Paper · Jul 3, 2026

Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics

Maxwell Stolarski, Wei-Bo Su

math.DGWider flowsarXiv:2607.03152
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Abstract

For each integer when , and for when , we construct an almost-calibrated Lagrangian mean curvature flow in , starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time with the explicit curvature blow up rate The tangent flow at the singularity is a transverse pair of cohomogeneity-one special Lagrangian cones, while the Type II blow-up limit is a smooth cohomogeneity-one special Lagrangian desingularization. This gives a quantitative construction of Type II blow-up for a fully nonlinear parabolic PDE arising from cohomogeneity-one Lagrangian mean curvature flow. Our construction is based on a modulation analysis around a shrinking family of cohomogeneity-one special Lagrangian desingularizations, using the perturbative spectral theory developed in the companion paper.

Comments: 110 pages, 1 figure. Comments welcome

Mean curvature flowLagrangian mean curvature flowSingularitiesHomogeneous spacesEigenvalues & spectrum

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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 Ancient solutions and translators of Lagrangian mean curvature flow (Jason D. Lotay, Felix Schulze and Gábor Székelyhidi, 2022)6Lotay 2022 Cohomogeneity-One Lagrangian Mean Curvature Flow (Jesse Madnick and Albert Wood, 2022)4Madnick 2022 Infinite-Time Singularities of the Lagrangian Mean Curvature Flow (Wei-Bo Su, Chung-Jun Tsai and Albert Wood, 2024)2Su 2024 Singularity formations in Lagrangian mean curvature flow (Yang Li and Gábor Székelyhidi, 2024)3Li 2024 Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class (Xiang Li, Yong Luo and Jun Sun, 2024)7Li 2024 Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow (Maxwell Stolarski and Wei-Bo Su, 2026)1Stolarski 2026 Infinite-Time Singularities with Vanishing Mean Curvature for Lagrangian Mean Curvature Flow in Gibbons–Hawking Spaces (Ping-Hung Lee and Chung-Jun Tsai, 2026)5Lee 2026 The mean curvature blows up at nondegenerate neck pinch singularities of Lagrangian mean curvature flow (Jason D. Lotay and Goncalo Oliveira, 2026)8Lotay 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

    Maxwell Stolarski and Wei-Bo SuWider flows PDF

    Topics Lagrangian mean curvature flow, Eigenvalues & spectrum, Singularities · Same authors Maxwell Stolarski, Wei-Bo Su · Shared terms “Lagrangian desingularization”, “special Lagrangian”, “time singularity”

  2. 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 Wei-Bo Su · Shared terms “special Lagrangian”, “time singularity”, “initial data”

  3. Singularity formations in Lagrangian mean curvature flow

    Yang Li and Gábor SzékelyhidiWider flows PDF

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

  4. Cohomogeneity-One Lagrangian Mean Curvature Flow

    Jesse Madnick and Albert Wood PDF

    Topics Lagrangian mean curvature flow, Homogeneous spaces, Singularities · Shared terms “special Lagrangian”, “cohomogeneity”, “almost calibrated”

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

    Ping-Hung Lee and Chung-Jun TsaiWider flows PDF

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

  6. Ancient solutions and translators of Lagrangian mean curvature flow

    Jason D. Lotay, Felix Schulze and Gábor SzékelyhidiWider flows PDF

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

  7. Type II Singularities of Lagrangian Mean Curvature Flow with Zero Maslov Class

    Xiang Li, Yong Luo and Jun SunWider flows PDF

    Topics Lagrangian mean curvature flow, Singularities · Shared terms “calibrated Lagrangian”, “almost calibrated”, “arbitrarily”

  8. 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”

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) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.