Paper · Oct 28, 2025

Ricci flow and the scalar curvature rigidity of Einstein manifolds

Klaus Kroencke

math.DGarXiv:2510.24529
PDF Abstract page

As on arXiv

Abstract

We review recent results relating linear stability to dynamical stability and the scalar curvature rigidity of Einstein manifolds. We discuss closed and open Einstein manifolds as well as complete noncompact Einstein manifolds which are asymptotically locally Euclidean and asymptotically hyperbolic. For these classes, the relation to the positive mass theorem will also be explained.

Comments: 14 pages

Ricci flowEinstein metricsScalar curvatureStability & uniquenessBoundaries & non-compactPhysics

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 2021, 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 A volume-renormalized mass for asymptotically hyperbolic manifolds (Mattias Dahl, Klaus Kroencke and Stephen McCormick, 2023)3Dahl 2023 Dynamical Stability and Instability of Poincaré–Einstein Manifolds (Klaus Kroencke and Louis Yudowitz, 2023)1Kroencke 2023 Scalar curvature rigidity of the four-dimensional sphere (Simone Cecchini, Jinmin Wang, Zhizhang Xie et al., 2024)8Cecchini 2024 Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature (Ping-Hung Lee and Chung-Jun Tsai, 2024)7Lee 2024 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)5Lee 2026 On the Stability of Einstein Manifolds with Boundary (Rasmus Jouttijärvi, 2026)2Jouttijärvi 2026 A continuous Positive Mass Theorem for perturbations of Euclidean space (Paula Burkhardt-Guim, 2026)4Burkhardt-Gu 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)6Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz PDF

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

  2. On the Stability of Einstein Manifolds with Boundary

    Rasmus Jouttijärvi PDF

    Topics Einstein metrics, Physics, Scalar curvature

  3. A volume-renormalized mass for asymptotically hyperbolic manifolds

    Mattias Dahl, Klaus Kroencke and Stephen McCormick PDF

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

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

    Paula Burkhardt-Guim PDF

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

  5. Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass

    Man-Chun Lee, Florian Litzinger and Miles Simon PDF

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

  6. Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set

    Fengjiang Li, Wenqi Li, Jianyu Ou et al. PDF

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness · Shared terms “Euclidean”

  7. 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, Scalar curvature, Stability & uniqueness · Shared terms “dynamical stability”, “linear stability”

  8. Scalar curvature rigidity of the four-dimensional sphere

    Simone Cecchini, Jinmin Wang, Zhizhang Xie et al. PDF

    Topics Scalar curvature, Stability & uniqueness · Shared terms “curvature rigidity”

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) 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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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. 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.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.

  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.