1,079 research outputs found
CANONICAL MEASURES AND KĂ„HLER-RICCI FLOW
We show that the Kähler-Ricci flow on an algebraic manifold of positive Kodaira dimension and semi-ample canonical line bundle converges to a unique canonical metric on its canonical model. It is also shown that there exists a canonical measure of analytic Zariski decomposition on an algebraic manifold of positive Kodaira dimension. Such a canonical measure is unique and invariant under birational transformations under the assumption o
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
On polars of mixed projection bodies
AbstractRecently, Lutwak established general Minkowski inequality, Brunn–Minkowski inequality and Aleksandrov–Fenchel inequality for mixed projection bodies. In this paper, following Lutwak, we established their polar forms. As applications, we prove some interrelated results
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