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

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 $L<sup>2$ space of $G$-equivariant functions. For $0&lt;a\ll1$, 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.

• The paper introduces a new approach by constructing explicit orthonormal eigenfunctions for the linearized self-shrinker operator as the desingularization parameter tends to zero.
• It employs matched asymptotic and gluing methods to tackle multi-scale spectral behavior and overcome challenges due to non-L2 proximity of the conical operator.
• The spectral insights enable construction of Type II blow-up solutions with detailed modulation analysis, advancing the understanding of singularity formation in Lagrangian mean curvature flow.

Spectral Theory of Finite-Time Singularities in Lagrangian Mean Curvature Flow

Introduction and Problem Setting

This work rigorously addresses the spectral analysis of finite-time singularities arising in Lagrangian mean curvature flow (LMCF) in $\mathbb{C}^n$ for $n \geq 3$, especially under $G$-equivariant symmetry reductions, which lead to cohomogeneity-one special Lagrangian solutions. The motivation is to understand the analytic underpinnings of Type II singular behaviors, where the geometric flow develops singularities modeled not by smooth self-similar solutions, but by conical special Lagrangian submanifolds, and the relevant asymptotics are controlled by highly non-trivial desingularizations rather than small $L^2$ perturbations.

The technical challenge addressed is that, in these settings, the linearized self-shrinker operator (the infinitesimal generator for the rescaled LMCF) on the desingularization fails to be a regular perturbation of the conical operator; the spectral problem becomes genuinely multi-scale, and the global spectrum is not amenable to classical perturbative or functional analytic techniques. Instead, new gluing and matched asymptotic methods must be developed to give precise spectral decomposition of the linearized operator on approximate, smooth, but non-$L^2$ models.

Analytic Framework and Operator Theory

The flow under consideration is the rescaled Lagrangian mean curvature flow, structurally

$$\frac{d}{d\tau} L(\tau) = \mathbf{H}_{L(\tau)} + \frac{1}{2} \mathbf{x}_{L(\tau)}^\perp,$$

with the associated linearized operator (after restricting to $G$-invariant settings and suitable function spaces) given by

$H_a = L_{s,a} - \frac{s}{2} \partial_s + 1,$

with $L_{s,a}$ the Laplace–Beltrami operator on the desingularized special Lagrangian $a \overline{L}$, acting on weighted $n \geq 3$0 spaces with respect to the Gaussian measure adapted to the self-shrinker structure.

A crucial technical contribution is the detailed construction of the spectral decomposition of $n \geq 3$1 as $n \geq 3$2, i.e., in the singular limit as the desingularization converges to the Lagrangian cone $n \geq 3$3. The main difficulty is that the spectrum of $n \geq 3$4 in this region is not perturbatively close (in any regular sense) to that of the conical operator $n \geq 3$5 due to the lack of $n \geq 3$6 proximity and the multi-scale geometry near the singularity.

Construction and Matching of Eigenfunctions

The key analytic advance is the construction of an explicit orthonormal basis of (approximate) eigenfunctions of $n \geq 3$7, uniformly controlled in the $n \geq 3$8 limit:

• Interior eigenfunctions are constructed by perturbatively expanding around the generalized kernel of the Laplace–Beltrami operator on the desingularization, solving an ODE for the profile in a small region near the singularity. This exploits the inductive structure of the kernel induced by scaling and translation modes.
• Exterior eigenfunctions are constructed on the conical region by using explicit generalized Laguerre polynomials, with precise matching to the weighted $n \geq 3$9 structure imposed by the flow. These solutions provide sharp asymptotic expansions at infinity.
• A gluing approach is employed to match interior and exterior eigenfunctions at an intermediate scale (dependent on $G$0), ensuring $G$1-smoothness and correct matching of derivatives, exploiting detailed pointwise and weighted estimates.

The main result is the existence, for any prescribed finite $G$2, of a basis $G$3 of eigenfunctions of $G$4 with corresponding eigenvalues $G$5, where the perturbation $G$6 is explicit and controlled: $G$7 with spectral gap: $G$8 for $G$9 orthogonal to the span of the first $L^2$0 eigenfunctions, uniformly as $L^2$1.

Geometric Implications and Scaling Modes

An important geometric identification within the spectral theory is the precise correspondence between the lowest eigenmode $L^2$2 and the scaling vector field on the desingularization. The analysis proves that $L^2$3 is $L^2$4-close to the normalized scaling deformation mode $L^2$5, where $L^2$6 is the potential associated to rescaling $L^2$7.

This identification is crucial for the construction of modulated solutions in the dynamical problem. It enables a precise decoupling of the dangerous scaling instability from the higher modes, which is essential for applying nonlinear stability analyses and constructing actual LMCF solutions with prescribed singularity formation rates and profiles.

Implications and Future Developments

The analytic foundation developed here supplies the spectral ingredients needed for constructing Type II blow-up solutions with detailed asymptotic expansions for the LMCF near conical singularities in higher dimensions. The results provide explicit control over the spectrum and deformations about asymptotically conical special Lagrangian singularities.

Practically, these techniques enable:

• The construction of finite-time singular LMCF solutions whose blow-up rates and profiles are governed by the spectral data of the desingularization, extending beyond the regime where flows are perturbatively close to self-similar shrinkers.
• Highly quantitative understanding of which perturbations can drive (or prevent) singularity formation at Type II scales, via explicit spectral gaps and mode projections.

Theoretically, this work is a step toward the systematic classification of singularity models in geometric flows beyond minimal or self-similar cases, and the development of a robust multi-scale perturbation theory in PDEs and geometric analysis. The gluing and spectral matching methods may find further applications in other settings involving degeneration or desingularization phenomena—such as in Ricci flow, semi-linear heat equations, and mean curvature flow for submanifolds with more general symmetry or topology.

The technical foundations laid here will be leveraged in a companion paper for the full construction of singular solutions with desired properties, and may pave the way for future exploration of generic behaviors and stability near singularities in Lagrangian and related geometric flows.

Conclusion

This paper establishes a rigorous spectral theory for the linearized self-shrinker operator on special Lagrangian desingularizations near singular cones under Lagrangian mean curvature flow, in the previously inaccessible non-$L^2$8 regime. By developing new gluing techniques and delicate spectral estimates, the work identifies the spectrum, scaling modes, and spectral gap, thereby providing the analytic infrastructure required to construct and analyze Type II singularities with explicit blow-up profiles and rates in geometric flows. This foundation also suggests pathways for further explorations in singularity formation, modulation methods, and stability theory for geometric evolution equations.