The rigidity theorem for complete Lagrangian self-shrinkers

Abstract

In this paper, we obtain a rigidity result of $2$-dimensional complete lagrangian self-shrinkers with constant squared norm $|\vec{H}|^{2}$ 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 $\mathbb{R}^{4}$.