On the Gluing Construction of Translating Solitons to Lagrangian Mean Curvature Flow

TL;DR

This paper presents a method to construct complex Lagrangian translating solitons in mean curvature flow by desingularizing intersections with special necks, allowing for solitons with multiple ends and non-trivial topology.

Contribution

It introduces a novel gluing construction for Lagrangian translating solitons using Lawlor necks to resolve intersections, expanding the known classes of such solitons.

Findings

Constructed Lagrangian translating solitons with arbitrarily many ends.

Demonstrated the possibility of non-contractible loops in the solitons.

Provided a new technique for desingularizing intersections in Lagrangian mean curvature flow.

Abstract

We construct Lagrangian translating solitons by desingularizing the intersection points between Lagrangian Grim Reaper cylinders with the same phase using special Lagrangian Lawlor necks. The resulting Lagrangian translating solitons could have arbitrarily many ends and non-contractible loops.