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

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.Comment: All comments welcome; improved introduction and minor edit

Geometric regularity of blow-up limits of the K\"ahler-Ricci flow

We establish geometric regularity for Type I blow-up limits of the K\"ahler-Ricci flow based at any sequence of Ricci vertices. As a consequence, the limiting flow is continuous in time in both Gromov-Hausdorff and Gromov-$W_1$ distance. In particular, the singular sets of each time slice and its tangent cones are close and of codimension no less than $4$.Comment: All comments welcome. arXiv admin note: text overlap with arXiv:2310.0794

Global ε\varepsilon-regularity for 4-dimensional Ricci flow with integral scalar curvature bound

Ge-Jiang (Geom Funct Anal 27:1231-1256, 2017) proved global $\varepsilon$-regularity for 4-dimensional Ricci flow with bounded scalar curvature. In this note, we extend this result to 4-dimensional Ricci flow with integral bound on the scalar curvature.Comment: 9 pages, all comments are welcom

L∞L^\infty estimates for K\"ahler-Ricci flow on K\"ahler-Einstein Fano manifolds: a new derivation

Assuming Perelman's estimates, we give a new proof of uniform $L^\infty$ estimate along normalized K\"ahler-Ricci flow on Fano manifolds with K\"ahler-Einstein metrics, using Chen-Cheng's auxiliary Monge-Amp\`ere equation and the Alexandrov-Bakelman-Pucci maximum principle. This proof does not use pluripotential theory.Comment: comments are welcom