Minimal translation surfaces in Sol\(_{3}\) (Q455015)

The three-dimensional Riemannian manifold \({\mathrm{Sol}}_3\), one of the Thurston geometries, can be viewed as \({\mathbb{R}}^3\) with the metric NEWLINE\[NEWLINE ds^2 = {\mathrm e}^{2z} dx^2 + {\mathrm e}^{-2z} dy^2 + dz^2. NEWLINE\]NEWLINE The isometry group of \({\mathrm{Sol}}_3\) has dimension three. The metric is left-invariant with respect to the operation NEWLINE\[NEWLINE (x,y,z)\circ(x',y',z') = (x+{\mathrm{e}}^{-z}x', y+{\mathrm{e}}^{z}y', z+z').NEWLINE\]NEWLINE The Lie group \(({\mathrm{Sol}}_3,\circ)\) is unimodular and solvable.NEWLINENEWLINEIn recent papers, surfaces with constant mean curvature in Thurston geometries were studied. The existence of surfaces with constant mean curvature in \({\mathrm{Sol}}_3\) and which are topologically a sphere was proved.NEWLINENEWLINEIn the present work, translation surfaces in \({\mathrm{Sol}}_3\) whose mean curvature vanishes (minimal surfaces) are studied. A translation surface \(M(\alpha,\beta)\) in \(({\mathrm{Sol}}_3,\circ)\) is a surface parametrized by \(x(s,t)=\alpha(s)\circ\beta(t)\), where \(\alpha: I\rightarrow {\mathrm{Sol}}_3\), \(\beta: J\rightarrow {\mathrm{Sol}}_3\) are curves in two coordinate planes of \({\mathbb{R}}^3\). Minimal translational surfaces of \({\mathrm{Sol}}_3\) are classified.