Rigidity theorems of complete self-shrinkers in the Euclidean space

We classify complete Lagrangian self-shrinking surfaces in ${\mathbb{R}}^4$ with constant squared norm of the mean curvature vector, without assuming an upper bound for the second fundamental form. Their images are a plane through the origin, the round cylinder $S^1(1)\times \mathbb{R}$, or the Clifford torus $S^1(1)\times S^1(1)$. More generally, a complete self-shrinking surface in ${\mathbb{R}}^4$ with constant $\mathrm{\mid }\vec{H}{\mathrm{\mid }}^2$ is either a plane, a round cylinder, or a complete minimal immersion into $S^3(\sqrt{2})$. The proof uses the function $\mathrm{\mid }x{\mathrm{\mid }}^2-\mathrm{\mid }\vec{H}{\mathrm{\mid }}^2$, the Bochner formula, and completeness.