In this paper we make an analysis of self-similar solutions for the mean curvature flow (MCF) by surfaces of revolution and ruled surfaces in $\mathbb{ R}^{ 3}$. We prove that self-similar solutions of the MCF by non-cylindrival surfaces and conical surfaces in $\mathbb{ R}^{ 3}$ are trivial. Moreover, we characterize the self-similar solutions of the MCF by surfaces of revolutions under a homothetic helicoidal motion in $\mathbb{ R}^{ 3}$ in terms of the curvature of the generating curve. Finally, we characterize the self-similar solutions for the MCF by cylindrical surfaces under a homothetic helicoidal motion in $\mathbb{ R}^3$. Explicit families of exact solutions for the MCF by cylindrical surfaces in $\mathbb{ R}^{ 3}$ are also given.
Categories: math.DG