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

TL;DR

This paper characterizes biconservative surfaces with parallel normalized mean curvature in 4D hyperbolic space, establishing intrinsic conditions for their existence, uniqueness, and describing their geometric properties.

Contribution

It provides the first intrinsic characterization of PNMC biconservative surfaces in -dimensional hyperbolic space, including existence, uniqueness, and a parametric family description.

Findings

Existence of a two-parameter family of such surfaces.

Intrinsic condition characterizes all PNMC biconservative surfaces.

Uniqueness of the immersion when the intrinsic condition is satisfied.

Abstract

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.