Paper · Nov 26, 2025

Flows of conformally coclosed -structures with dilaton

Spiro Karigiannis, Sébastien Picard, Caleb Suan

math.DGarXiv:2511.21055v2The Journal of Geometric Analysis 36 (2026), 171, 48pp
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Abstract

We study flows of -structures guided by the principle of dimensional reduction: natural geometric flows in -geometry reduce to natural flows in complex geometry. Our main examples are the -Laplacian coflow, which lifts the Kähler–Ricci flow, and a 7-dimensional lift of the anomaly flow on complex threefolds. The -lift of the anomaly flow deforms conformally coclosed -structures. We compare the -anomaly flow to the -Laplacian coflow, and investigate short-time existence and fixed points.

Comments: 35 pages, 2 tables. Version 2: Minor revisions and remarks added following referee's report. Final version, to appear in The Journal of Geometric Analysis

Ricci flowKähler–Ricci flowGeneralized & G₂ flows3-manifoldsKähler geometryHarmonic maps & DeTurck trick

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Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Laplacian coflows of -structures on contact Calabi–Yau 7-manifolds

    Henrique N. Sá Earp, Julieth Saavedra and Caleb Suan PDF

    Topics Harmonic maps & DeTurck trick, Generalized & G₂ flows, Kähler–Ricci flow · Same author Caleb Suan · Shared terms “Laplacian coflow”, “dimensional reduction”

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

    Sébastien Picard and Caleb Suan PDF

    Topics Generalized & G₂ flows, Kähler–Ricci flow, Kähler geometry · Same authors Sébastien Picard, Caleb Suan · Shared terms “Laplacian coflow”

  3. BRIDGES Lectures: Flows of geometric structures, especially -structures

    Spiro Karigiannis PDF

    Topics Harmonic maps & DeTurck trick, Generalized & G₂ flows · Same author Spiro Karigiannis · Shared terms “geometric flows”, “short time”, “time existence”

  4. The Strominger System and Flows by the Ricci Tensor

    Sébastien Picard PDF

    Topics 3-manifolds, Kähler geometry · Same author Sébastien Picard · Shared terms “anomaly flow”, “threefold”, “geometric flows”

  5. Almost Hermitian geometry and generalized Ricci flow

    Anna Fino and Julieth Saavedra PDF

    Topics Harmonic maps & DeTurck trick, Generalized & G₂ flows · Shared terms “short time”, “time existence”

  6. Geometric Flows of -Structures on 3-Sasakian 7-Manifolds

    Aaron Kennon and Jason D. Lotay PDF

    Topics Generalized & G₂ flows · Shared terms “Laplacian coflow”, “geometric flows”

  7. Geometric flows from unified string theories

    Duong H. Phong PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “complex geometry”, “geometric flows”

  8. Vaisman manifolds and transversally Kähler-Einstein metrics

    Vladimir Slesar and Gabriel-Eduard Vîlcu PDF

    Topics Harmonic maps & DeTurck trick, Kähler–Ricci flow, Kähler geometry · Shared terms “short time”, “time existence”

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) 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) 3-manifolds Dimension three is where Ricci flow has done its most complete work. It proves geometrization, describes the flow's long-time behaviour, and now… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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. 10A.1
    A Zoo of Three-ManifoldsBook 10A · Three-Manifolds and Geometrization

    The 3-sphere, lens spaces, the Poincaré sphere, the Hopf fibration and the shape of space.

  3. 10A.2
    Finite Rotation Groups and Spherical Space FormsBook 10A · Three-Manifolds and Geometrization

    Klein's classification by counting poles, the binary groups, and every quotient of the round 3-sphere.