Paper · Sep 7, 2022

Flows of -Structures associated to Calabi-Yau Manifolds

Sébastien Picard, Caleb Suan

math.DGarXiv:2209.03411v2
PDF Abstract page

As on arXiv

Abstract

We establish a correspondence between a parabolic complex Monge-Ampère equation and the -Laplacian flow for initial data produced from a Kähler metric on a complex - or -fold. By applying estimate for the complex Monge-Ampère equation, we show that for this class of initial data the -Laplacian flow exists for all time and converges to a torsion-free -structure induced by a Kähler Ricci-flat metric. Similar results are obtained for the -Laplacian coflow, and in this case the coflow is related to the Kähler-Ricci flow.

Comments: 27 Pages; v2 (Minor Changes)

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

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 Diameter estimates in Kähler geometry (Bin Guo, Duong H. Phong, Jian Song et al., 2022)8Guo 2022 The generalized Kähler Calabi-Yau problem (Vestislav Apostolov, Xin Fu, Jeffrey Streets et al., 2022)6Apostolov 2022 The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds (Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni, 2023)7Bedulli 2023 The L^(∞) estimate for parabolic complex Monge-Ampère equations (Qizhi Zhao, 2023)5Zhao 2023 Laplacian coflows of G₂-structures on contact Calabi–Yau 7-manifolds (Henrique N. Sá Earp, Julieth Saavedra and Caleb Suan, 2024)1Earp 2024 Complex Monge-Ampère equation for positive (p,p)-forms on compact Kähler manifolds (Mathew George, 2024)3George 2024 Flows of conformally coclosed G₂-structures with dilaton (Spiro Karigiannis, Sébastien Picard and Caleb Suan, 2025)2Karigiannis 2025 Parabolic Monge-Ampère equations on a family of Calabi-Yau manifolds (Jiyuan Han, 2026)4Han 2026
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 Ricci-flat & Calabi–Yau, Generalized & G₂ flows, Kähler–Ricci flow · Same author Caleb Suan · Shared terms “Laplacian coflow”

  2. Flows of conformally coclosed -structures with dilaton

    Spiro Karigiannis, 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. 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 “complex Monge”, “Ampère equation”, “Monge Ampère”

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

    Jiyuan Han PDF

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

  5. The estimate for parabolic complex Monge-Ampère equations

    Qizhi Zhao PDF

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “complex Monge”, “parabolic complex”, “Ampère equation”

  6. 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 · Shared terms “initial data”, “flow exists”

  7. The parabolic quaternionic Calabi-Yau equation on hyperkähler manifolds

    Lucio Bedulli, Giovanni Gentili and Luigi Vezzoni PDF

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

  8. Diameter estimates in Kähler geometry

    Bin Guo, Duong H. Phong, Jian Song et al. PDF

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

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. 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.