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 [Self-similar solutions and translating solitons for Lagrangian mean curvature flow, J. Differential Geom.84 (2010) 127–161] 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.
Self-similar curve shortening flow in hyperbolic 2-space
