Minimal Translation Surfaces in Euclidean Space

Abstract

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 $$a,b\in \mathbb {R}$$ where the surface is of the form $$\phi (s,t)=\beta _{a,b}(s)+\beta _{a,b}(t)$$ .