Extrinsic characterizations of biconservative surfaces in the $4$-dimensional hyperbolic space

TL;DR

This paper classifies non-constant mean curvature biconservative surfaces with parallel normalized mean curvature vector in four-dimensional hyperbolic space, describing their local extrinsic structure via a generating curve in a hypersurface.

Contribution

It provides a local extrinsic description and classification of non-CMC, PNMC biconservative surfaces in hyperbolic space, completing the understanding in four-dimensional space forms.

Findings

Surfaces are generated by a curve in a totally geodesic hypersurface.

Classification depends on the type of a certain vector field (null, spacelike, timelike).

The classification of such surfaces in four-dimensional space forms is now complete.

Abstract

Biconservative submanifolds arise as a natural relaxation of the biharmonic condition and play an important role in the submanifold theory. In this paper, we study non-CMC biconservative surfaces with parallel normalized mean curvature vector field (PNMC surfaces) in the four-dimensional hyperbolic space $\mathbb{H}^4$, for which we consider the hyperboloid model. We provide a local extrinsic description of such surfaces, showing that they are generated by a directrix curve lying in a totally geodesic hypersurface $\mathbb{H}^3$ of $\mathbb{H}^4$, through a certain normal flow. This extrinsic classification of non-CMC, PNMC biconservative surfaces in $\mathbb{H}^4$ splits naturally into three cases according to the type of a certain vector field, which can be non-zero null, spacelike or timelike. Together with the previous results, the classification of non-CMC, PNMC surfaces in four-dimensional space forms is now completed, from intrinsic and extrinsic point of view.