Paper · Jan 21, 2026

Quantitative Spectral Stability for an Embedded Annulus under Coupled Curve Shortening and D Ricci Flows

Mohammadjavad Habibivostakolaei

math.DGarXiv:2601.14575
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Abstract

We study the spectral stability of Dirichlet eigenvalues on an embedded annulus whose boundary evolves by curve shortening flow while the ambient surface evolves under the two dimensional Ricci flow using variational formulas, Rellich–type identities, and harmonic capacity methods, we relate eigenvalue variations to geometric deficit and modulus. We establish quantitative bounds comparing the spectrum of the evolving annulus with that of a flat cylinder of equal modulus. As a consequence, we obtain geometric stability and a spectral gap estimate controlled by the deficit functional.

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Submitted
Jan 21, 2026
Subjects
math.DG, math.SP

Comments: This manuscript includes 31 pages, 4 figures, and 2 tables. All comments and suggestions are kindly welcome

Ricci flowCurve shorteningSurfacesStability & uniquenessEigenvalues & spectrum

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 Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces (Hiuri dos Reis, Benedito Leandro and Rafael Novais, 2022)2dos Reis 2022 Non-uniqueness of curve shortening flow (Luke Thomas Peachey, 2022)3Peachey 2022 Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature (Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al., 2022)4Khan 2022 Monotonicity of the modulus under curve shortening flow (Arjun Sobnack and Peter M. Topping, 2024)1Sobnack 2024 Spectrum of the drift Laplacian on Ricci expanders (Helton Leal, Matheus Vieira and Detang Zhou, 2024)7Leal 2024 Curve shortening flows on surfaces that are not convex at infinity (Naotoshi Fujihara, 2024)6Fujihara 2024 Curve shortening flow with an ambient force field (Sam Cuthbertson, Glen Wheeler and Valentina-Mira Wheeler, 2024)8Cuthbertson 2024 Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity (Mohammadjavad Habibivostakolaei, Abbas M. Sherif and Yen-Kheng Lim, 2026)5Habibivostak 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Monotonicity of the modulus under curve shortening flow

    Arjun Sobnack and Peter M. ToppingWider flows

    Topics Curve shortening · Shared terms “ambient surface”

  2. Rotational Solitons for the Curve Shortening Flow on Revolution Surfaces

    Hiuri dos Reis, Benedito Leandro and Rafael NovaisWider flows

    Topics Curve shortening, Surfaces

  3. Non-uniqueness of curve shortening flow

    Luke Thomas PeacheyWider flows

    Topics Curve shortening, Stability & uniqueness

  4. Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

    Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al.

    Topics Eigenvalues & spectrum, Surfaces · Shared terms “Dirichlet”

  5. Conformal Killing–Yano Ricci solitons: Structure, compatibility, and rigidity

    Mohammadjavad Habibivostakolaei, Abbas M. Sherif and Yen-Kheng Lim

    Topics Eigenvalues & spectrum, Stability & uniqueness · Same author Mohammadjavad Habibivostakolaei · Shared terms “spectral”, “identity”

  6. Curve shortening flows on surfaces that are not convex at infinity

    Naotoshi FujiharaWider flows

    Topics Curve shortening, Surfaces

  7. Spectrum of the drift Laplacian on Ricci expanders

    Helton Leal, Matheus Vieira and Detang Zhou

    Topics Eigenvalues & spectrum, Surfaces

  8. Curve shortening flow with an ambient force field

    Sam Cuthbertson, Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Curve shortening

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) Curve shortening Wider flows Plane curves moving by their curvature. Convex curves shrink to round points (Gage–Hamilton), and every embedded curve becomes convex first (Grayson). Start with: The heat equation shrinking convex plane curves (Gage et al., 1986) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988) 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. 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.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

  3. 12C.4
    Loops, Disks and Curve ShrinkingBook 12C · Extinction and the Proof

    Perelman III §§2–3 and Morgan–Tian §§18.3–18.7: π₃ as spheres of loops, minimal disks and the −2π estimate, curve shrinking, ramps in M × S¹ and the estimates that make the argument rigorous.