Almost strict domination and anti-de Sitter 3-manifolds

We define a condition called almost strict domination for pairs of
representations $\rho_1:\pi_1(S_{g,n})\to \textrm{PSL}(2,\mathbb{R})$,
$\rho_2:\pi_1(S_{g,n})\to G$, where $G$ is the isometry group of a Hadamard
manifold $(X,\nu)$, and prove it holds if and only if one can find a
$(\rho_1,\rho_2)$-equivariant spacelike maximal surface in a certain
pseudo-Riemannian manifold, unique up to fixing some parameters. The proof
amounts to setting up and solving an interesting variational problem that
involves infinite energy harmonic maps. Adapting a construction of Tholozan, we
construct all such representations and parametrize the deformation space.
When $(X,\nu)=(\mathbb{H},\sigma)$, an almost strictly dominating pair is
equivalent to the data of an anti-de Sitter 3-manifold with specific
properties. The results on maximal surfaces provide a parametrization of the
deformation space of such $3$-manifolds as a union of components in a
$\textrm{PSL}(2,\mathbb{R})\times \textrm{PSL}(2,\mathbb{R})$ relative
representation variety.