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(Sg,n)-> PSL(2,R)$\rho _1:\pi _1(S_{g,n})\rightarrow \textrm {PSL}(2,\mathbb {R})$, rho 2:pi 1(Sg,n)-> G$\rho _2:\pi _1(S_{g,n})\rightarrow G$, where G$G$ is the isometry group of a Hadamard manifold, and prove that it holds if and only if one can find a (rho 1,rho 2)$(\rho _1,\rho _2)$-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 parametrise the deformation space. When G=PSL(2,R)$G=\textrm {PSL}(2,\mathbb {R})$, 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 parametrisation of the deformation space of such 3-manifolds as a union of components in a PSL(2,R)xPSL(2,R)$\textrm {PSL}(2,\mathbb {R})\times \textrm {PSL}(2,\mathbb {R})$ relative representation variety.