Minimal translation surfaces in hyperbolic space

Abstract

In the half-space model of hyperbolic space, that is, $${\mathbb{R}^3_{+}=\{(x,y,z)\in\mathbb{R}^3;z > 0\}}$$ with the hyperbolic metric, a translation surface is a surface that writes as z = f(x) + g(y) or y = f(x) + g(z), where f and g are smooth functions. We prove that the only minimal translation surfaces (zero mean curvature in all points) are totally geodesic planes.