Almost strict domination and anti-de Sitter 3-manifolds

We define a condition called almost strict domination for pairs of representations ${\rho }_1:{\pi }_1(S_{g,n})\to \text{PSL}(2,\mathbb{R})$, ${\rho }_2:{\pi }_1(S_{g,n})\to G$, 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 $\text{PSL}(2,\mathbb{R})\times \text{PSL}(2,\mathbb{R})$ relative representation variety.