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 lying in a totally geodesic hyperbolic surface ${\mathbb{H}}^2$ in ${\mathbb{H}}^3$.

我们构建的是简单连接，完整，非$C\mathrm{中号}C$ 3维双曲空间中的双保守表面 ${\mathbb{H}}^3$，以内在和外在的方式。我们获得了此类曲面的三个族，并且对于每个曲面，平均曲率函数的梯度不消失的点集很密集，并且具有两个相连的分量。在内在方法中，我们首先构造一个简单连接的完整抽象表面，然后证明它承认在其中具有独特的双保守沉浸性。${\mathbb{H}}^3$。外在地工作，我们使用显式参数方程的图像和胶合过程来获得我们的表面。它们由分别位于2个仿射平行平面中的圆（分别为双曲线或抛物线）组成，并与位于完全测地双曲面上的某条曲线接触${\mathbb{H}}^2$ 在 ${\mathbb{H}}^3$。