Infinite energy equivariant harmonic maps, domination, and anti-de Sitter $3$-manifolds

TL;DR

This paper extends the theory of equivariant harmonic maps to infinite energy cases, analyzes their asymptotics, and applies these results to domination problems in geometric group theory, leading to new anti-de Sitter 3-manifolds.

Contribution

It generalizes existence and uniqueness results for harmonic maps to non-compact infinite energy settings and applies these to domination problems and the construction of anti-de Sitter 3-manifolds.

Findings

Extended harmonic map theory to infinite energy cases.

Proved strict domination of representations in length spectrum.

Constructed new anti-de Sitter 3-manifolds.

Abstract

We generalize a well-known existence and uniqueness result for equivariant harmonic maps due to Corlette, Donaldson, and Labourie to a non-compact infinite energy setting and analyze the asymptotic behaviour of the harmonic maps. When the relevant representation is Fuchsian and has hyperbolic monodromy, our construction recovers a family of harmonic maps originally studied by Wolf. We employ these maps to solve a domination problem for representations. In particular, following ideas laid out by Deroin-Tholozan, we prove that any representation from a finitely generated free group to the isometry group of a CAT$(-1)$ Hadamard manifold is strictly dominated in length spectrum by a large collection of Fuchsian ones. As an intermediate step in the proof, we obtain a result of independent interest: parametrizations of certain Teichm{\"u}ller spaces by holomorphic quadratic differentials. The main consequence of the domination result is the existence of a new collection of anti-de Sitter $3$-manifolds. We also present an application to the theory of minimal immersions into the Grassmanian of timelike planes in $\mathbb{R}^{2,2}$.