Translation surfaces of Weingaryen type in 3-space (Q2866247)

The authors consider surfaces \(S\) in the Euclidean space \({\mathbf E}^3\) or non-degenerate surfaces in the Minkowski space \({\mathbf E}^3_1\) (with signature convention \((+,+,-)\)). Denote by \(K\) and \(H\) the Gaussian and the mean curvature functions of \(S\). If the Jacobian determinant of the pair (\(K\),\(H\)) vanishes, then \(S\) is called Weingarten surface.NEWLINENEWLINE The authors define translation surfaces: in \({\mathbf E}^3\), as parameterized by \(x(s,t)=(s,t,f(s)+g(t))\); in \({\mathbf E}^3_1\), as parameterized by \(x(s,t)=(s,t,f(s)+g(t))\) or \(x(s,t)=(f(s)+g(t),s,t)\).NEWLINENEWLINE The main results of the paper characterize the translation surfaces which are also Weingarten. In \({\mathbf E}^3\) (respectively in \({\mathbf E}^3_1\)), these are: a plane, a generalized cylinder, a minimal translation surface of Scherk or an orthogonal elliptic (respectively elliptic or hyperbolic) paraboloid.