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Ancient solutions to the Ricci flow with isotropic curvature conditions

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Abstract

We show that every n-dimensional, \(\kappa \)-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for \(n=4\) or \(n\ge 12\) has weakly PIC\(_2\) and bounded curvature. Combining this with the results in Brendle and Naff (Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions, arXiv:2005.05830), we prove that any such solution is isometric to either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton. Also, we classify all complex 2-dimensional, \(\kappa \)-noncollapsed, complete ancient solutions to the Kähler Ricci flow with weakly PIC.

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Authors and Affiliations

  1. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Jae Ho Cho

  2. Institute of Geometry and Physics, University of Science and Technology of China, No. 96 Jinzhai Road, Hefei, 230026, Anhui, China

    Yu Li

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  1. Jae Ho Cho
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Yu Li is supported by YSBR-001 and a research fund from USTC.

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Cho, J.H., Li, Y. Ancient solutions to the Ricci flow with isotropic curvature conditions. Math. Ann. 387, 1009–1041 (2023). https://doi.org/10.1007/s00208-022-02478-6

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