Complete biconservative surfaces in the hyperbolic space $\mathbb{H}^3$

TL;DR

This paper constructs and classifies complete, simply connected, non-constant mean curvature biconservative surfaces in hyperbolic 3-space, revealing three families with dense gradient sets and explicit geometric structures.

Contribution

It introduces three explicit families of non-constant mean curvature biconservative surfaces in hyperbolic space, using intrinsic and extrinsic methods, and describes their geometric properties.

Findings

Three families of such surfaces are constructed.

The set where the mean curvature gradient does not vanish is dense and has two components.

Surfaces are composed of conic sections lying in parallel planes touching a curve in a hyperbolic surface.

Abstract

We construct simply connected, complete, non-$CMC$ biconservative surfaces in the $3$-dimensional hyperbolic space $\mathbb{H}^3$ in an intrinsic and extrinsic way. We obtain three families of such surfaces, and, for each surface, the set of points where the gradient of the mean curvature function does not vanish is dense and has two connected components. In the intrinsic approach, we first construct a simply connected, complete abstract surface and then prove that it admits a unique biconservative immersion in $\mathbb{H}^3$. Working extrinsically, we use the images of the explicit parametric equations and a gluing process to obtain our surfaces. They are made up of circles (or hyperbolas, or parabolas, respectively) which lie in $2$-affine parallel planes and touch a certain curve in a totally geodesic hyperbolic surface $\mathbb{H}^2$ in $\mathbb{H}^3$.