Helicoidal Surfaces of Prescribed Mean Curvature in \({{\mathbb R}}^3\)

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