Paper · Jun 19, 2026

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

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Abstract

Let $C$ be a $G$ -invariant special Lagrangian cone admitting a scaled family of $G$ -invariant special Lagrangian desingularizations $a L$ which converge to $C$ as $a ↘ 0$ . We study the linearized self-shrinker operator on $a L$ in a Gaussian weighted $L^{2}$ space of $G$ -equivariant functions. For $0 < a ≪ 1$ , 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.

Comments: 64 pages, comments welcome

Mean curvature flow Lagrangian mean curvature flow Self-shrinkers & translators Singularities Eigenvalues & spectrum

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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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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)

In the guidebook

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Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives
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Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods
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Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

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

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