Infinite-Time Singularities of the Lagrangian Mean Curvature Flow

Abstract: In this paper, we construct solutions of Lagrangian mean curvature flow which exist and are embedded for all time, but form an infinite-time singularity and converge to an immersed special Lagrangian as $t\to\infty$. In particular, the flow decomposes the initial data into a union of special Lagrangians intersecting at one point. This result shows that infinite-time singularities can form in the Thomas--Yau semi-stable' situation. A precise polynomial blow-up rate of the second fundamental form is also shown. The infinite-time singularity formation is obtained by a perturbation of an approximate family $N^{\varepsilon(t)}$ constructed by gluing in special LagrangianLawlor necks' of size $\varepsilon(t)$, where the dynamics of the neck size $\varepsilon(t)$ are driven by the obstruction for the existence of nearby special Lagrangians to $N<sup>{\varepsilon(t)}$. This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.