Dominating surface group representations and deforming closed AdS 3-manifolds

In a previous paper by Deroin-Tholozan, the authors construct a map $\mathbf{\Psi}_\rho$ from the Teichm\"uller space of $S$ to itself and prove that, when $M$ has sectional curvature $\leq -1$, the image of $\mathbf{\Psi}_\rho$ lies (almost always) in the domain $\mathrm{Dom}(\rho)$ of Fuchsian representations stricly dominating $\rho$. Here we prove that $\mathbf{\Psi}_\rho: \mathrm{Teich}(S) \to \mathrm{Dom}(\rho)$ is a homeomorphism. As a consequence, we are able to describe the topology of the deformation space of anti-de Sitter structures on closed 3-manifolds.