Minimal translation surfaces in hyperbolic space

TL;DR

This paper proves that in hyperbolic space, the only minimal translation surfaces are totally geodesic planes, providing a classification of such surfaces in the half-space model.

Contribution

It establishes a complete classification of minimal translation surfaces in hyperbolic space, showing they are exclusively totally geodesic planes.

Findings

Only totally geodesic planes are minimal translation surfaces.

Classification of translation surfaces in hyperbolic space.

No other minimal translation surfaces exist besides planes.

Abstract

In the half-space model of hyperbolic space, that is, $\r^3_{+}=\{(x,y,z)\in\r^3;z>0\}$ 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.