  In this paper, we obtain several classification results of $2$-dimensional
complete Lagrangian translators and lagrangian self-expanders with constant
squared norm $|\vec{H}|^{2}$ 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 $\mathbb{C}^{2}$.
