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

For each integer $K\ge 2$ when $n\ge 4$, 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 $$\underset{L_K(t)}{\sup }\mathrm{\mid }{\mathbf{A}}_{L_K(t)}\mathrm{\mid }\sim (T-t{)}^{-K\mathrm{/}2}\text{as }t\nearrow T\mathrm{.}$$ 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.