Geometry & Topology Volume 29, issue 2 (2025)

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ISSN (electronic): 1364-0380

ISSN (print): 1465-3060

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$O(2)$-symmetry of 3D steady gradient Ricci solitons

Yi Lai

Geometry & Topology 29 (2025) 687–789

DOI: 10.2140/gt.2025.29.687

Bibliography

1	G Anderson, B Chow, A pinching estimate for solutions of the linearized Ricci flow system on 3-manifolds, Calc. Var. Partial Differential Equations 23 (2005) 1 MR2133658
2	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
3	I Bakas, S Kong, L Ni, Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations, J. Reine Angew. Math. 663 (2012) 209 MR2889711
4	R H Bamler, Ricci flow with surgery, Diplomarbeit, Ludwig-Maximilians-Universität München (2007)
5	R H Bamler, Long-time behavior of 3-dimensional Ricci flow, A : Generalizations of Perelman’s long-time estimates, Geom. Topol. 22 (2018) 775 MR3748680
6	R H Bamler, Long-time behavior of 3-dimensional Ricci flow, D : Proof of the main results, Geom. Topol. 22 (2018) 949 MR3748683
7	R H Bamler, Compactness theory of the space of super Ricci flows, Invent. Math. 233 (2023) 1121 MR4623543
8	R H Bamler, B Kleiner, On the rotational symmetry of 3-dimensional κ-solutions, J. Reine Angew. Math. 779 (2021) 37 MR4319063
9	R H Bamler, B Kleiner, Uniqueness and stability of Ricci flow through singularities, Acta Math. 228 (2022) 1 MR4448680
10	T Bourni, M Langford, G Tinaglia, Collapsing ancient solutions of mean curvature flow, J. Differential Geom. 119 (2021) 187 MR4318294
11	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), Lond. Math. Soc. Lect. Note Ser. 463, Cambridge Univ. Press (2021) 50 MR4420785
12	S Brendle, Rotational symmetry of self-similar solutions to the Ricci flow, Invent. Math. 194 (2013) 731 MR3127066
13	S Brendle, Rotational symmetry of Ricci solitons in higher dimensions, J. Differential Geom. 97 (2014) 191 MR3231974
14	S Brendle, K Choi, Uniqueness of convex ancient solutions to mean curvature flow in ℝ³, Invent. Math. 217 (2019) 35 MR3958790
15	S Brendle, K Choi, Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions, Geom. Topol. 25 (2021) 2195 MR4310889
16	S Brendle, P Daskalopoulos, K Naff, N Sesum, Uniqueness of compact ancient solutions to the higher-dimensional Ricci flow, J. Reine Angew. Math. 795 (2023) 85 MR4542417
17	S Brendle, G Huisken, C Sinestrari, Ancient solutions to the Ricci flow with pinched curvature, Duke Math. J. 158 (2011) 537 MR2805067
18	S Brendle, K Naff, Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions, Geom. Topol. 27 (2023) 153 MR4584263
19	R Bryant, Ricci flow solitons in dimension three with SO(3)-symmetries, preprint (2005)
20	D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
21	Y Burago, M Gromov, G Perelman, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3 MR1185284
22	H D Cao, Existence of gradient Kähler–Ricci solitons, from: "Elliptic and parabolic methods in geometry" (editors B Chow, R Gulliver, S Levy, J Sullivan), Peters (1996) 1 MR1417944
23	H D Cao, Recent progress on Ricci solitons, from: "Recent advances in geometric analysis" (editors Y I Lee, C S Lin, M P Tsui), Adv. Lect. Math. 11, International (2010) 1 MR2648937
24	H D Cao, G Catino, Q Chen, C Mantegazza, L Mazzieri, Bach-flat gradient steady Ricci solitons, Calc. Var. Partial Differential Equations 49 (2014) 125 MR3148109
25	H D Cao, Q Chen, On locally conformally flat gradient steady Ricci solitons, Trans. Amer. Math. Soc. 364 (2012) 2377 MR2888210
26	H D Cao, C He, Infinitesimal rigidity of collapsed gradient steady Ricci solitons in dimension three, Comm. Anal. Geom. 26 (2018) 505 MR3844113
27	G Catino, P Mastrolia, D D Monticelli, Classification of expanding and steady Ricci solitons with integral curvature decay, Geom. Topol. 20 (2016) 2665 MR3556348
28	P Y Chan, Curvature estimates for steady Ricci solitons, Trans. Amer. Math. Soc. 372 (2019) 8985 MR4029719
29	P Y Chan, B Zhu, On a dichotomy of the curvature decay of steady Ricci solitons, Adv. Math. 404 (2022) 108458 MR4423811
30	B L Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009) 363 MR2520796
31	B L Chen, X P Zhu, Complete Riemannian manifolds with pointwise pinched curvature, Invent. Math. 140 (2000) 423 MR1757002
32	X Chen, Y Wang, On four-dimensional anti-self-dual gradient Ricci solitons, J. Geom. Anal. 25 (2015) 1335 MR3319974
33	K Choi, R Haslhofer, O Hershkovits, Classification of noncollapsed translators in ℝ⁴, Camb. J. Math. 11 (2023) 563 MR4629007
34	K Choi, R Haslhofer, O Hershkovits, A nonexistence result for wing-like mean curvature flows in ℝ4, Geom. Topol. 28 (2024) 3095
35	B Chow, S C Chu, D Glickenstein, C Guenther, J Isenberg, T Ivey, D Knopf, P Lu, F Luo, L Ni, The Ricci flow : techniques and applications, I : Geometric aspects, 135, Amer. Math. Soc. (2007) MR2302600
36	B Chow, S C Chu, D Glickenstein, C Guenther, J Isenberg, T Ivey, D Knopf, P Lu, F Luo, L Ni, The Ricci flow : techniques and applications, III : Geometric-analytic aspects, 163, Amer. Math. Soc. (2010) MR2604955
37	B Chow, Y Deng, Z Ma, On four-dimensional steady gradient Ricci solitons that dimension reduce, Adv. Math. 403 (2022) 108367 MR4405369
38	B Chow, P Lu, L Ni, Hamilton’s Ricci flow, 77, Amer. Math. Soc. (2006) MR2274812
39	S C Chu, Type II ancient solutions to the Ricci flow on surfaces, Comm. Anal. Geom. 15 (2007) 195 MR2301253
40	P Daskalopoulos, R Hamilton, N Sesum, Classification of ancient compact solutions to the Ricci flow on surfaces, J. Differential Geom. 91 (2012) 171 MR2971286
41	P Daskalopoulos, N Sesum, Eternal solutions to the Ricci flow on ℝ², Int. Math. Res. Not. 2006 (2006) 83610 MR2264733
42	Y Deng, X Zhu, Three-dimensional steady gradient Ricci solitons with linear curvature decay, Int. Math. Res. Not. 2019 (2019) 1108 MR3915297
43	A Deruelle, Smoothing out positively curved metric cones by Ricci expanders, Geom. Funct. Anal. 26 (2016) 188 MR3494489
44	W Du, R Haslhofer, Hearing the shape of ancient noncollapsed flows in ℝ⁴, Comm. Pure Appl. Math. 77 (2024) 543 MR4666631
45	L C Evans, Partial differential equations, 19, Amer. Math. Soc. (1998) MR1625845
46	V A Fateev, The sigma model (dual) representation for a two-parameter family of integrable quantum field theories, Nuclear Phys. B 473 (1996) 509 MR1403533
47	M Feldman, T Ilmanen, D Knopf, Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169 MR2058261
48	R S Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255 MR0664497
49	R S Hamilton, The Ricci flow on surfaces, from: "Mathematics and general relativity" (editor J A Isenberg), Contemp. Math. 71, Amer. Math. Soc. (1988) 237 MR0954419
50	R S Hamilton, Eternal solutions to the Ricci flow, J. Differential Geom. 38 (1993) 1 MR1231700
51	R S Hamilton, The formation of singularities in the Ricci flow, from: "Surveys in differential geometry, II" (editor S T Yau), International (1995) 7 MR1375255
52	A Hatcher, Notes on basic 3-manifold topology, unpublished book draft (2001)
53	A Hatcher, Algebraic topology, Cambridge Univ. Press (2002) MR1867354
54	H J Hein, A Naber, New logarithmic Sobolev inequalities and an 𝜖-regularity theorem for the Ricci flow, Comm. Pure Appl. Math. 67 (2014) 1543 MR3245102
55	D Hoffman, T Ilmanen, F Martín, B White, Graphical translators for mean curvature flow, Calc. Var. Partial Differential Equations 58 (2019) 117 MR3962912
56	G Huisken, C Sinestrari, Convex ancient solutions of the mean curvature flow, J. Differential Geom. 101 (2015) 267 MR3399098
57	B Kotschwar, L Wang, Rigidity of asymptotically conical shrinking gradient Ricci solitons, J. Differential Geom. 100 (2015) 55 MR3326574
58	Y Lai, Ricci flow under local almost non-negative curvature conditions, Adv. Math. 343 (2019) 353 MR3881661
59	Y Lai, Producing 3D Ricci flows with nonnegative Ricci curvature via singular Ricci flows, Geom. Topol. 25 (2021) 3629 MR4372638
60	Y Lai, A family of 3D steady gradient solitons that are flying wings, J. Differential Geom. 126 (2024) 297 MR4704550
61	G M Lieberman, Second order parabolic differential equations, World Sci. (1996) MR1465184
62	J Lott, Dimensional reduction and the long-time behavior of Ricci flow, Comment. Math. Helv. 85 (2010) 485 MR2653690
63	J Morgan, G Tian, Ricci flow and the Poincaré conjecture, 3, Amer. Math. Soc. (2007) MR2334563
64	J Morgan, G Tian, Completion of the proof of the geometrization conjecture, preprint (2008) arXiv:0809.4040
65	O Munteanu, C J A Sung, J Wang, Poisson equation on complete manifolds, Adv. Math. 348 (2019) 81 MR3925015
66	G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002) arXiv:math/0211159
67	P Petersen, Riemannian geometry, 171, Springer (1998) MR1480173
68	W X Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989) 223 MR1001277
69	M Simon, P M Topping, Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces, Geom. Topol. 25 (2021) 913 MR4251438
70	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
71	X J Wang, Convex solutions to the mean curvature flow, Ann. of Math. 173 (2011) 1185 MR2800714
72	Z Zhao, X Zhu, Rigidity of the Bryant Ricci soliton, preprint (2022) arXiv:2212.02889
73	J Zhu, SO(2) symmetry of the translating solitons of the mean curvature flow in ℝ⁴, Ann. PDE 8 (2022) 6 MR4400118
74	J Zhu, Rotational symmetry of uniformly 3-convex translating solitons of mean curvature flow in higher dimensions, Ann. Sc. Norm. Super. Pisa Cl. Sci. (2023)