We analyse the blow-down solutions for $n$-dimensional $(n\ge 4)$ noncompact $\kappa $-noncollapsed steady gradient Ricci solitons $(M, g)$ with $\operatorname{Rm}\ge 0$ and $\operatorname{Ric}>0$ away from a compact set of $M$. As one of main results, we prove that the $(n-1)$-dimensional compact split limit ancient Ricci flows of type I and type II cannot occur simultaneously from the blow-down of $(M, g)$. Consequently, we prove that $(M,g)$ with $\operatorname{Rm}\ge 0$ must be isometric to the Bryant Ricci soliton up to scaling, if there exists a sequence of normally rescaled Ricci flows of $(M,g)$, which converges subsequently to a family of shrinking quotient cylinders. The latter improves a previous result of Brendle.
Nous étudions des solutions blow-down pour $(M,g)$ soliton de Ricci gradient stable non-compact et $\kappa $-non effondré de dimension $n\ge 4$ à $\operatorname{Rm}\ge 0$ et $\operatorname{Ric}>0$ en dehors d’un compact de $M$. L’une de nos principales conclusions est la preuve que, lors du blow-down de $(M,g)$, les flots de Ricci anciens limites compacts scindés de dimension $(n-1)$ ne peuvent pas être simultanément de type I et de type II. Par conséquent, nous démontrons que si une suite de flots de Ricci normalement dilatés de $(M,g)$ converge successivement vers une famille de cylindres quotients contractants, alors $(M,g)$ à $\operatorname{Rm}\ge 0$, doit être isométrique au solition de Bryant à une dilatation près. Ce dernier résultat améliore un résultat antérieur dû à Brendle.
Accepté le :
Publié le :
Keywords: Steady Ricci soliton, Ricci flow, ancient $\kappa $-solution, Bryant Ricci soliton
Ziyi Zhao 1 ; Xiaohua Zhu 2
CC-BY 4.0
Ziyi Zhao; Xiaohua Zhu. Steady gradient Ricci solitons with nonnegative curvature away from a compact set. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 35 (2026) no. 3, pp. 801-837. doi: 10.5802/afst.1864
@article{AFST_2026_6_35_3_801_0,
author = {Ziyi Zhao and Xiaohua Zhu},
title = {Steady gradient {Ricci} solitons with nonnegative curvature away from a compact set},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {801--837},
year = {2026},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 35},
number = {3},
doi = {10.5802/afst.1864},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1864/}
}
TY - JOUR AU - Ziyi Zhao AU - Xiaohua Zhu TI - Steady gradient Ricci solitons with nonnegative curvature away from a compact set JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2026 SP - 801 EP - 837 VL - 35 IS - 3 PB - Université de Toulouse, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1864/ DO - 10.5802/afst.1864 LA - en ID - AFST_2026_6_35_3_801_0 ER -
%0 Journal Article %A Ziyi Zhao %A Xiaohua Zhu %T Steady gradient Ricci solitons with nonnegative curvature away from a compact set %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2026 %P 801-837 %V 35 %N 3 %I Université de Toulouse, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1864/ %R 10.5802/afst.1864 %G en %F AFST_2026_6_35_3_801_0
[1] Unique asymptotics of compact ancient solutions to three-dimensional Ricci flow, Commun. Pure Appl. Math., Volume 75 (2022) no. 5, pp. 1032-1073 | DOI | Zbl | MR
[2] A family of non-collapsed steady Ricci solitons in even dimensions greater or equal to four (2017) | arXiv | Zbl
[3] Structure theory of non-collapsed limits of Ricci flows (2020) | arXiv | Zbl
[4] On the rotational symmetry of 3-dimensional ${\kappa }$-solutions, J. Reine Angew. Math., Volume 779 (2021), pp. 37-55 | DOI | Zbl | MR
[5] Manifolds with positive curvature operators are space forms, Ann. Math. (2), Volume 167 (2008) no. 3, pp. 1079-1097 | DOI | Zbl | MR
[6] Rotational symmetry of self-similar solutions to the Ricci flow, Invent. Math., Volume 194 (2013) no. 3, pp. 731-764 | DOI | Zbl | MR
[7] Rotational symmetry of Ricci solitons in higher dimensions, J. Differ. Geom., Volume 97 (2014) no. 2, pp. 191-214 | DOI | Zbl | MR
[8] Ancient solutions to the Ricci flow in dimension $3$, Acta Math., Volume 225 (2020) no. 1, pp. 1-102 | DOI | Zbl | MR
[9] Uniqueness of compact ancient solutions to three-dimensional Ricci flow, Invent. Math., Volume 226 (2021) no. 2, pp. 579-651 | DOI | Zbl | MR
[10] Manifolds with $1/4$-pinched curvature are space forms, J. Am. Math. Soc., Volume 22 (2009) no. 1, pp. 287-307 | DOI | Zbl | MR
[11] Ricci flow solitons in dimension three with $ {SO} (3)$-symmetries, preprint, Duke University, 2005 (https://sites.math.duke.edu/~bryant/3DRotSymRicciSolitons.pdf)
[12] The structure of complete manifolds of nonnegative curvature, Bull. Am. Math. Soc., Volume 74 (1968), pp. 1147-1150 | DOI | Zbl | MR
[13] Strong uniqueness of the Ricci flow, J. Differ. Geom., Volume 82 (2009) no. 2, pp. 363-382 | DOI | Zbl | MR
[14] The Ricci flow: techniques and applications. Part III: Geometric-analytic aspects, Mathematical Surveys and Monographs, 163, American Mathematical Society, 2010 | Zbl | MR
[15] On four-dimensional steady gradient Ricci solitons that dimension reduce, Adv. Math., Volume 403 (2022), 108367, 61 pages | DOI | Zbl | MR
[16] Hamilton’s Ricci flow, Graduate Studies in Mathematics, 77, American Mathematical Society, 2006 | Zbl | DOI
[17] Asymptotic behavior of positively curved steady Ricci solitons, Trans. Am. Math. Soc., Volume 370 (2018) no. 4, pp. 2855-2877 | DOI | Zbl | MR
[18] Classification of gradient steady Ricci solitons with linear curvature decay, Sci. China, Math., Volume 63 (2020) no. 1, pp. 135-154 | DOI | Zbl | MR
[19] Higher dimensional steady Ricci solitons with linear curvature decay, J. Eur. Math. Soc., Volume 22 (2020) no. 12, pp. 4097-4120 | DOI | Zbl | MR
[20] The formation of singularities in the Ricci flow, Surveys in differential geometry. Vol. II: Proceedings of the conference on geometry and topology held at Harvard University, Cambridge, MA, USA, April 23-25, 1993 (Surveys in Differential Geometry. Supplement to the Journal of Differential Geometry), Volume 2, International Press, 1995, pp. 7-136 | Zbl | MR
[21] On $\kappa $-solutions and canonical neighborhoods in 4d Ricci flow, J. Reine Angew. Math., Volume 811 (2024), pp. 257-265 | DOI | Zbl | MR
[22] Notes on Perelman’s papers, Geom. Topol., Volume 12 (2008) no. 5, pp. 2587-2855 | DOI | Zbl
[23] A family of 3d steady gradient solitons that are flying wings, J. Differ. Geom., Volume 126 (2024) no. 1, pp. 297-328 | DOI | Zbl | MR
[24] Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set, Ann. Math. (2), Volume 125 (1987), pp. 171-207 | DOI | Zbl | MR
[25] Ricci flow and the Poincaré conjecture, Clay Mathematics Monographs, 3, American Mathematical Society; Clay Mathematics Institute, 2007 | Zbl | MR
[26] Smooth metric measure spaces with non-negative curvature, Commun. Anal. Geom., Volume 19 (2011) no. 3, pp. 451-486 | DOI | Zbl | MR
[27] Closed type I ancient solutions to Ricci flow, Recent advances in geometric analysis. Proceeding of the international conference on geometric analysis, Taipei, Taiwan, June 18–22, 2007 (Advanced Lectures in Mathematics), Volume 11, International Press; Higher Education Press, 2010, pp. 147-150 | Zbl
[28] The entropy formula for the Ricci flow and its geometric applications (2002) | arXiv | Zbl
[29] Ricci flow with surgery on three-manifolds (2003) | arXiv | Zbl
[30] On the completeness of gradient Ricci solitons, Proc. Am. Math. Soc., Volume 137 (2009) no. 8, pp. 2755-2759 | DOI | Zbl | MR
[31] Rigidity of the Bryant Ricci soliton (2022) | arXiv | Zbl
[32] No compact split limit Ricci flow of type II from the blow-down, J. Reine Angew. Math., Volume 826 (2025), pp. 1-22 | DOI | Zbl
Cité par Sources :