The rigidity theorem for complete Lagrangian self-shrinkers

In this paper, we obtain a rigidity result of $2$-dimensional complete lagrangian self-shrinkers with constant squared norm $\mathrm{\mid }\vec{H}{\mathrm{\mid }}^2$ of the mean curvature vector in the Euclidean space ${\mathbb{R}}^4$. The same idea is also used to give a similar result of Lagrangian $\xi$-submanifolds in ${\mathbb{R}}^4$.