Paper · Apr 4, 2023

Linear stability of compact shrinking Ricci solitons

Huai-Dong Cao, Meng Zhu

math.DGarXiv:2304.01453v4Math. Ann. (2024)
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Abstract

In this paper, we continue investigating the second variation of Perelman's -entropy for compact shrinking Ricci solitons. In particular, we improve some of our previous work in "H.-D. Cao and M. Zhu, Math. Ann. 353 (2012), No. 3, 747-763", as well as the more recent work in "M. Mehrmohamadi and A. Razavi, arXiv:2104.08343", and obtain a necessary and sufficient condition for a compact shrinking Ricci soliton to be linearly stable. Our work also extends similar results of Hamilton, Ilmanen and the first author in "arXiv:math.DG/0404165" (see also "H.-D. Cao and C. He, J. Reine Angew. Math. 2015 (2015), no. 709, 229-246.") for positive Einstein manifolds to the compact shrinking Ricci soliton case.

Comments: Final version; to appear in Math. Ann

Ricci solitonsEinstein metricsStability & uniqueness

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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. The Stability of Generalized Ricci Solitons

    Kuan-Hui Lee PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons · Shared terms “linear stability”, “linearly stable”, “variation”

  2. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda PDF

    Topics Stability & uniqueness, Ricci solitons · Shared terms “compact shrinking”, “shrinking Ricci”, “Einstein”

  3. Linear Stability of Steady and Expanding Kähler-Ricci Solitons

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    Topics Stability & uniqueness, Ricci solitons · Shared terms “linear stability”, “linearly stable”

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

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

  5. Ricci solitons as critical points of quadratic curvature functionals

    Atreyee Bhattacharya and Sayoojya Prakash PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  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, Stability & uniqueness, Ricci solitons

  7. Dynamical Stability and Instability of Poincaré–Einstein Manifolds

    Klaus Kroencke and Louis Yudowitz PDF

    Topics Einstein metrics, Stability & uniqueness, Ricci solitons

  8. The weighted ambient metric for manifolds with density

    Ayush Khaitan PDF

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

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 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.