Paper · Jun 4, 2024

Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

Jiawei Liu, Miles Simon

math.DGarXiv:2406.02351v2
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Abstract

In this paper we prove localised weighted curvature integral estimates for solutions to the Ricci flow in the setting of a smooth four dimensional Ricci flow or a closed -dimensional Kähler Ricci flow. These integral estimates improve and extend the integral curvature estimates shown by the second author in an earlier paper. If the scalar curvature is uniformly bounded in the spatial sense for some then the estimates imply a uniform bound on the spatial norm of the Riemannian curvature tensor. Stronger integral estimates are shown to hold if one further assumes a weak non-inflating condition, or we restrict to closed manifolds.

Ricci flowKähler–Ricci flow4-manifoldsKähler geometryScalar curvature

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    Jiawei Liu and Miles Simon PDF

    Topics 4-manifolds, Scalar curvature · Same authors Jiawei Liu, Miles Simon · Shared terms “bounded scalar”, “inflating”, “restrict”

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    Topics 4-manifolds, Scalar curvature · Shared terms “bounded scalar”, “estimates improve”, “curvature estimates”

  3. Ricci flow of -metrics in four dimensions

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    Topics 4-manifolds, Scalar curvature · Same author Miles Simon · Shared terms “closed manifolds”

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

    Jiawei Liu and Xi Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jiawei Liu

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

    Jiawei Liu, Shiyu Zhang and Xi Zhang PDF

    Topics Kähler–Ricci flow, Kähler geometry · Same author Jiawei Liu

  6. Curvature estimates for four-dimensional complete gradient expanding Ricci solitons

    Huai-Dong Cao and Tianbo Liu PDF

    Topics 4-manifolds, Scalar curvature · Shared terms “curvature estimates”, “four dimensional”, “curvature tensor”

  7. Gromov-Hausdorff Limits of Noncollapsed Kähler-Ricci Flows and the Geometry of Ricci Shrinkers

    Yu Li and Junsheng Zhang PDF

    Topics 4-manifolds, Kähler–Ricci flow, Scalar curvature · Shared terms “bounded scalar”, “dimensional Ricci”, “four dimensional”

  8. Rigidity of four-dimensional Kähler-Ricci solitons

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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) 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. 5A.6
    Where This Track LeadsBook 5A · Complex Analysis and Conformal Geometry

    The Kähler–Ricci flow, and when to come back to it.

  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.