Special Cases of Weingarten Translation Surfaces in Euclidean and Minkowski 3-space

Some results on Weingarten translation surfaces and more specific translation surfaces with vanishing second Gaussian curvature, in Euclidean and Minkowski 3-space, are presented. For surfaces with non-degenerate second fundamental form the second Gaussian and second mean curvature are, respectively, defined as the Gaussian and mean curvature of the second fundamental form. Taking these curvatures into account a more elaborate definition of Weingarten surfaces can be stated. As a consequence a broader class of Weingarten surfaces can be examined. Special examples of Weingarten translation surfaces are investigated, namely those surfaces for which the second Gaussian curvature is zero. The minimal translation surfaces of Scherk are among these examples. Other non-trivial examples of surfaces that satisfy the curvature condition examined, can be expressed in terms of the less known LambertW function. Most attention will be paid to this last kind of special cases.status: publishe