Classification and Construction of Minimal Translation Surfaces in Euclidean Space

Abstract

A translation surface of Euclidean space ${\mathbb {R}}^3$ is the sum of two regular curves $\alpha$ and $\beta$, called the generating curves. In this paper we classify the minimal translation surfaces of ${\mathbb {R}}^3$ 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 $\Psi (s,t)=\alpha (s)+\alpha (t)$, where $\alpha$ is a curve parameterized by arc length s, its curvature $\kappa$ is a positive solution of the autonomous ODE $(y')^2+y^4+c_3y^2+c_1^2y^{-2}+c_1c_2=0$ and its torsion is $\tau (s)=c_1/\kappa (s)^2$. Here $c_1\not =0$, $c_2$ and $c_3$ are constants such that the cubic equation $-\lambda ^3+c_2\lambda ^2-c_3\lambda +c_1=0$ has three real roots $\lambda _1$, $\lambda _2$ and $\lambda _3$.