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 $L^2$ space of $G$-equivariant functions. For $0<a\ll 1$, 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.

64 pages, comments welcome