Paper · May 22, 2026

Eigenvalue Estimates for Schrödinger Operators on Ricci Shrinkers

Xu Cheng, Franciele Conrado, Neilha Pinheiro, Detang Zhou

math.DGarXiv:2605.23199
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Abstract

Let be a complete Ricci shrinker satisfying and let denote its scalar curvature. For a confined function on , we obtain a lower bound for the lowest eigenvalue of the Schrödinger operator , expressed in terms of an integral quantity involving and the shrinker entropy, and the equality case is characterized by the potential functions. We further generalize this estimate to complete Riemannian manifolds via Perelman's -functional. We also study the drifted Schrödinger operator on smooth metric measure spaces. In particular, on Ricci shrinkers, we derive a lower bound for its lowest eigenvalue, with equality if and only if is affine.

Comments: 18 pages. Comments are welcome

Ricci solitonsScalar curvatureLimit spaces & RCDEigenvalues & spectrum

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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 Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons (Xu Cheng, Ernani Ribeiro and Detang Zhou, 2022)7Cheng 2022 The Wasserstein distance for Ricci shrinkers (Franciele Conrado and Detang Zhou, 2023)3Conrado 2023 Eigenvalue estimates on shrinkers (S. Brendle and R. Tsiamis, 2024)2Brendle 2024 Spectrum of the drift Laplacian on Ricci expanders (Helton Leal, Matheus Vieira and Detang Zhou, 2024)4Leal 2024 An eigenvalue estimate for self-shrinkers in a Ricci shirinker (Franciele Conrado and Detang Zhou, 2025)1Conrado 2025 Spectral bounds for f-Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers (Fei He, 2026)8He 2026 The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers (Junrong Yan, 2026)5Yan 2026 Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid (Wenqi Li, Jianyu Ou, Guoqiang Wu 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. An eigenvalue estimate for self-shrinkers in a Ricci shirinker

    Franciele Conrado and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Same authors Franciele Conrado, Detang Zhou · Shared terms “eigenvalue estimate”, “Ricci shrinkers”, “lower bound”

  2. Eigenvalue estimates on shrinkers

    S. Brendle and R. Tsiamis PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “eigenvalue estimate”

  3. The Wasserstein distance for Ricci shrinkers

    Franciele Conrado and Detang Zhou PDF

    Topics Ricci solitons · Same authors Franciele Conrado, Detang Zhou · Shared terms “Ricci shrinkers”, “Let”

  4. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Same author Detang Zhou · Shared terms “potential function”

  5. The Weyl Law Meets Large-Scale Regularity on Ricci Shrinkers

    Junrong Yan PDF

    Topics Eigenvalues & spectrum, Ricci solitons · Shared terms “Schrödinger operator”, “Ricci shrinkers”

  6. Kähler–Ricci Shrinkers with Constant Scalar Curvature Are Rigid

    Wenqi Li, Jianyu Ou, Guoqiang Wu et al. PDF

    Topics Scalar curvature, Ricci solitons · Same author Detang Zhou · Shared terms “Ricci shrinkers”, “shrinker satisfying”, “Let”

  7. On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

    Xu Cheng, Ernani Ribeiro and Detang Zhou PDF

    Topics Ricci solitons · Same authors Xu Cheng, Detang Zhou · Shared terms “potential function”

  8. Spectral bounds for -Laplace-type operators with applications to Betti numbers on gradient Ricci shrinkers

    Fei He PDF

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

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) 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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997) Eigenvalues & spectrum Perelman's λ-functional is the lowest eigenvalue of −4Δ+ R, and it never decreases along Ricci flow. Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002)

In the guidebook

  1. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.

  2. 9B.4
    Convergence of ManifoldsBook 9B · Comparison, Convergence and Heat on Manifolds

    Gromov–Hausdorff and Cheeger–Gromov convergence, and the compactness theorem.

  3. 11B.3
    Compactness of Ricci FlowsBook 11B · Solitons, Compactness and Singularities

    Hamilton’s compactness theorem and the missing injectivity radius bound.