### Abstract

As indicated by the third author [Proc. Amer. Math. Soc. 146 (2018), pp. 4899-4903], there is a gap in the previous version of this paper by the first two authors [Three-dimensional noncompact κ-solutions that are Type I forward and backward, arXiv:1606.02698v1 (2016)]. We provide in this version an argument to fix the aforementioned gap. The main proposition, whose proof uses Perelman's techniques, is implied by Ding [Pacific J. Math. 240 (2009), pp. 289-308] and is covered by Hallgren [The nonexistence of noncompact Type-I ancient 3-d κ-solutions of Ricci flow with positive curvature, arXiv:1801.08643 (2018)] and the third author [Proc. Amer. Math. Soc. 146 (2018), pp. 4899-4903]. Our approach, however, is different from theirs. In addition, we prove a necessary and sufficient condition for a three-dimensional κ-solution to form a forward singularity. We hope that this condition is helpful in the classification of all three-dimensional κ-solutions. Up to now, the only main progress on such a classification, as conjectured by Perelman, is by Brendle [Invent. Math. 194 (2013), pp. 731-764].

Original language | English (US) |
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Pages (from-to) | 2595-2600 |

Number of pages | 6 |

Journal | Proceedings of the American Mathematical Society |

Volume | 148 |

Issue number | 6 |

DOIs | |

State | Published - Jan 1 2020 |

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## Cite this

*Proceedings of the American Mathematical Society*,

*148*(6), 2595-2600. https://doi.org/10.1090/proc/14327