Almost strict domination and anti‐de Sitter 3‐manifolds

Abstract

We define a condition called almost strict domination for pairs of representations , , where is the isometry group of a Hadamard manifold, and prove that it holds if and only if one can find a ‐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 , 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 relative representation variety.