Intrinsic Characterizations of Biconservative Surfaces in the Four-Dimensional Hyperbolic Space

In this paper, we extend the investigation of biconservative surfaces with parallel normalized mean curvature vector fields (PNMC) in the four-dimensional space forms, focusing on the hyperbolic space \({\mathbb {H}}^4\) ${\mathbb{H}}^4$ , the last remaining case to explore. We establish that an abstract surface admits a PNMC biconservative immersion in \({\mathbb {H}}^4\) ${\mathbb{H}}^4$ 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. In addition, we provide three characterizations of the intrinsic condition to explore the geometric properties of these surfaces.