Paper · Apr 25, 2017

Matrix Inequality for the Laplace Equation

Jiewon Park

math.DGarXiv:1704.07530v2
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Abstract

Since Li and Yau obtained the gradient estimate for the heat equation, related estimates have been extensively studied. With additional curvature assumptions, matrix estimates that generalize such estimates have been discovered for various time-dependent settings, including the heat equation on a Kähler manifold, Ricci flow, Kähler-Ricci flow, and mean curvature flow, to name a few. As an elliptic analogue, Colding proved a sharp gradient estimate for the Green function on a manifold with nonnegative Ricci curvature. In this paper we prove a related matrix inequality on manifolds with suitable curvature and volume growth assumptions.

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Submitted
Apr 25, 2017
Last revised
Apr 26, 2017 (v2)
Subjects
math.DG

Comments: 11 pages; comments welcome

Ricci flowKähler–Ricci flowMean curvature flowKähler geometry

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 2016, plus the canon.

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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. Matrix Li-Yau-Hamilton Estimates for Nonlinear Heat Equations

    Xin-An Ren

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “heat equation”, “Yau”

  2. Matrix Li-Yau-Hamilton Estimates under Kähler-Ricci Flow

    Xiaolong Li, Hao-Yue Liu and Xin-An Ren

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “heat equation”, “Yau”

  3. Two Comments on "Laplace comparison on Kahler Ricci flow and convergence (arXiv:2509.14820v1)"

    Xiuxiong Chen and Bing Wang

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

  4. Laplace comparison on Kähler Ricci flow and convergence

    Gang Tian, Qi S. Zhang, Zhenlei Zhang et al.

    Topics Kähler–Ricci flow, Kähler geometry · Shared terms “Laplace”, “Colding”

  5. Miyaoka-Yau inequality for compact Kähler manifolds with semi-positive canonical bundle

    Ryosuke Nomura

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

  6. Kähler manifolds with negative holomorphic sectional curvature, Kähler-Ricci flow approach

    Ryosuke Nomura

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

  7. Immortal solutions of the Kähler-Ricci flow

    Valentino Tosatti

    Topics Kähler–Ricci flow, Kähler geometry

  8. Global Ricci Curvature Behaviour for the Kähler-Ricci Flow with Finite Time Singularities

    Alexander Bednarek

    Topics Kähler–Ricci flow, Kähler geometry

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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  3. 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.