As on arXiv
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.
Comments: All comments welcome; improved introduction and minor edits
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Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
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A new proof of Perelman's scalar curvature and diameter estimates for the Kähler-Ricci flow on Fano manifolds
Topics Harnack inequalities, Kähler–Ricci flow, Scalar curvature · Same authors Wangjian Jian, Jian Song, Gang Tian · Shared terms “time solution”, “Harnack estimate”, “Fano”
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Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles II
Topics Kähler–Ricci flow, (Non)collapsing, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”, “fibre collapsing”
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Type I estimates for the Kähler-Ricci flow II
Topics Kähler–Ricci flow, Limit spaces & RCD, Scalar curvature · Same authors Wangjian Jian, Jian Song · Shared terms “diameter bound”, “time solution”, “finite time”
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Finite-Time Singularities of the Kähler–Ricci Flow on Fano Bundles
Topics Kähler–Ricci flow, (Non)collapsing, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”, “Fano”
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Geometric regularity of blow-up limits of the Kähler-Ricci flow
Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same authors Wangjian Jian, Jian Song, Gang Tian · Shared terms “Gromov Hausdorff”
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Type I estimates for the Kahler-Ricci flow I
Topics Kähler–Ricci flow, Scalar curvature, Kähler geometry · Same authors Wangjian Jian, Jian Song · Shared terms “time singularity”, “finite time”
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Horosymmetric limits of Kähler-Ricci flow on Fano -manifolds
Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “Fano”, “Gromov Hausdorff”
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A Note On Kähler-Ricci Flow on Fano Threefolds
Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Same author Gang Tian · Shared terms “develop Type”, “Fano”, “Gromov Hausdorff”
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Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11B.2Ancient Solutions and the Harnack Inequality
Comparing curvature across space and time.
- 12B.4Compactness and Universal Estimates for κ-Solutions
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.
- 12B.3The Structure of κ-Solutions
Necks and caps, compactness, and universal estimates.