Paper · Dec 20, 2023

Dynamical Stability and Instability of Poincaré–Einstein Manifolds

Klaus Kroencke, Louis Yudowitz

math.DGarXiv:2312.13011
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Abstract

We prove dynamical stability and instability theorems for Poincaré-Einstein metrics under the Ricci flow. Our key tool is a variant of the expander entropy for asymptotically hyperbolic manifolds, which Dahl, McCormick and the first author established in a recent article. It allows us to characterize stability and instability in terms of a local positive mass theorem and in terms of volume comparison for nearby metrics.

Comments: 25 pages

Ricci flowRicci solitonsEinstein metricsStability & uniquenessBoundaries & non-compactPhysics

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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.

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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. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke PDF

    Topics Einstein metrics, Physics, Boundaries & non-compact · Same author Klaus Kroencke · Shared terms “dynamical stability”, “asymptotically hyperbolic”, “mass theorem”

  2. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick PDF

    Topics Physics, Boundaries & non-compact, Ricci solitons · Same author Klaus Kroencke · Shared terms “hyperbolic manifolds”, “asymptotically hyperbolic”, “expander entropy”

  3. On the Stability of Einstein Manifolds with Boundary

    Rasmus Jouttijärvi PDF

    Topics Einstein metrics, Physics, Boundaries & non-compact

  4. A continuous Positive Mass Theorem for perturbations of Euclidean space

    Paula Burkhardt-Guim PDF

    Topics Physics, Boundaries & non-compact · Shared terms “mass theorem”, “positive mass”

  5. The Stability of Generalized Ricci Solitons

    Kuan-Hui Lee PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons · Shared terms “dynamical stability”

  6. Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Einstein metrics, Stability & uniqueness · Shared terms “dynamical stability”

  7. Novel Boundary Conditions for the Ricci Flow

    Rasmus Jouttijärvi PDF

    Topics Einstein metrics, Boundaries & non-compact

  8. Linear stability of compact shrinking Ricci solitons

    Huai-Dong Cao and Meng Zhu PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Einstein metrics Metrics whose Ricci curvature is a constant multiple of the metric. They are the fixed points of normalised Ricci flow, and the most natural metrics… 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. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.

  2. 12B.2
    The Asymptotic Soliton and Three-Dimensional ShrinkersBook 12B · κ-Solutions and Surgery

    Perelman's Proposition 11.2 proved with Morgan and Tian: the reduced-length bounds on κ-solutions, the limit of rescalings, the equality case, and the classification of shrinking solitons in dimensions two and three.

  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.