Nondegeneracy of CMC bubbles in the hyperbolic space, and applications

Let $K$ be a smooth function on the hyperbolic space. We deal with $K$-bubbles, that are embedded surfaces parametrized by $\mathbb{S}^2$, having hyperbolic mean curvature $K$ at each point.
It is well known that no $K$-bubbles exist if $|K|< 1$; if $|K| > 1$ is a given constant, then there exists a $9$-dimensional smooth manifold $Z$ of $K$-bubbles, each of them being a round sphere of hyperbolic radius ${\rm artanh\,}(1/|K|)$. We prove a crucial nondegeneracy result involving the manifold $Z$. As an application, we provide sufficient (and almost necessary) conditions on a prescribed function $\phi$, which ensure the existence of a regular curve, parametrized by $\varepsilon \approx 0$, of $K + \varepsilon \phi$ bubbles.
This is joint work with Gabriele Cora, Università di Udine,
[GC-RM], Bubbles with constant mean curvature, and almost constant mean curvature, in the hyperbolic space, Calc. Var. Partial Differential Equations, to appear. Preprint arXiv:2008.03227 (2020).