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arXiv · 2310.07945

Finite time singularities of the Kähler-Ricci flow

Wangjian Jian·Jian Song·Gang Tian
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Abstract

We establish the scalar curvature and distance bounds, extending Perelman's work on the Fano Kähler-Ricci flow to general finite time solutions of the Kähler-Ricci flow. These bounds are achieved by our Li-Yau type and Harnack estimates for weighted Ricci potential functions of the Kähler-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 Kähler-Ricci flow on a Fano fibre bundle. We also apply our estimates to show that every solution of the Kähler-Ricci flow with Calabi symmetry must develop Type I singularities, including both cases of high codimensional contractions and fibre collapsing.

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Wangjian Jian, Jian Song, Gang Tian. 2023-10-27. Finite time singularities of the Kähler-Ricci flow. https://arxiv.org/abs/2310.07945

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