Paper · Oct 6, 2026

HKT geometry on locally conformally hyperkähler manifolds

Giovanni Gentili, Luigi Vezzoni

math.DGarXiv:2610.07897
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Abstract

We study the HKT geometry of compact locally conformally hyperkähler manifolds. We prove that on these manifolds the quaternionic Monge-Ampère equation is always solvable, extending results of Alesker and of Dinew-Sroka, and we confirm the conjecture on the maximal existence time of the HKT-Ricci flow. On quaternionic Hopf surfaces we construct non-trivial HKT solitons and prove convergence of the normalized flow for symmetric initial data.

Ricci flowRicci-flat & Calabi–YauKähler geometryHomogeneous spaces

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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 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds (Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni, 2023)1Bedulli 2023 Generalized Ricci flow on aligned homogeneous spaces (Valeria Gutiérrez, 2024)8Gutiérrez 2024 Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds (Mathew George, 2024)3George 2024 Quaternionic Kähler manifolds fibered by solvsolitons (Vicente Cortés, Alejandro Gil-García and Markus Röser, 2025)6Cortés 2025 Higher-Dimensional Vacuum Einstein Equations: Symmetry, New Solutions, and Ricci Solitons (M. M. Akbar and M. Self, 2025)5Akbar 2025 Ricci Flow on ALF manifolds (Dain Kim and Tristan Ozuch, 2025)4Kim 2025 Hermitian curvature flow and HKT geometry (Beatrice Brienza and Jeffrey Streets, 2026)2Brienza 2026 Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds (Jiyuan Han, 2026)7Han 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

    Lucio Bedulli, 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. Hermitian curvature flow and HKT geometry

    Beatrice Brienza and Jeffrey Streets PDF

    Topics Homogeneous spaces, Kähler geometry · Shared terms “HKT geometry”, “quaternionic”, “Hopf surfaces”

  3. 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”, “normalized flow”

  4. Ricci Flow on ALF manifolds

    Dain Kim and Tristan Ozuch PDF

    Topics Ricci-flat & Calabi–Yau, Kähler geometry · Shared terms “hyperkähler”, “conformally”

  5. Higher-Dimensional Vacuum Einstein Equations: Symmetry, New Solutions, and Ricci Solitons

    M. M. Akbar and M. Self PDF

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces

  6. Quaternionic Kähler manifolds fibered by solvsolitons

    Vicente Cortés, Alejandro Gil-García and Markus Röser PDF

    Topics Homogeneous spaces, Kähler geometry · Shared terms “quaternionic”, “solvable”

  7. 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”

  8. Generalized Ricci flow on aligned homogeneous spaces

    Valeria Gutiérrez PDF

    Topics Ricci-flat & Calabi–Yau, Homogeneous spaces

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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) 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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  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.