Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane

TL;DR

This paper constructs new examples of translating solitons for mean curvature flow in complex Euclidean plane using solutions of the curve shortening flow, expanding known classifications and identifying unique Hamiltonian stationary cases.

Contribution

It introduces a broad family of translating solitons in complex Euclidean plane, generalizing previous examples and characterizing the unique Hamiltonian stationary soliton.

Findings

New families of translating solitons constructed

Characterization of the unique Hamiltonian stationary soliton

Extension of previous classifications in dimension two

Abstract

Using certain solutions of the curve shortening flow, including self-shrinking and self-expanding curves or spirals, we construct and characterize many new examples of translating solitons for mean curvature flow in complex Euclidean plane. They generalize the Joyce, Lee and Tsui ones \cite{JLT} in dimension two. The simplest (non trivial) example in our family is characterized as the only (non totally geodesic) Hamiltonian stationary Lagrangian translating soliton for mean curvature flow in complex Euclidean plane.