Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds

Mathew George

Paper · Nov 10, 2024

Complex Monge-Ampère equation for positive $(p, p)$ -forms on compact Kähler manifolds

Mathew George

math.AParXiv:2411.06497v3

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Abstract

A complex Monge-Ampère equation for differential $(p, p)$ -forms is introduced on compact Kähler manifolds. For any $1 \leq p < n$ , we show the existence of smooth solutions unique up to adding constants. For $p = 1$ , this corresponds to the Calabi-Yau theorem proved by S. T. Yau, and for $p = n - 1$ , this gives the Monge-Ampère equation for $(n - 1)$ plurisubharmonic functions studied by Tosatti-Weinkove. For other $p$ values, this defines a non-linear PDE that falls outside of the general framework of Caffarelli-Nirenberg-Spruck. Further, we define a geometric flow for higher-order forms that preserves their cohomology classes, and extends the Kähler-Ricci flow naturally to $(p, p)$ -forms. As a consequence of our main theorem, we show that this flow exists in a maximal time interval and can be shown to converge under some assumptions. A modified flow is introduced and the convergence of the associated normalized flow is shown.

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Submitted: Nov 10, 2024
Last revised: Nov 18, 2025 (v3)
Subjects: math.AP, math.AG, math.DG

Ricci flow Kähler–Ricci flow Ricci-flat & Calabi–Yau Kähler geometry

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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) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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.

Complex Monge-Ampère equation for positive $(p, p)$ -forms on compact Kähler manifolds

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