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Finite time singularities of the K\"ahler-Ricci flow

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arxiv 2310.07945 v2 pith:IER2UMSV submitted 2023-10-11 math.DG

Finite time singularities of the K\"ahler-Ricci flow

Wangjian Jian , Jian Song , Gang Tian This is my paper
classification math.DG
keywords ahler-ricciflowtypefibrefinitesolutiontimebounds
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We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano K\"ahler-Ricci flow to general finite time solutions of the K\"ahler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the K\"ahler-Ricci flow. We further prove that the Type I blow-ups of the finite time solution always sub-converge in Gromov-Hausdorff sense to an ancient solution on a family of analytic normal varieties with suitable choices of base points. As a consequence, the Type I diameter bound is proved for almost every fibre of collapsing solutions of the K\"ahler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the K\"ahler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

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Cited by 3 Pith papers

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