Complete biconservative surfaces in the hyperbolic space [formula omitted

Abstract

We construct simply connected, complete, non-CMC biconservative surfaces in the 3-dimensional hyperbolic space H3, 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 H3. 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 lying in a totally geodesic hyperbolic surface H2 in H3.