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

T0 review · 0 major / 4 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read Lagrangian mean curvature flows can form Type II singularities with exact curvature blow-up rate (T-t)^{-K/2}, even when started arbitrarily close to special Lagrangians.

desk verdict Solid, first quantitative Type II rates for LMCF via modulation + box; rests cleanly on the companion spectral theory and is worth engaging. read the letter →

arxiv 2607.03152 v1 pith:OUFOMQMY submitted 2026-07-03 math.DG math.AP

Maxwell Stolarski , Wei-Bo Su This is my paper · ORCID
classification math.DGmath.AP MSC 53E1053C3835K55
keywords LagrangianmeancurvatureflowTypeIIsingularityspecialcohomogeneity-onemodulationanalysisspectralgapblow-upratealmost-calibrated
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper constructs almost-calibrated, group-invariant Lagrangian mean curvature flows in complex n-space that develop finite-time singularities whose curvature blows up at a precise rate of order (T-t)^{-K/2}, for integers K at least 2 (with a short list of allowed K when n=3). The flows begin from data whose Lagrangian angle can be made arbitrarily close to constant, so they sit arbitrarily near the special Lagrangian condition. At the singular time the parabolic-scale tangent flow is a pair of special Lagrangian cones, while a faster Type II rescaling converges to a smooth special Lagrangian that desingularizes those cones. The construction works by modulating around a one-parameter family of shrinking special Lagrangian desingularizations and controlling the error with spectral gap estimates, thereby giving the first quantitative blow-up rates for this fully nonlinear geometric flow.

What carries the argument

A modulation analysis that expands the rescaled profile as a linear combination of the first K scale-dependent eigenfunctions of the linearized operator H_a about a shrinking family of special Lagrangian desingularizations, plus an orthogonal remainder controlled by the spectral gap; a Ważewski box argument then produces a solution that stays inside a carefully chosen shrinking set of coefficient bounds for all future time.

What would settle it

Either exhibit a zero-Maslov almost-calibrated Lagrangian mean curvature flow whose curvature blows up at a rate strictly between two consecutive powers (T-t)^{-k/2}, or prove that no such spectral gap exists for the linearized operator on the cohomogeneity-one special Lagrangians used in the construction.

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Extended reading notes

Core claim

For each admissible pair (n,K) there exists a smooth, non-compact, properly embedded, exact, almost-calibrated, G-invariant Lagrangian mean curvature flow in C^n that forms a finite-time Type II singularity at the origin with curvature satisfying 0 < liminf (T-t)^{K/2} sup|A| ≤ limsup (T-t)^{K/2} sup|A| < ∞, whose tangent flow is a transverse pair of cohomogeneity-one special Lagrangian cones and whose Type II blow-up is a smooth cohomogeneity-one special Lagrangian desingularization; the oscillation of the Lagrangian angle of the initial data can be taken arbitrarily small.

Load-bearing premise

The whole argument depends on the existence of a quantitative spectral theory for the linearized operator about every small desingularization, including a spectral gap for the remainder after the first K modes; that theory is taken from a companion paper and is not proved here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The paper constructs, for admissible pairs (n,K) (n≥4 and K≥2, or n=3 and K=2,3,4), a smooth almost-calibrated G-invariant Lagrangian mean curvature flow L_K(t) in C^n that develops a finite-time Type II singularity at the origin with the precise curvature rate sup |A| ∼ (T-t)^{-K/2}. The tangent flow is a transverse pair of cohomogeneity-one special Lagrangian cones, the Type II blow-up is a smooth cohomogeneity-one SL desingularization aL, and the initial Lagrangian-angle oscillation can be taken arbitrarily small. The argument reduces the G-invariant flow to a fully nonlinear scalar PDE for a profile function, expands the solution in the scale-dependent eigenfunctions of the companion paper [SS26], derives modulation equations and energy/Hölder estimates that keep the remainder small, and closes existence by a Ważewski box argument whose exit map would otherwise retract a K-ball onto its boundary.

Significance. If correct, this is the first construction of finite-time LMCF singularities with a quantitatively precise curvature blow-up rate, answering the rate question left open by earlier Type-II results under the zero-Maslov assumption. It also supplies the first application of modulation/spectral methods to a fully nonlinear parabolic PDE and handles a non-L^2-integrable deformation of the tangent cone by working in the scale-dependent spaces L^2_a. The explicit rate, uniqueness of the tangent flow, and arbitrarily small initial angle oscillation make the examples useful for testing conjectures on singularity formation and for future gluing or surgery constructions. The dependence on the companion spectral theory is cleanly modular and does not diminish the geometric contribution of the present work.

minor comments (4)
  1. The dependence on the companion paper [SS26] is correctly acknowledged, but a short self-contained statement of the precise spectral-gap constant and the C^1-dependence of the eigenfunctions on a (beyond the citations to Theorems 1.1 and 1.2 of [SS26]) would make the present manuscript easier to read in isolation.
  2. In Definition 5.1 the box parameters η±, γ, κ_in/out, ε_in/par/out, A_out are introduced with a long list of inequalities; a brief table or remark collecting the admissible ranges already verified in Propositions 5.3–5.4 would improve readability.
  3. Remark 9 sketches a compact immersed zero-Maslov version via figure-eight caps and pseudolocality. The sketch is plausible but informal; either expand it into a short appendix with the necessary cutoff estimates or flag it more clearly as a heuristic outline.
  4. A few typographical inconsistencies appear (e.g., “Ważewski” vs. “Wazewski”, occasional missing spaces around ∼). These are purely cosmetic.

Circularity Check

1 steps flagged · score 2.0 of 10

No significant circularity: existence construction is independent once companion spectral theory is granted as a black-box input; rate is by design of the K-mode ansatz, not a fitted prediction.

  1. self citation load bearing [Abstract; Theorem 2.8 / [SS26, Theorem 1.1]; Proposition 2.10; Sections 3–4 (modulation, energy, Hölder); Section 5 (box argument)]
    "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 [SS26]. ... Theorem 2.8 (Eigenfunctions and Spectral Gap, [SS26, Theorem 1.1]). ... the spectral gap estimate (28) ..."

    The entire nonlinear analysis (geometric decomposition via Inverse Function Theorem, modulation ODEs for a and b_i, L^2_a control of the remainder v via the gap, and the weighted C^{2,α} estimates that close the box) treats the existence of the eigenfunctions ϕ_{k,a}, the eigenvalues 1-k+ẽ_k, and the spectral gap ⟨v,H_a v⟩≤(-K+ϵ)∥v∥^{2}_{L^{2}_a} as given black-box inputs from the same authors’ companion. This is load-bearing self-citation, but the companion is an independent linear analysis and does not itself rely on the present existence result, so the circularity is mild and does not force the final claim by definition.

full rationale

The paper is a pure existence construction via Ważewski box (Section 5) around the approximate solution (3) built from scale-dependent eigenfunctions of the linearized operator. All modulation equations (Theorem 3.1), energy estimates (Section 3), weighted Hölder estimates (Section 4, including the inner-region Liouville argument of Proposition 4.15), geometric decomposition (Proposition 2.10), and the topological exit map close under the smallness hypotheses of the box B once the spectral gap (28) and quantitative eigenfunction estimates of the companion [SS26, Theorem 1.1] are granted. That companion is an independent analytic paper on the linearised operator H_a (not a re-use of the present nonlinear result), so the self-citation is load-bearing but not circular. No parameters are fitted to data and then re-predicted; the curvature rate (T-t)^{-K/2} is forced by the deliberate choice of the K-th eigenmode in the approximate solution (3) and the resulting ODE for a(τ). No uniqueness theorems are imported to forbid alternatives, no ansatz is smuggled beyond the standard cohomogeneity-one reduction, and no known empirical pattern is merely renamed. The derivation is therefore self-contained against its stated inputs, with only the ordinary (and acknowledged) dependence on prior linear analysis by the same authors.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper is a pure-existence construction. Free parameters are the smallness thresholds (a*, ε, δ, τ0, s0, Υ, etc.) chosen sufficiently small so that absorption works; they are not fitted to data. Axioms are standard parabolic theory plus the spectral package of the companion. No new physical entities are postulated.

free parameters (2)
  • smallness thresholds a*, ε_in, ε_par, ε_out, δ_in, τ0, s0, Υ, A_out, η±, γ, κ_in/out
    All chosen sufficiently small or large (depending only on n,K and previous parameters) so that the spectral gap, absorption, barriers, and box exit map close. Explicit dependence is tracked but the numerical values are existential.
  • K (mode number) and admissible range
    Integer parameter selecting the leading eigenmode; the range (K≥2 for n≥4; K=2,3,4 for n=3) is forced by the numerology that makes the exponents positive.
assumptions (4)
  • domain assumption Perturbative spectral theory for the linearised operator H_a on the weighted spaces L^2_a (eigenfunctions ϕ_{k,a}, eigenvalues 1-k+λ̃_k, spectral gap (28))
    Taken as black box from companion [SS26, Thm 1.1, Prop 1.2]; used from geometric decomposition onward.
  • standard math Standard short-time existence, uniqueness, and parabolic Schauder/energy estimates for the quasilinear/fully nonlinear profile equation (14)/(18)
    Invoked for the rescaled flow and for the local energy estimates of Lemmas 3.3-3.7.
  • domain assumption Liouville theorems for ancient/immortal solutions of the heat equation on the Lagrangian catenoid and on the transverse pair of planes (Lemmas 4.12-4.14)
    Used in the blow-up argument for the weighted C^0 estimate (Prop 4.15) that controls the inner region.
  • standard math Wazewski topological principle / homology obstruction to continuous retraction of ball onto sphere
    Core of the box argument in §5.2 that produces a global solution inside B.

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Pith. "Pith review of Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics." pith.science (2026). https://pith.science/paper/OUFOMQMY

@misc{pith2026260703152,
  author       = {Pith},
  title        = {Pith review of: Finite-Time Singularities of Lagrangian Mean Curvature Flow with Quantitatively Precise Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OUFOMQMY}},
  note         = {Machine review of arXiv:2607.03152}
}
abstract

For each integer $K\geq2$ when $n\geq4$, and for $K=2,3,4$ when $n=3$, we construct an almost-calibrated Lagrangian mean curvature flow $L_K(t)$ in $\mathbb{C}^{n}$, starting from initial data arbitrarily close to being special Lagrangian, which develops a finite-time Type II singularity at time $T$ with the explicit curvature blow up rate \[ \sup_{L_{K}(t)} |\mathbf{A}_{L_{K}(t)}| \sim (T-t)^{-K/2} \qquad \text{as } t\nearrow T . \] 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.

Figures

Figures reproduced from arXiv: 2607.03152 by the authors.

Figure 1
Figure 1. A heuristic compactification of the profile curve. The solid figure￾eight-type curve is the profile curve of a compact immersed zero-Maslov La￾grangian. The dotted circles indicate interior barriers for the lobes, chosen sufficiently large so that these lobes cannot collapse before the neck pinch at the origin occurs [PITH_FULL_IMAGE:figures/full_fig_p096_1.png] view at source ↗

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Works this paper leans on

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