Minimal Translation Surfaces in Euclidean Space

A translation surface in Euclidean space is a surface that is the sum of two regular curves (alpha ) and (beta ). In this paper we characterize all minimal translation surfaces. In the case that (alpha ) and (beta ) are non-planar curves, we prove that the curvature (kappa ) and the torsion (tau ) of both curves must satisfy the equation (kappa ^2 tau = C) where C is constant. We show that, up to a rigid motion and a dilation in the Euclidean space and, up to reparametrizations of the curves generating the surfaces, all minimal translation surfaces are described by two real parameters (a,bin mathbb {R}) where the surface is of the form (phi (s,t)=beta _{a,b}(s)+beta _{a,b}(t)).