Paper · Mar 5, 2023

The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

Lucio Bedulli, Giovanni Gentili, Luigi Vezzoni

math.DGarXiv:2303.02689v3
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Abstract

We show that the parabolic quaternionic Monge-Ampère equation on a compact hyperkähler manifold has always a long-time solution which once normalized converges smoothly to a solution of the quaternionic Monge-Ampère equation. This is the same setting in which Dinew and Sroka prove the conjecture of Alesker and Verbitsky. We also introduce an analogue of the Chern-Ricci flow in hyperhermitian manifolds.

Comments: This is a new version of the paper "The Calabi-Yau theorem on Hypercomplex manifolds", which was withdrawn due to a crucial mistake in a key Lemma. In this new version we fix the mistake by introducing the extra assumption of the existence of a background hyperkähler metric

Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–YauKähler geometryLong-time existence & convergence

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  1. HKT geometry on locally conformally hyperkähler manifolds

    Giovanni Gentili and Luigi Vezzoni PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Same authors Giovanni Gentili, Luigi Vezzoni · Shared terms “hyperKähler manifold”, “quaternionic”, “Ampère equation”

  2. The behavior of the second Ricci flow on complex parallelizable manifolds

    Lucio Bedulli and Luigi Vezzoni PDF

    Topics Generalized & G₂ flows · Same authors Lucio Bedulli, Luigi Vezzoni · Shared terms “Chern Ricci”

  3. The generalized Kähler Calabi-Yau problem

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

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Long-time existence & convergence · Shared terms “hyperKähler manifold”

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

    Sébastien Picard and Caleb Suan PDF

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Long-time existence & convergence · Shared terms “Ampère equation”, “Monge Ampère”

  5. On the Calabi estimate of geometric flows of Hermitian metrics

    Marco Gallo and Luigi Vezzoni PDF

    Topics Generalized & G₂ flows, Kähler geometry · Same author Luigi Vezzoni · Shared terms “Calabi”, “Chern Ricci”

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

    Jiyuan Han PDF

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

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

    Mathew George PDF

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

  8. Regularity of solutions to Monge–Ampère equations on compact Hermitian manifolds

    Quang-Tuan Dang PDF

    Topics Generalized & G₂ flows, Kähler geometry · Shared terms “Ampère equation”, “Monge”, “Chern”

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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) Generalized & G₂ flows Ricci-type flows for geometry beyond Kähler metrics. Generalized Ricci flow couples a metric to a closed 3-form, pluriclosed and Chern–Ricci flows… Start with: A parabolic flow of pluriclosed metrics (Streets et al., 2009) 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)

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    A heat equation for the metric: spheres, cylinders and scaling.