Translation surfaces of linear Weingarten type

TL;DR

This paper proves that translation surfaces satisfying a linear relation between mean and Gauss curvature must have either constant mean curvature or constant Gaussian curvature, with an extension to Lorentzian spaces.

Contribution

It provides a simple proof that such surfaces must have either constant mean or Gaussian curvature, extending previous results and methods to Lorentzian ambient spaces.

Findings

Translation surfaces satisfying $aH+bK=c$ have either $H$ or $K$ constant.

The proof method is simplified compared to previous approaches.

Results extend to Lorentzian ambient spaces.

Abstract

We give a relatively simple proof that a translation surface in Euclidean space that satisfies a relation of type $aH+bK=c$, for some real numbers $a,b,c$, where $H$ and $K$ are the mean curvature and the Gauss curvature of the surface, respectively, must have $a=0$ or $b=0$, and thus, $K$ is constant or $H$ is constant. Our method of proof extends to the Lorentzian ambient space.