Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space

Given a constant $k>1$, let $Z$ be the family of round spheres of radius $\text{artanh}(k^{-1})$ in the hyperbolic space ${\mathbb{H}}^3$, so that any sphere in $Z$ has mean curvature $k$. We prove a crucial nondegeneracy result involving the manifold $Z$. As an application, we provide sufficient conditions on a prescribed function $\varphi$ on ${\mathbb{H}}^3$, which ensure the existence of a ${\mathcal{C}}^1$-curve, parametrized by $\epsilon \approx 0$, of embedded spheres in ${\mathbb{H}}^3$ having mean curvature $k+\epsilon \varphi$ at each point.