Translation surfaces of finite Chen type in hyperbolic space $\mathbb H^3$

In the hyperbolic space $\mathbb {H}^3$, defined as $\mathbb {R}_{+}^3 = \{(x,y,z)\in \mathbb {R}^3 : z \gt 0 \}$ equipped with the hyperbolic metric, a translation surface is expressed as $z = f(x)+g(y)$ or $y=f(x)+g(z)$, where $f$ and $g$ are smooth functions. This paper explores the classification of translation surfaces in $\mathbb {H}^3$ under the condition $\varDelta r_i = \lambda_i r_i$, $\lambda _i \in \mathbb {R}$, $i=1,2,3$.