Lagrangian translators under mean curvature flow

Abstract

We provide a new construction of Lagrangian surfaces in C2 in terms of two planar curves. When we take such curves as appropriate solutions of the curve shortening problem, including self-shrinking and self-expanding curves or spirals, we will obtain translating solitons which generalize the Joyce, Lee and Tsui ones in dimension two D. Joyce, Y.-I. Lee, and M.-P. Tsui. Self-similar solutions and translating solitons for Lagrangian mean curvature flow. J. Differential Geom. 84 (2010), no. 1, 127-161. Finally, we characterize locally all examples in terms of an analytical condition on the Hermitian product of the position vector of the immersion and the translating vector that allows us separation of variables. As a consequence we get the classification of the Hamiltonian stationary Lagrangian translating solitons for Lagrangian mean curvature flow in complex Euclidean plane. This work is based in I. Castro and A. M. Lerma. Translating solitons for Lagrangian mean curvature flow in complex Euclidean plane. Internat. J. Math. 23 (2012), no. 10, 1250101-1-1250101-16.