Examples of constant mean curvature surfaces obtained from harmonic maps to the two sphere (Q1374875)

In this paper families of surfaces with constant mean curvature in quotients of the Euclidean space are obtained from their Gauss maps. We start by considering a compact Riemann surface \(M\) of positive genus \(g\leq 3\) with three commuting anticonformal involutions \(u_1, u_2, u_3\). Following \textit{L. Lemaire} [J. Differ. Geom. 13, 51-78 (1978; Zbl 0388.58003)], we construct a smooth harmonic map \(\varphi: M\to \mathbb{S}^2\) minimizing the energy functional in the class of equivariant maps with respect to \(\{u_1, u_2, u_3\}\) and with respect to the symmetries around the three coordinate planes in \(\mathbb{R}^3\). From this harmonic map we obtain an unbranched constant mean curvature immersion \(\psi:M \to \mathbb{R}^3/ \Gamma\) whose Gauss map is \(\varphi\). Here \(\Gamma\) is a discrete subgroup of translations of rank \(g\) generated by orthogonal vectors. We also prove that the immersions are embeddings and exhibit a geometrical characterization of the examples.