Paper · Oct 1, 2026

Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds

Jiyuan Han

math.DGarXiv:2610.01202v2
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Abstract

We prove a uniform -estimate for the parabolic Monge–Ampère equation under uniform Hörmander-Tian type estimates with respect to the initial and prescribed measures. We apply this estimate to rescaled Kähler–Ricci flows on a family of polarized Calabi–Yau manifolds over the punctured disk. An interpolation argument gives uniform Hörmander-Tian type estimates along the flows. We also establish uniform diameter bounds and Gromov–Hausdorff precompactness for this family of flows. In particular, we obtain an alternative proof of the diameter estimate by Li-Tosatti.

Comments: More remarks are added in v.2

Ricci flowKähler–Ricci flowRicci-flat & Calabi–YauKähler geometryLimit spaces & RCD

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  1. Complex Monge-Ampère equation for positive -forms on compact Kähler manifolds

    Mathew George PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”, “Tosatti”

  2. Diameter estimates in Kähler geometry

    Bin Guo, Duong H. Phong, Jian Song et al. PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Shared terms “diameter estimates”, “Ampère equation”, “Monge Ampère”

  3. Flows of -Structures associated to Calabi-Yau Manifolds

    Sébastien Picard and Caleb Suan PDF

    Topics Ricci-flat & Calabi–Yau, Kähler–Ricci flow, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”

  4. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”

  5. The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

    Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”

  6. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”

  7. The generalized Kähler Calabi-Yau problem

    Vestislav Apostolov, Xin Fu, Jeffrey Streets et al. PDF

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

  8. On a general class of free boundary Monge-Ampère equations

    Tristan C. Collins and Benjy Firester PDF

    Topics Limit spaces & RCD, Kähler geometry · Shared terms “Ampère equation”, “Monge Ampère”

Read around it

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) 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

  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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  3. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.