Almost strict domination and anti-de Sitter 3-manifolds

TL;DR

This paper introduces almost strict domination for pairs of surface group representations, linking it to maximal surfaces in pseudo-Riemannian manifolds and providing a parametrization of anti-de Sitter 3-manifold deformation spaces.

Contribution

It establishes a new equivalence between almost strict domination and the existence of maximal surfaces, and constructs a parametrization of the deformation space of anti-de Sitter 3-manifolds.

Findings

Almost strict domination characterizes certain representation pairs.

Maximal surfaces are unique and parametrized by the deformation space.

Deformation space of anti-de Sitter 3-manifolds is described as a union of components.

Abstract

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.