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 {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 $phi$ on $mathbb{H}^3$, which ensure the existence of a ${cal C}^1$-curve, parametrized by $arepsilonapprox 0$, of embedded spheres in $mathbb{H}^3$ having mean curvature $k +arepsilonphi$ at each point.