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arxiv: 2606.21541 · v1 · pith:ZB7BTRRHnew · submitted 2026-06-19 · 🧮 math.DG · math.AP· math.SP

Spectral Analysis for Finite-Time Singularities of Lagrangian Mean Curvature Flow

Maxwell Stolarski , Wei-Bo Su This is my paper

Pith reviewed 2026-06-26 13:17 UTC · model grok-4.3

classification 🧮 math.DG math.APmath.SP
keywords special Lagrangian conedesingularizationLagrangian mean curvature flowlinearized self-shrinker operatoreigenvalue convergencespectral gapG-equivariant functionsself-shrinkers
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The pith

For small desingularizations of special Lagrangian cones, the linearized operator has eigenfunctions converging to the cone's spectrum with a gap on the complement.

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

The paper proves that given a G-invariant special Lagrangian cone with a converging family of desingularizations, the linearized self-shrinker operator on the small desingularized surface admits eigenfunctions whose eigenvalues approach those of the limiting cone. It constructs any finite number of such modes and shows a spectral gap on the orthogonal complement in the space of G-equivariant functions. The lowest mode is shown to correspond to the scaling variation of the desingularization. This provides the spectral control necessary to study the stability of these surfaces under Lagrangian mean curvature flow near the singular cone.

Core claim

Assuming the existence of G-invariant special Lagrangian desingularizations a L-bar converging to the cone C as a approaches 0, the analysis of the linearized self-shrinker operator in the Gaussian weighted L2 space shows that for small a, any prescribed finite set of eigenfunctions can be constructed with eigenvalues converging to the conical operator's eigenvalues, a spectral gap estimate holds on the orthogonal complement, and the lowest eigenfunction is identified with the scaling mode of the desingularization.

What carries the argument

The linearized self-shrinker operator on the desingularized special Lagrangian in the Gaussian-weighted L2 space restricted to G-equivariant functions.

If this is right

  • Any finite number of prescribed modes from the conical operator can be realized approximately on the desingularization for small a.
  • A spectral gap separates the approximated modes from the rest of the spectrum.
  • The lowest eigenfunction corresponds exactly to the scaling mode induced by the desingularization parameter.
  • This spectral structure forms the basis for constructing Type II blow-up solutions of the flow.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result allows perturbation methods to produce solutions that blow up in finite time with Type II singularities.
  • Similar spectral analysis could be applied to other classes of cones or flows where desingularizations exist.
  • Explicit examples of such desingularizations would permit numerical checks of the eigenvalue convergence rates.

Load-bearing premise

The assumption that there exists a scaled family of G-invariant special Lagrangian desingularizations converging to the cone as the scale goes to zero.

What would settle it

Finding a specific cone and desingularization family where the eigenvalues of the linearized operator on a L-bar do not converge to the conical eigenvalues as a goes to zero, or where no gap appears.

read the original abstract

Let $\mathcal C$ be a $G$-invariant special Lagrangian cone admitting a scaled family of $G$-invariant special Lagrangian desingularizations $a \overline L$ which converge to $\mathcal C$ as $a\searrow 0$. We study the linearized self-shrinker operator on $a\overline L$ in a Gaussian weighted $L^2$ space of $G$-equivariant functions. For $0<a\ll1$, we construct any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, and we prove a spectral gap estimate on the orthogonal complement of these modes. We also identify the lowest eigenfunction with the scaling mode of the special Lagrangian desingularization. This spectral basis provides the analytic foundation for the construction of Type II blow-up solutions of Lagrangian mean curvature flow in the companion paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript assumes the existence of a scaled family of G-invariant special Lagrangian desingularizations a ¯L converging to a G-invariant special Lagrangian cone C as a ↓ 0. It studies the linearized self-shrinker operator on a ¯L in the Gaussian-weighted L^{2} space of G-equivariant functions. For 0 < a ≪ 1, it constructs any prescribed finite number of eigenfunctions whose eigenvalues converge to those of the limiting conical operator, proves a spectral gap estimate on the orthogonal complement of these modes, and identifies the lowest eigenfunction with the scaling mode of the desingularization. These results supply the analytic foundation for Type II blow-up constructions of Lagrangian mean curvature flow in a companion paper.

Significance. If the results hold, the work supplies key spectral tools for analyzing stability and finite-time singularities in Lagrangian mean curvature flow near conical singularities, directly enabling Type II blow-up constructions via desingularization. The finite-mode approximation, spectral gap on the complement, and explicit identification of the scaling mode are load-bearing for controlling the linearized dynamics. The paper's conditional framing on the desingularization family is clearly stated, permitting the spectral analysis to rest on standard theory for the linearized operator in the weighted space.

minor comments (2)
  1. [§1] §1 (Introduction): the precise definition of the Gaussian-weighted inner product and the precise domain of G-equivariant functions should be stated explicitly before the main theorems, rather than deferred to later sections.
  2. The statement that eigenvalues 'converge to those of the limiting conical operator' would benefit from a short reminder of the spectrum of the conical operator (e.g., reference to its explicit eigenvalues or prior computation) to make the approximation claim self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the manuscript and the recommendation for minor revision. The report provides no specific major comments or points requiring detailed response.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation is self-contained and non-circular. It explicitly states the existence of the scaled family of G-invariant special Lagrangian desingularizations aL-bar converging to the cone C as a setup assumption in the abstract and proceeds by applying standard spectral theory to the linearized self-shrinker operator on these surfaces in the Gaussian-weighted L2 space of G-equivariant functions. The constructions of approximating eigenfunctions, the spectral gap on the orthogonal complement, and identification of the lowest eigenfunction with the scaling mode are derived directly from this operator without any reduction to fitted inputs, self-definitions, or load-bearing self-citations. The mention of the companion paper is a forward reference for application, not a justification of the present results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Review is based solely on the abstract; no explicit free parameters, axioms, or invented entities are identifiable from the given text beyond the standard background of special Lagrangian geometry and mean curvature flow.

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discussion (0)

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Reference graph

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