Helicoidal surfaces of prescribed mean curvature in ${\mathbb{R}}^3$

Given a function $\mathcal{H}\in C^1({\mathbb{S}}^2)$, an $\mathcal{H}$-surface $\mathrm{\Sigma }$ is a surface in the Euclidean space ${\mathbb{R}}^3$ whose mean curvature $H_{\mathrm{\Sigma }}$ satisfies $H_{\mathrm{\Sigma }}=\mathcal{H}\circ \eta$, where $\eta$ is the Gauss map of $\mathrm{\Sigma }$. The purpose of this paper is to use a phase space analysis to give some classification results for helicoidal $\mathcal{H}$-surfaces, when $\mathcal{H}$ is rotationally symmetric, that is, $\mathcal{H}\circ \eta =\mathfrak{h}\circ \nu$, for some $\mathfrak{h}\in C^1([-1,1])$, where $\nu$ is the angle function of the surface. We prove a classification theorem for the case where $\mathfrak{h}(t)$ is even and increasing for $t\in [0,1]$. Finally, we provide examples of helicoidal $\mathcal{H}$-surfaces in cases where $\mathfrak{h}$ vanishes at some point.