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 $\hat{I}¾$-submanifolds in ${\mathbb{R}}^4$.