The L^(∞) estimate for parabolic complex Monge-Ampère equations

Qizhi Zhao

Paper · Jun 29, 2023

The $L^{\infty}$ estimate for parabolic complex Monge-Ampère equations

Qizhi Zhao

math.DGarXiv:2306.16730

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Abstract

Following the recent development by Guo-Phong-Tong and Chen-Cheng, we derived the $L^{\infty}$ estimate for Kähler-Ricci flows under a weaker assumption. The technique also extends to more general cases coming from different geometric backgrounds.

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Submitted: Jun 29, 2023
Subjects: math.DG
Journal: Analysis & PDE 18 (2025) 1875-1896
DOI: 10.2140/apde.2025.18.1875

Ricci flow Kähler–Ricci flow Kähler geometry

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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) Kähler geometry Complex manifolds with compatible Riemannian metrics. Here Ricci curvature is the complex Hessian of a log-determinant, so Ricci flow becomes a… 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.
5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.
10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

[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] 5A.5
Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry
Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

[3] 10A.12
Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization
How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

The $L^{\infty}$ estimate for parabolic complex Monge-Ampère equations

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The L∞ estimate for parabolic complex Monge-Ampère equations

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The $L^{\infty}$ estimate for parabolic complex Monge-Ampère equations