Classification of Lagrangian translators and Lagrangian self-expanders in ${\mathbb{C}}^2$

In this paper, we obtain several classification results of $2$-dimensional complete Lagrangian translators and lagrangian self-expanders with constant squared norm $\mathrm{\mid }\vec{H}{\mathrm{\mid }}^2$ of the mean curvature vector in ${\mathbb{C}}^2$ by using a new Omori-Yau type maximum principle which was proved by Chen and Qiu \cite{CQ}. The same idea is also used to give a similar result of Lagrangian $\xi$-translators in ${\mathbb{C}}^2$.