Minimal translation surfaces in $Sol_3$

TL;DR

This paper investigates minimal translation surfaces in the homogeneous space Sol_3, focusing on those with zero mean curvature, and characterizes their geometric properties based on the group operation and generating curves.

Contribution

It provides a classification and analysis of minimal translation surfaces in Sol_3, a space with unique geometric structure, which was not previously well-understood.

Findings

Characterization of zero mean curvature translation surfaces in Sol_3

Explicit descriptions of generating curves for minimal surfaces

Insights into the geometric structure of minimal surfaces in Sol_3

Abstract

In the homogeneous space Sol$_3$, a translation surface is parameterized by $x(s,t)=\alpha(s)\ast\beta(t)$, where $\alpha$ and $\beta$ are curves contained in coordinate planes and $\ast$ denotes the group operation of Sol$_3$. In this paper we study translation surfaces in Sol$_3$ whose mean curvature vanishes.