A rigidity theorem of self-expander

In this paper, we completely classify $3$-dimensional complete self-expanders with constant norm $S$ of the second fundamental form and constant $f_3$ in Euclidean space ${\mathbb{R}}^4$, where $h_{ij}$ are components of the second fundamental form, $S={\sum }_{i,j}{h}_{ij}^2$ and $f_3={\sum }_{i,j,k}{h}_{ij}{h}_{jk}{h}_{ki}$.