Du, Wenkui; Haslhofer, Robert; Šešum, Nataša. Classification of bubble-sheet ovals in ℝ4. Geometry & topology, Tome 29 (2025) no. 2, pp. 931-1016. doi: 10.2140/gt.2025.29.931

[1] S J Altschuler, L F Wu, Translating surfaces of the non-parametric mean curvature flow with prescribed contact angle, Calc. Var. Partial Differential Equations 2 (1994) 101 | DOI

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

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

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

[5] 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

[6] S Angenent, P Daskalopoulos, N Šešum, Dynamics of convex mean curvature flow, preprint (2023)

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

[8] T Bourni, M Langford, G Tinaglia, Ancient mean curvature flows out of polytopes, Geom. Topol. 26 (2022) 1849 | DOI

[9] S Brendle, A sharp bound for the inscribed radius under mean curvature flow, Invent. Math. 202 (2015) 217 | DOI

[10] S Brendle, Ancient solutions to the Ricci flow in dimension 3, Acta Math. 225 (2020) 1 | DOI

[11] S Brendle, K Choi, Uniqueness of convex ancient solutions to mean curvature flow in R3, Invent. Math. 217 (2019) 35 | DOI

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

[13] S Brendle, K Naff, Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions, Geom. Topol. 27 (2023) 153 | DOI

[14] S Brendle, P Daskalopoulos, N Šešum, Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Invent. Math. 226 (2021) 579 | DOI

[15] S Brendle, P Daskalopoulos, K Naff, N Šešum, Uniqueness of compact ancient solutions to the higher-dimensional Ricci flow, J. Reine Angew. Math. 795 (2023) 85 | DOI

[16] J Cheeger, R Haslhofer, A Naber, Quantitative stratification and the regularity of mean curvature flow, Geom. Funct. Anal. 23 (2013) 828 | DOI

[17] O Chodosh, K Choi, C Mantoulidis, F Schulze, Mean curvature flow with generic initial data, Invent. Math. 237 (2024) 121 | DOI

[18] K Choi, R Haslhofer, Classification of ancient noncollapsed flows in R4, preprint (2024)

[19] B Choi, R Haslhofer, O Hershkovits, A note on the selfsimilarity of limit flows, Proc. Amer. Math. Soc. 149 (2021) 1239 | DOI

[20] K Choi, R Haslhofer, O Hershkovits, Ancient low-entropy flows, mean-convex neighborhoods, and uniqueness, Acta Math. 228 (2022) 217 | DOI

[21] K Choi, R Haslhofer, O Hershkovits, B White, Ancient asymptotically cylindrical flows and applications, Invent. Math. 229 (2022) 139 | DOI

[22] K Choi, R Haslhofer, O Hershkovits, Classification of noncollapsed translators in R4, Camb. J. Math. 11 (2023) 563 | DOI

[23] K Choi, R Haslhofer, O Hershkovits, A nonexistence result for wing-like mean curvature flows in R4, Geom. Topol. 28 (2024) 3095 | DOI

[24] B Chow, P Lu, L Ni, Hamilton’s Ricci flow, 77, Amer. Math. Soc. (2006) | DOI

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

[26] T H Colding, W P Minicozzi Ii, Uniqueness of blowups and Łojasiewicz inequalities, Ann. of Math. 182 (2015) 221 | DOI

[27] T H Colding, W P Minicozzi Ii, Differentiability of the arrival time, Comm. Pure Appl. Math. 69 (2016) 2349 | DOI

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

[29] W Du, R Haslhofer, On uniqueness and nonuniqueness of ancient ovals, (2021)

[30] W Du, R Haslhofer, A nonexistence result for rotating mean curvature flows in R4, J. Reine Angew. Math. 802 (2023) 275 | DOI

[31] W Du, R Haslhofer, Hearing the shape of ancient noncollapsed flows in R4, Comm. Pure Appl. Math. 77 (2024) 543 | DOI

[32] R S Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986) 153

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

[34] R Haslhofer, On κ–solutions and canonical neighborhoods in 4d Ricci flow, J. Reine Angew. Math. 811 (2024) 257 | DOI

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

[36] R Haslhofer, B Kleiner, On Brendle’s estimate for the inscribed radius under mean curvature flow, Int. Math. Res. Not. 2015 (2015) 6558 | DOI

[37] D Hoffman, T Ilmanen, F Martín, B White, Graphical translators for mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019) 117 | DOI

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

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

[40] T Ilmanen, Problems in mean curvature flow, electronic reference (2003)

[41] Y Lai, A family of 3D steady gradient solitons that are flying wings, J. Differential Geom. 126 (2024) 297 | DOI

[42] X Li, Y Zhang, Ancient solutions to the Ricci flow in higher dimensions, Comm. Anal. Geom. 30 (2022) 2011 | DOI

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

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

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

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

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

[48] J Zhu, SO(2) symmetry of the translating solitons of the mean curvature flow in R4, Ann. PDE 8 (2022) 6 | DOI

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