Classificação e construção de superfícies mínimas de translação em formas espaciais - Silva, Marcos Gomes da (2022)
A translation surface of Euclidean space is the sum of two regular curves and , called the generating curves. In this paper we classify the minimal translation surfaces of and we give a method of construction of explicit examples. Besides the plane and the minimal surfaces of Scherk type, it is proved that up to reparameterizations of the generating curves, any minimal translation surface is described as , where is a curve parameterized by arc length s, its curvature is a positive solution of the autonomous ODE and its torsion is Here and are constants such that the cubic equation has three real roots , and . Furthermore in the half-space model of hyperbolic space, that is, with the hyperbolic metric, a translation surface that writes as or , where f and g are smooth functions, we prove that the only minimal translation surfaces are totally geodesic planes.