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We generalize a well-known existence and uniqueness result for equivariantharmonic maps due to Corlette, Donaldson, and Labourie to a non-compactinfinite energy setting and analyze the asymptotic behaviour of the harmonicmaps. When the relevant representation is Fuchsian and has hyperbolicmonodromy, our construction recovers a family of harmonic maps originallystudied by Wolf.  We employ these maps to solve a domination problem for representations. Inparticular, following ideas laid out by Deroin-Tholozan, we prove that anyrepresentation from a finitely generated free group to the isometry group of aCAT$(-1)$ Hadamard manifold is strictly dominated in length spectrum by a largecollection of Fuchsian ones. As an intermediate step in the proof, we obtain aresult of independent interest: parametrizations of certain Teichm{\"u}llerspaces by holomorphic quadratic differentials. The main consequence of thedomination result is the existence of a new collection of anti-de Sitter$3$-manifolds. We also present an application to the theory of minimalimmersions into the Grassmanian of timelike planes in $\mathbb{R}^{2,2}$.

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