Paper · Apr 12, 2016

From the Kähler-Ricci flow to moving free boundaries and shocks

Robert J. Berman, Chinh H. Lu

math.DGarXiv:1604.03259
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Abstract

We show that the twisted Kähler-Ricci flow on a complex manifold X converges to a flow of moving free boundaries, in a certain scaling limit. This leads to a new phenomenon of singularity formation and topology change which can be seen as a complex generalization of the extensively studied formation of shocks in Hamilton-Jacobi equations and hyperbolic conservation laws (notably, in the adhesion model in cosmology). In particular we show how to recover the Hele-Shaw flow (Laplacian growth) of growing 2D domains from the Ricci flow. As we briefly indicate the scaling limit in question arises as the zero-temperature limit of a certain many particle system on X.

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Submitted
Apr 12, 2016
Subjects
math.DG, math-ph, math.AP

Comments: 49 pages

Ricci flowKähler–Ricci flowSingularitiesKähler geometryFree boundary & capillaryPhysics

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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 2016, 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 modified Kähler-Ricci flow, II (Haotian Wu and Zhou Zhang, 2022)5Wu 2022 Kähler-Ricci flow and conformal submersion (Hoan Nguyen, 2023)4Nguyen 2023 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)3Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)6Liu 2024 Kähler-Ricci Flow: from divisors to cusps (Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu, 2026)1Di Nezza 2026 Geometric smoothing by the Kähler-Ricci Flow (Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu, 2026)2Di Nezza 2026 Quantitative avoidance for free boundary flows and applications (Yueheng Bao and Robert Haslhofer, 2026)8Bao 2026 Uniqueness of tangent flows in free boundary flow (Sourav Ghosh, 2026)7Ghosh 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Kähler-Ricci Flow: from divisors to cusps

    Eleonora Di Nezza, Vincent Guedj and Chinh H. Lu

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Same author Chinh H. Lu

  2. Geometric smoothing by the Kähler-Ricci Flow

    Eleonora Di Nezza, Vincent Guedj and Chinh Hoang Lu

    Topics Kähler–Ricci flow, Kähler geometry · Same author Chinh H. Lu

  3. Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle

    Jiawei Liu and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  4. Kähler-Ricci flow and conformal submersion

    Hoan Nguyen

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “formation”

  5. The modified Kähler-Ricci flow, II

    Haotian Wu and Zhou Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities

  6. Regularizing property of the twisted conical Kähler-Ricci flow

    Jiawei Liu, Shiyu Zhang and Xi Zhang

    Topics Kähler–Ricci flow, Kähler geometry, Singularities · Shared terms “twisted Kähler”

  7. Uniqueness of tangent flows in free boundary flow

    Sourav GhoshWider flows

    Topics Free boundary & capillary, Singularities

  8. Quantitative avoidance for free boundary flows and applications

    Yueheng Bao and Robert HaslhoferWider flows

    Topics Free boundary & capillary, Singularities

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) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Kähler geometry Complex manifolds with compatible Riemannian metrics whose Kähler form is closed. Here Ricci curvature is the complex Hessian of a log-determinant… 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. 11B.4
    Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities

    Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

  3. 11B.6
    Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities

    The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.