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No compact split limit Ricci flow of type II from the blow-down

  • Ziyi Zhao and Xiaohua Zhu EMAIL logo
Published/Copyright: June 5, 2025

Abstract

By Perelman’s ℒ-geodesic theory, we study the blow-down solutions on a noncompact 𝜅-noncollapsed steady gradient Ricci soliton ( M n , g ) ( n 4 ) with nonnegative curvature operator and positive Ricci curvature away from a compact set of 𝑀. We prove that any compact split ancient solution of codimension one from the blow-down of ( M , g ) is of type I. The result is a generalization of our previous work from n = 4 to any dimension.

Award Identifier / Grant number: 2023YFA1009900

Award Identifier / Grant number: 2020YFA0712800

Award Identifier / Grant number: 12271009

Funding statement: Xiaohua Zhu partially supported by National Key R&D Program of China 2023YFA1009900 and 2020YFA0712800, and NSFC 12271009.

Acknowledgements

The authors would like to thank the referee for many valuable comments on their paper.

References

[1] A. Appleton, A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four, preprint (2017), https://arxiv.org/abs/1708.00161. Search in Google Scholar

[2] R. H. Bamler, P.-Y. Chan, Z. Ma and Y. Zhang, An optimal volume growth estimate for noncollapsed steady gradient Ricci solitons, Peking Math. J. 6 (2023), no. 2, 353–364. 10.1007/s42543-023-00060-wSearch in Google Scholar

[3] S. Brendle, Rotational symmetry of self-similar solutions to the Ricci flow, Invent. Math. 194 (2013), no. 3, 731–764.10.1007/s00222-013-0457-0Search in Google Scholar

[4] S. Brendle, P. Daskalopoulos and N. Sesum, Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Invent. Math. 226 (2021), no. 2, 579–651. 10.1007/s00222-021-01054-0Search in Google Scholar

[5] S. Brendle and R. Schoen, Manifolds with 1 / 4 -pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009), no. 1, 287–307. 10.1090/S0894-0347-08-00613-9Search in Google Scholar

[6] R. Bryant, Ricci flow solitons in dimension three with S O ( 3 ) -symmetries, preprint, Duke University, 2005. Search in Google Scholar

[7] B.-L. Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009), no. 2, 363–382. 10.4310/jdg/1246888488Search in Google Scholar

[8] B. Chow, Y. Deng and Z. Ma, On four-dimensional steady gradient Ricci solitons that dimension reduce, Adv. Math. 403 (2022), Article ID 108367. 10.1016/j.aim.2022.108367Search in Google Scholar

[9] B. Chow, P. Lu and L. Ni, Hamilton’s Ricci flow, Grad. Stud. Math. 77, American Mathematical Society, Providence 2006. Search in Google Scholar

[10] Y. Deng and X. Zhu, Asymptotic behavior of positively curved steady Ricci solitons, Trans. Amer. Math. Soc. 370 (2018), no. 4, 2855–2877. 10.1090/tran/7235Search in Google Scholar

[11] Y. Deng and X. Zhu, Three-dimensional steady gradient Ricci solitons with linear curvature decay, Int. Math. Res. Not. IMRN 2019 (2019), no. 4, 1108–1124. 10.1093/imrn/rnx155Search in Google Scholar

[12] Y. Deng and X. Zhu, Classification of gradient steady Ricci solitons with linear curvature decay, Sci. China Math. 63 (2020), no. 1, 135–154. 10.1007/s11425-019-1548-0Search in Google Scholar

[13] Y. Deng and X. Zhu, Higher dimensional steady Ricci solitons with linear curvature decay, J. Eur. Math. Soc. (JEMS) 22 (2020), no. 12, 4097–4120. 10.4171/jems/1003Search in Google Scholar

[14] R. S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II, International Press, Cambridge (1995), 7–136. 10.4310/SDG.1993.v2.n1.a2Search in Google Scholar

[15] B. Kleiner and J. Lott, Notes on Perelman’s papers, Geom. Topol. 12 (2008), no. 5, 2587–2855. 10.2140/gt.2008.12.2587Search in Google Scholar

[16] Y. Lai, A family of 3D steady gradient solitons that are flying wings, J. Differential Geom. 126 (2024), no. 1, 297–328. 10.4310/jdg/1707767339Search in Google Scholar

[17] J. Morgan and G. Tian, Ricci flow and the Poincaré conjecture, Clay Math. Monogr. 3, American Mathematical Society, Providence 2007. Search in Google Scholar

[18] L. Ni, Closed type I ancient solutions to Ricci flow, Recent advances in geometric analysis, Adv. Lect. Math. (ALM) 11, International Press, Somerville (2010), 147–150. Search in Google Scholar

[19] G. Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002), https://arxiv.org/abs/math/0211159. Search in Google Scholar

[20] W.-X. Shi, Deforming the metric on complete Riemannian manifolds, J. Differential Geom. 30 (1989), no. 1, 223–301. 10.4310/jdg/1214443292Search in Google Scholar

[21] R. Ye, On the 𝑙-function and the reduced volume of Perelman. I, Trans. Amer. Math. Soc. 360 (2008), no. 1, 507–531. 10.1090/S0002-9947-07-04405-4Search in Google Scholar

[22] Z.-H. Zhang, On the completeness of gradient Ricci solitons, Proc. Amer. Math. Soc. 137 (2009), no. 8, 2755–2759. 10.1090/S0002-9939-09-09866-9Search in Google Scholar

[23] Z. Y. Zhao and X. H. Zhu, 4d steady gradient Ricci solitons with nonnegative curvature away from a compact set, preprint (2023), https://arxiv.org/abs/2310.12529. Search in Google Scholar

[24] Z. Y. Zhao and X. H. Zhu, Steady gradient Ricci solitons with nonnegative curvature operator away from a compact set, preprint (2023), https://arxiv.org/abs/2402.00316. Search in Google Scholar

Received: 2024-05-20
Revised: 2025-03-27
Published Online: 2025-06-05
Published in Print: 2025-09-01

© 2025 Walter de Gruyter GmbH, Berlin/Boston

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