Paper · Aug 14, 2026

Hermitian curvature flow and HKT geometry

Beatrice Brienza, Jeffrey Streets

math.DGarXiv:2608.14358
PDF Abstract page

As on arXiv

Abstract

We identify a Hermitian curvature flow which preserves HKT geometry, and whose fixed points are HKT-Einstein metrics, equivalent to a flow suggested by Verbitsky in the context of the quaternionic Monge-Ampère equation. We exhibit a fundamental regularity obstruction and a monotonicity formula for the Chern scalar curvature. We formulate a maximal existence time conjecture for this flow, and give a conditional resolution. We establish the existence conjecture in dimension four. We show the existence of a divergent sequence of HKT-Einstein metrics on quaternionic Hopf surfaces. These are the first non-homogeneous examples in the literature, and indicate the delicacy of the convergence question. Finally we classify which strong HKT structures arising from bi-invariant metrics on Lie groups are also HKT-Einstein.

Generalized & G₂ flowsEinstein metrics4-manifoldsKähler geometryScalar curvatureHomogeneous spaces

Its neighbourhood

Connected papers

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 Four-dimensional generalized Ricci flows with nilpotent symmetry (Steven Gindi and Jeffrey Streets, 2021)2Gindi 2021 Positive Hermitian Curvature Flow on special linear groups and perfect solitons (James Stanfield, 2021)6Stanfield 2021 Scalar curvature, entropy, and generalized Ricci flow (Jeffrey Streets, 2022)8Streets 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)7Apostolov 2022 Hermitian metrics on complex non-Kähler manifolds (Daniele Angella, 2025)5Angella 2025 Positive Hermitian curvature flow on 2-step nilpotent Lie groups (Ettore Lo Giudice, 2025)4Giudice 2025 Sasaki with torsion manifolds and string backgrounds (Beatrice Brienza, Anna Fino, Udhav Fowdar et al., 2026)3Brienza 2026 HKT geometry on locally conformally hyperkähler manifolds (Giovanni Gentili and Luigi Vezzoni, 2026)1Gentili 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. HKT geometry on locally conformally hyperkähler manifolds

    Giovanni Gentili and Luigi Vezzoni PDF

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

  2. Four-dimensional generalized Ricci flows with nilpotent symmetry

    Steven Gindi and Jeffrey Streets PDF

    Topics Generalized & G₂ flows, 4-manifolds, Homogeneous spaces · Same author Jeffrey Streets · Shared terms “Lie”

  3. Sasaki with torsion manifolds and string backgrounds

    Beatrice Brienza, Anna Fino, Udhav Fowdar et al. PDF

    Topics Generalized & G₂ flows, Einstein metrics, Kähler geometry · Same author Beatrice Brienza · Shared terms “Hermitian”, “equivalent”, “preserves”

  4. Positive Hermitian curvature flow on 2-step nilpotent Lie groups

    Ettore Lo Giudice PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Hermitian curvature”, “Lie groups”, “invariant metrics”

  5. Hermitian metrics on complex non-Kähler manifolds

    Daniele Angella PDF

    Topics Generalized & G₂ flows, Einstein metrics, Scalar curvature · Shared terms “Hermitian”, “Chern scalar”

  6. Positive Hermitian Curvature Flow on special linear groups and perfect solitons

    James Stanfield PDF

    Topics Generalized & G₂ flows, Homogeneous spaces · Shared terms “Hermitian curvature”, “Lie groups”

  7. The generalized Kähler Calabi-Yau problem

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

    Topics Generalized & G₂ flows, Kähler geometry · Same author Jeffrey Streets · Shared terms “fixed point”, “monotonicity formula”, “formulate”

  8. Scalar curvature, entropy, and generalized Ricci flow

    Jeffrey Streets PDF

    Topics Generalized & G₂ flows, Scalar curvature · Same author Jeffrey Streets · Shared terms “monotonicity formula”

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.

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) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 4-manifolds Four is the first dimension where Ricci flow's singularities are not fully understood. It is also the dimension of the smooth Poincaré conjecture… Start with: The topology of four-dimensional manifolds (Freedman, 1982) 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. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

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

  2. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  3. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.