Paper · Jul 6, 2026

On the Stability of Einstein Manifolds with Boundary

Rasmus Jouttijärvi

math.DGarXiv:2607.05002
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Abstract

We study the stability problem for Einstein manifolds with boundary with respect to the Einstein-Hilbert action. The geometric boundary conditions we are using arise naturally from studying the calculus of variations associated with Ricci flow. Upon the introduction of a boundary, the space of TTg tensors no longer arise naturally as the defining space for the stability condition. Thus we must settle for the larger subspace of tensors preserving the scalar curvature, the total volume and the Bianchi gauge condition, which we call the space of TVg tensors. As a test-case, we shall discuss stability of the Riemannian Schwarzschild anti-deSitter family of metrics; a problem known as "a Black hole in a box".

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Submitted
Jul 6, 2026
Subjects
math.DG

Comments: 32 pages, 2 figures

Ricci flowEinstein metricsScalar curvatureStability & 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.

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 On curvature related geometric properties of Hayward black hole spacetime (Absos Ali Shaikh, Shyamal Kumar Hui, Biswa Ranjan Datta et al., 2023)5Shaikh 2023 Dynamical Stability and Instability of Poincaré–Einstein Manifolds (Klaus Kroencke and Louis Yudowitz, 2023)3Kroencke 2023 Novel Boundary Conditions for the Ricci Flow (Rasmus Jouttijärvi, 2024)1Jouttijärvi 2024 Ricci flow and the scalar curvature rigidity of Einstein manifolds (Klaus Kroencke, 2025)2Kroencke 2025 Ricci solitons as critical points of quadratic curvature functionals (Atreyee Bhattacharya and Sayoojya Prakash, 2026)7Bhattacharya 2026 Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass (Man-Chun Lee, Florian Litzinger and Miles Simon, 2026)4Lee 2026 A continuous Positive Mass Theorem for perturbations of Euclidean space (Paula Burkhardt-Guim, 2026)6Burkhardt-Gu 2026 Rigidity of Constant-Scalar-Curvature Ricci Shrinkers with Totally Geodesic Focal Set (Fengjiang Li, Wenqi Li, Jianyu Ou et al., 2026)8Li 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Novel Boundary Conditions for the Ricci Flow

    Rasmus Jouttijärvi

    Topics Einstein metrics, Boundaries & non-compact · Same author Rasmus Jouttijärvi · Shared terms “boundary condition”, “Hilbert action”, “Einstein Hilbert”

  2. Ricci flow and the scalar curvature rigidity of Einstein manifolds

    Klaus Kroencke

    Topics Einstein metrics, Physics, Scalar curvature

  3. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz

    Topics Einstein metrics, Physics, Boundaries & non-compact

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

    Man-Chun Lee, Florian Litzinger and Miles Simon

    Topics Physics, Scalar curvature, Boundaries & non-compact

  5. On curvature related geometric properties of Hayward black hole spacetime

    Absos Ali Shaikh, Shyamal Kumar Hui, Biswa Ranjan Datta et al.

    Topics Einstein metrics, Physics · Shared terms “black hole”

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

    Paula Burkhardt-Guim

    Topics Physics, Scalar curvature, Boundaries & non-compact

  7. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness

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

    Fengjiang Li, Wenqi Li, Jianyu Ou et al.

    Topics Einstein metrics, Scalar curvature, Stability & uniqueness

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.