Ruled Surfaces of Generalized Self-Similar Solutions of the Mean Curvature Flow

In Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation $$H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda $$ H = α ⟨ N , x ⟩ + λ , where N is the Gauss map, $${\mathbf {x}}$$ x is the position vector, and $$\alpha $$ α and $$\lambda $$ λ are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.