A rigidity theorem for self-expanders

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