Almost strict domination and anti-de Sitter 3-manifolds

Abstract

We define a condition called almost strict domination for pairs of representations ${\rho }_1:{\pi }_1\left(S_{g,n}\right)\to \mathrm{PSL}(2,R)$, ${\rho }_2:{\pi }_1\left(S_{g,n}\right)\to G$, where $G$ is the isometry group of a Hadamard manifold, and prove that it holds if and only if one can find a $({\rho }_1,{\rho }_2)$-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 $G=\mathrm{PSL}(2,R)$, 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 $\mathrm{PSL}(2,R)\times \mathrm{PSL}(2,R)$ relative representation variety.