Rigidity of $4$-dimensional complete self-shrinkers in $\mathbb{R}^{5}

We show that any $4$-dimensional complete self-shrinker in $\mathbb{R}^{5}$ with constant squared norm $S$ of the second fundamental form, $f_{3}=0$ and constant $f_{4}$ is isometric to $\mathbb{R}^{4}$, where $h_{ij}$ are components of the second fundamental form, $S=\sum h_{ij}^{2}$, $f_{3}=\sum h_{ij}h_{jk}h_{ki}$ and $f_{4}=\sum h_{ij}h_{jk}h_{kl}h_{li}$. As an application, we obtain a classification result.