Paper · Jun 21, 2024

Laplacian coflows of -structures on contact Calabi–Yau 7-manifolds

Henrique N. Sá Earp, Julieth Saavedra, Caleb Suan

math.DGarXiv:2406.15254v2
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Abstract

We explore three versions of the Laplacian coflow of -structures on circle fibrations over Calabi–Yau 3-folds, interpreting their dimensional reductions to the Kähler geometry of the base. Precisely, we reduce Ansätze for the Laplacian coflow, modified or not by de Turck's trick, both on trivial products and on contact Calabi–Yau 7-manifolds, obtaining in each case a natural modification of the Kähler–Ricci flow.

Comments: to appear in Math Z

Ricci flowKähler–Ricci flowGeneralized & G₂ flowsRicci-flat & Calabi–Yau3-manifoldsKähler geometryHarmonic maps & DeTurck trick

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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 Flows of G₂-Structures associated to Calabi-Yau Manifolds (Sébastien Picard and Caleb Suan, 2022)2Picard 2022 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)5Guo 2022 Geometric Flows of G₂-Structures on 3-Sasakian 7-Manifolds (Aaron Kennon and Jason D. Lotay, 2022)6Kennon 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)4Apostolov 2022 Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds (Mathew George, 2024)7George 2024 BRIDGES Lectures: Flows of geometric structures, especially G₂-structures (Spiro Karigiannis, 2025)8Karigiannis 2025 Flows of conformally coclosed G₂-structures with dilaton (Spiro Karigiannis, Sébastien Picard and Caleb Suan, 2025)1Karigiannis 2025 Almost Hermitian geometry and generalized Ricci flow (Anna Fino and Julieth Saavedra, 2026)3Fino 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Flows of conformally coclosed -structures with dilaton

    Spiro Karigiannis, Sébastien Picard 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 Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Kähler–Ricci flow · Same author Caleb Suan · Shared terms “Laplacian coflow”

  3. Almost Hermitian geometry and generalized Ricci flow

    Anna Fino and Julieth Saavedra PDF

    Topics Harmonic maps & DeTurck trick, Generalized & G₂ flows · Same author Julieth Saavedra

  4. The generalized Kähler Calabi-Yau problem

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

    Topics Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Kähler–Ricci flow

  5. 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 “fibration”

  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”

  7. 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 “modified”

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

    Spiro Karigiannis PDF

    Topics Harmonic maps & DeTurck trick, Generalized & G₂ flows · Shared terms “Laplacian”, “modified”

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) 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)

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.