On the uniqueness of Minimal Translation Surfaces in the Heisenberg Group

Abstract

In this paper, we establish a complete classification of minimal translation surfaces within the 3-dimensional Heisenberg group ${\mathcal{H}}_3$. While translation surfaces in ${\mathcal{H}}_3$ yield diverse families of solutions depending on the choice of generating curves, we prove that, up to isometries of ${\mathcal{H}}_3$, there exists a unique class of translation surfaces.