A rigidity theorem for self-expanders

Abstract

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