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 Lagrangian `Lawlor 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 $N^{\varepsilon(t)}$. This is inspired by the work of Brendle and Kapouleas regarding ancient solutions of the Ricci flow.