We prove that every unstable equivariant minimal surface in R n produces a maximal representation of a surface group into n i=1 PSL(2, R) together with an unstable minimal surface in the corresponding product of closed hyperbolic surfaces. To do so, we lift the surface in R n to a surface in a product of R-trees, then deform to a surface in a product of closed hyperbolic surfaces. We show that instability in one context implies instability in the other two.

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 Teichmüller 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 R 2,2 .

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With respect to every Riemannian metric, the Teichmüller metric, and the Thurston metric on Teichmüller space, we show that there exist measured foliations on surfaces whose extremal length functions are not convex. The construction uses harmonic maps to R-trees and minimal surfaces in R n .

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