Paper · Jul 29, 2019

On the linear stability of nearly-Kähler -manifolds

Changliang Wang, M. Y. -K. Wang

math.DGarXiv:1907.12512
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Abstract

We show that a strict, nearly Kähler -manifold with either second or third Betti number nonzero is linearly unstable with respect to the -entropy of Perelman and hence is dynamically unstable for the Ricci flow.

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Submitted
Jul 29, 2019
Subjects
math.DG

Comments: 18 pages. There will be a different paper with an alternative proof that is joint with Uwe Semmelmann

Ricci flowKähler geometryStability & uniqueness

Its neighbourhood

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    Topics Kähler geometry, Stability & uniqueness · Shared terms “linear stability”, “dynamically unstable”, “Perelman”

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  5. Commutator formulas for gradient Ricci shrinker and their application to linear stability

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  6. Linear stability of compact shrinking Ricci solitons

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  7. Dynamic instability of under Ricci flow

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    Topics Kähler geometry, Stability & uniqueness · Shared terms “unstable”, “dynamically”

  8. Linear stability of the blowdown Ricci shrinker in 4D

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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 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) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

In the guidebook

  1. 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.

  2. 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.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.