Intrinsic characterizations of biconservative surfaces in the 4-dimensional hyperbolic space

In this paper, we extend the investigation of biconservative surfaces with parallel normalized mean curvature vector fields (PNMC) in the 4-dimensional space forms, focusing on the hyperbolic space $\mathbb{H}$^4, the last remaining case to explore. We establish that an abstract surface admits a PNMC biconservative immersion in $\mathbb{H}$^4 if and only if it satisfies a certain intrinsic condition; if such an immersion exists, it is unique. We further analyze these abstract surfaces, showing that they form a two-parameter family. Additionally, we provide three characterizations of the intrinsic condition to explore the geometric properties of these surfaces.