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.