Kähler manifolds with almost non-negative curvature

Man-Chun Lee; Luen-Fai Tam

Paper · Oct 6, 2019

Kähler manifolds with almost non-negative curvature

Man-Chun Lee, Luen-Fai Tam

math.DGarXiv:1910.02531v2

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Abstract

In this paper, we construct local and global solutions to the Kähler-Ricci flow from a non-collapsed Kähler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kähler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete Kähler manifolds with nonnegative orthogonal bisectional curvature, nonnegative Ricci curvature and maximal volume growth.

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Submitted: Oct 6, 2019
Last revised: Nov 4, 2020 (v2)
Subjects: math.DG, math.CV
Journal: Geom. Topol. 25 (2021) 1979-2015
DOI: 10.2140/gt.2021.25.1979

Comments: 32 pages

Ricci flow Kähler–Ricci flow (Non)collapsing Kähler geometry Limit spaces & RCD Boundaries & non-compact Long-time existence & convergence

Its neighbourhood

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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 2016, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

On existence and curvature estimates of Ricci flow
Man-Chun Lee and Luen-Fai Tam2017

Topics Kähler–Ricci flow, (Non)collapsing, Boundaries & non-compact · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “bisectional curvature”, “complete noncompact”, “maximal volume”
Chern-Ricci flows on noncompact complex manifolds
Man-Chun Lee and Luen-Fai Tam2017

Topics Kähler–Ricci flow, (Non)collapsing, Boundaries & non-compact · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “bisectional curvature”, “maximal volume”, “complex manifold”
Longtime existence of Kähler Ricci flow and holomorphic sectional curvature
Shaochuang Huang, Man-Chun Lee, Luen-Fai Tam et al.2018

Topics Kähler–Ricci flow, Long-time existence & convergence, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam
Kähler-Ricci Shrinkers and Ancient Solutions with Nonnegative Orthogonal Bisectional Curvature
Xiaolong Li and Lei Ni2019

Topics Kähler–Ricci flow, Kähler geometry · Shared terms “orthogonal bisectional”, “nonnegative orthogonal”, “bisectional curvature”
Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics
Shaochuang Huang, Man-Chun Lee and Luen-Fai Tam2019

Topics Long-time existence & convergence, Boundaries & non-compact, Kähler geometry · Same authors Man-Chun Lee, Luen-Fai Tam · Shared terms “complete noncompact”, “complex manifold”, “Topping”
Gromov-Hausdorff limits of immortal Kähler-Ricci flows
Man-Chun Lee, Valentino Tosatti and Junsheng Zhang2026

Topics Kähler–Ricci flow, Limit spaces & RCD, Long-time existence & convergence · Same author Man-Chun Lee · Shared terms “Hausdorff limit”, “Gromov Hausdorff”
Kähler-Ricci Flow preserves negative anti-bisectional curvature
Gabriel Khan and Fangyang Zheng2020

Topics Kähler–Ricci flow, Limit spaces & RCD, Kähler geometry · Shared terms “bisectional curvature”, “orthogonal”
Complete Kähler-Einstein metric on Stein manifolds with negative curvature
Man-Chun Lee2019

Topics Kähler–Ricci flow, Kähler geometry · Same author Man-Chun Lee · Shared terms “negative curvature”, “complete Kähler”

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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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Kähler–Ricci flow Ricci flow started from a Kähler metric stays Kähler and reduces to a single scalar equation, a parabolic complex Monge–Ampère equation. Start with: Deformation of Kähler matrics to Kähler-Einstein metrics on compact Kähler manifolds (Cao, 1985) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978)

In the guidebook

5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.
9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

[1] 5A.6
Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry
The Kähler–Ricci flow, and when to come back to it.

[2] 9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

[3] 12A.4
κ-NoncollapsingBook 12A · Entropy and Noncollapsing
The theorem that excludes the cigar.

Kähler manifolds with almost non-negative curvature

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