Paper · Jun 27, 2026

Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces

Ping-Hung Lee, Chung-Jun Tsai

math.DGWider flowsarXiv:2606.28767
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Abstract

We construct infinite-time singularities with vanishing mean curvature for Lagrangian mean curvature flow in Gibbons–Hawking spaces. We consider circle-invariant Lagrangian -spheres whose quotient curves are concave and are -close to a collection of consecutive collinear segments. We prove that the corresponding flow exists smoothly for all time and converges to the associated -chain of special Lagrangian spheres. Although the mean curvature converges uniformly to zero, the second fundamental form becomes unbounded. More precisely, is comparable to as . The proof is based on a one-parameter family of barrier curves and a detailed analysis of their asymptotics. In this way, we refine the infinite-time convergence picture arising in the work of Lotay and Oliveira by proving curvature blow-up and estimating its rate in this semi-stable case.

Comments: 31 pages

Mean curvature flowLagrangian mean curvature flowSingularities

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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 Mean Curvature Flows of Two-Convex Lagrangians (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)4Tsai 2023 Entire solutions of two-convex Lagrangian mean curvature flows (Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao Wang, 2023)3Tsai 2023 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)1Su 2024 Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature (Ping-Hung Lee and Chung-Jun Tsai, 2024)2Lee 2024 A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase (Arunima Bhattacharya and Jeremy Wall, 2024)7Bhattacharya 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)8Li 2024 Lagrangian mean curvature flow of surfaces with mean curvature bound (Sourav Ghosh, 2025)6Ghosh 2025 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)5Lotay 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

    Wei-Bo Su, Chung-Jun Tsai and Albert Wood PDF

    Topics Lagrangian mean curvature flow, Singularities · Same author Chung-Jun Tsai · Shared terms “infinite time”, “time singularity”, “special Lagrangian”

  2. Dynamical Stability of Minimal Lagrangians in Kähler-Einstein Manifolds of Non-Positive Curvature

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

    Topics Lagrangian mean curvature flow · Same authors Ping-Hung Lee, Chung-Jun Tsai · Shared terms “exists smoothly”, “flow exists”

  3. Entire solutions of two-convex Lagrangian mean curvature flows

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai · Shared terms “special Lagrangian”

  4. Mean Curvature Flows of Two-Convex Lagrangians

    Chung-Jun Tsai, Mao-Pei Tsui and Mu-Tao WangWider flows PDF

    Topics Lagrangian mean curvature flow · Same author Chung-Jun Tsai

  5. The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow

    Jason D. Lotay and Goncalo OliveiraWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “curvature blow”, “becomes unbounded”

  6. Lagrangian mean curvature flow of surfaces with mean curvature bound

    Sourav GhoshWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  7. A Priori Estimates for Singularities of the Lagrangian Mean Curvature Flow with Supercritical Phase

    Arunima Bhattacharya and Jeremy WallWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities

  8. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “special Lagrangian”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Lagrangian mean curvature flow Wider flows In a Calabi–Yau manifold, mean curvature flow keeps Lagrangian submanifolds Lagrangian. Its hoped-for limits are special Lagrangians, the volume… Start with: A canonical way to deform a Lagrangian submanifold (Smoczyk, 1996) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993)

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