AbstractIn 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.
Existence of self-similar solutions of the inverse mean curvature flow