Complete $$\lambda $$-surfaces in $${\mathbb {R}}^3$$

The purpose of this paper is to study complete $$\lambda $$ -surfaces in Euclidean space $${\mathbb {R}}^3$$ . A classification for 2-dimensional $$\mathbb R^3$$ with constant squared norm the second fundamental form given, which confirms a conjecture Guang (Self-shrinkers and translating solitons mean curvature flow, 2016, p 74).