Ancient solutions to the Kähler Ricci flow

Li, Yu¹

¹ Institute of Geometry and Physics, University of Science and Technology of China, Hefei, Anhui Province, China, Hefei National Laboratory, Hefei, Anhui Province, China

Geometry & topology, Tome 28 (2024) no. 7, pp. 3257-3283.

Résumé

We give a complete classification of all $\kappa$–noncollapsed, complete ancient solutions to the Kähler Ricci flow with nonnegative bisectional curvature.

Détail

DOI : 10.2140/gt.2024.28.3257

Keywords: Ricci flow, Kähler Ricci flow, ancient solutions

Bibliographie

[1] R H Bamler, E Cabezas-Rivas, B Wilking, The Ricci flow under almost non-negative curvature conditions, Invent. Math. 217 (2019) 95 | DOI

[2] A L Besse, Einstein manifolds, 10, Springer (1987) | DOI

[3] A Borel, J P Serre, Impossibilité de fibrer un espace euclidien par des fibres compactes, C. R. Acad. Sci. Paris 230 (1950) 2258

[4] J E Borzellino, Riemannian geometry of orbifolds, PhD thesis, University of California, Los Angeles (1992)

[5] S Brendle, A generalization of Hamilton’s differential Harnack inequality for the Ricci flow, J. Differential Geom. 82 (2009) 207

[6] S Brendle, Einstein manifolds with nonnegative isotropic curvature are locally symmetric, Duke Math. J. 151 (2010) 1 | DOI

[7] S Brendle, Ricci flow and the sphere theorem, 111, Amer. Math. Soc. (2010) | DOI

[8] S Brendle, Ricci flow with surgery on manifolds with positive isotropic curvature, Ann. of Math. 190 (2019) 465 | DOI

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

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

[11] S Brendle, G Huisken, C Sinestrari, Ancient solutions to the Ricci flow with pinched curvature, Duke Math. J. 158 (2011) 537 | DOI

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

[13] S Brendle, R Schoen, Manifolds with 1∕4–pinched curvature are space forms, J. Amer. Math. Soc. 22 (2009) 287 | DOI

[14] P Buser, H Karcher, Gromov’s almost flat manifolds, 81, Soc. Math. France (1981) 148

[15] E Cabezas-Rivas, B Wilking, How to produce a Ricci flow via Cheeger–Gromoll exhaustion, J. Eur. Math. Soc. 17 (2015) 3153 | DOI

[16] H D Cao, On Harnack’s inequalities for the Kähler–Ricci flow, Invent. Math. 109 (1992) 247 | DOI

[17] H D Cao, Limits of solutions to the Kähler–Ricci flow, J. Differential Geom. 45 (1997) 257

[18] H D Cao, On dimension reduction in the Kähler–Ricci flow, Comm. Anal. Geom. 12 (2004) 305 | DOI

[19] X Cao, Q S Zhang, The conjugate heat equation and ancient solutions of the Ricci flow, Adv. Math. 228 (2011) 2891 | DOI

[20] J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327 | DOI

[21] J Cheeger, M Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, II, J. Differential Geom. 32 (1990) 269

[22] B L Chen, Strong uniqueness of the Ricci flow, J. Differential Geom. 82 (2009) 363

[23] B L Chen, X P Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Differential Geom. 74 (2006) 177

[24] J H Cho, Y Li, Ancient solutions to the Ricci flow with isotropic curvature conditions, Math. Ann. 387 (2023) 1009 | DOI

[25] B Chow, P Lu, A bound for the order of the fundamental group of a complete noncompact Ricci shrinker, Proc. Amer. Math. Soc. 144 (2016) 2623 | DOI

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

[27] Y Deng, X Zhu, A note on compact κ–solutions of Kähler–Ricci flow, Proc. Amer. Math. Soc. 148 (2020) 3073 | DOI

[28] Y Deng, X Zhu, Rigidity of κ–noncollapsed steady Kähler–Ricci solitons, Math. Ann. 377 (2020) 847 | DOI

[29] M Fernández-López, E García-Río, A remark on compact Ricci solitons, Math. Ann. 340 (2008) 893 | DOI

[30] H L Gu, A new proof of Mok’s generalized Frankel conjecture theorem, Proc. Amer. Math. Soc. 137 (2009) 1063 | DOI

[31] M Hallgren, Nonexistence of noncompact type-I ancient three-dimensional κ–solutions of Ricci flow with positive curvature, Commun. Contemp. Math. 21 (2019) 1850049 | DOI

[32] S Kobayashi, K Nomizu, Foundations of differential geometry, I, Interscience (1963)

[33] X Li, L Ni, Kähler–Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature, J. Math. Pures Appl. 138 (2020) 28 | DOI

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

[35] Y Li, B Wang, Heat kernel on Ricci shrinkers, Calc. Var. Partial Differential Equations 59 (2020) 194 | DOI

[36] J Lott, Some geometric properties of the Bakry–Émery–Ricci tensor, Comment. Math. Helv. 78 (2003) 865 | DOI

[37] N Mok, The uniformization theorem for compact Kähler manifolds of nonnegative holomorphic bisectional curvature, J. Differential Geom. 27 (1988) 179

[38] O Munteanu, J Wang, Positively curved shrinking Ricci solitons are compact, J. Differential Geom. 106 (2017) 499 | DOI

[39] A Naber, Noncompact shrinking four solitons with nonnegative curvature, J. Reine Angew. Math. 645 (2010) 125 | DOI

[40] L Ni, Ancient solutions to Kähler–Ricci flow, Math. Res. Lett. 12 (2005) 633 | DOI

[41] 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

[42] G Perelman, The entropy formula for the Ricci flow and its geometric applications, preprint (2002)

[43] G Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, preprint (2003)

[44] G Perelman, Ricci flow with surgery on three-manifolds, preprint (2003)

[45] B Wilking, A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities, J. Reine Angew. Math. 679 (2013) 223 | DOI

[46] J A Wolf, Spaces of constant curvature, McGraw-Hill (1967)

[47] W Wylie, Complete shrinking Ricci solitons have finite fundamental group, Proc. Amer. Math. Soc. 136 (2008) 1803 | DOI

[48] Y Zhang, On three-dimensional type I κ–solutions to the Ricci flow, Proc. Amer. Math. Soc. 146 (2018) 4899 | DOI

[49] Z Zhang, On the finiteness of the fundamental group of a compact shrinking Ricci soliton, Colloq. Math. 107 (2007) 297 | DOI