9 research outputs found
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- distance. In particular, the singular sets of each time slice and
its tangent cones are close and of codimension no less than .Comment: All comments welcome. arXiv admin note: text overlap with
arXiv:2310.0794
Global -regularity for 4-dimensional Ricci flow with integral scalar curvature bound
Ge-Jiang (Geom Funct Anal 27:1231-1256, 2017) proved global
-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
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
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