Paper · May 17, 2024

Rigidity of eigenvalues of shrinking Ricci solitons

Chang Li, Huaiyu Zhang, Xi Zhang

math.DGarXiv:2405.10495
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Abstract

In this paper, we study the rigidity of eigenvalues of shring Ricci solitons. It is known that the drifted Laplacian on shrinking Ricci solitons has discrete spectrum, its eigenvalues have a lower bound and a rigidity result holds. Firstly, we show that if the eigenvalue is close to this lower bound, then the -soliton must be the trivial Gaussian soliton . Secondly, we show similar results for the and eigenvalue under a non-collapsing condition. Lastly, we give an alomost rigidity for the eigenvalue with general . Part of our results could be viewed as an soliton (could be noncompact) analog of Theorem 1.1 (which only holds for compact manifolds) in Peterson (Invent. Math. 138 (1999): 1-21).

Ricci solitons(Non)collapsingStability & uniquenessBoundaries & non-compactEigenvalues & spectrum

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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. Eigenvalue lower bounds and splitting for modified Ricci flow

    Tobias Holck Colding and William P. Minicozzi PDF

    Topics Eigenvalues & spectrum · Shared terms “lower bound”, “Laplacian”

  2. Eigenvalue estimates on shrinkers

    S. Brendle and R. Tsiamis PDF

    Topics Eigenvalues & spectrum, Boundaries & non-compact, Ricci solitons

  3. Higher-dimensional flying wing Steady Ricci Solitons

    Pak-Yeung Chan, Yi Lai and Man-Chun Lee PDF

    Topics Eigenvalues & spectrum, (Non)collapsing, Boundaries & non-compact

  4. The spectral rigidity of Ricci soliton and Einstein-type manifolds

    Ping Li, Xiaomei Sun and Anqiang Zhu PDF

    Topics Eigenvalues & spectrum, Stability & uniqueness, Ricci solitons · Shared terms “shrinking Ricci”, “Laplacian”

  5. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “discrete spectrum”, “Laplacian”, “Lastly”

  6. Rigidity results for shrinking and expanding Ricci solitons

    Benedito Leandro and Jeferson Poveda PDF

    Topics Boundaries & non-compact, Stability & uniqueness, Ricci solitons · Shared terms “shrinking Ricci”

  7. An eigenvalue estimate for self-shrinkers in a Ricci shirinker

    Franciele Conrado and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “drifted Laplacian”, “lower bound”

  8. On noncompact warped product Ricci solitons

    Valter Borges PDF

    Topics Eigenvalues & spectrum, Boundaries & non-compact, Stability & uniqueness · Shared terms “Laplacian”

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) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982) 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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

In the guidebook

  1. 9B.3
    Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds

    Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

  2. 12A.4
    κ-NoncollapsingBook 12A · Entropy and Noncollapsing

    The theorem that excludes the cigar.

  3. 12A.8
    Local Noncollapsing from the Reduced VolumeBook 12A · Entropy and Noncollapsing

    Perelman's §§7.3 and 8: noncollapsing that needs only local curvature control, the small-and-large-vector split, the maximum principle for the reduced distance, and the form that survives surgery.