Infinite energy equivariant harmonic maps, domination, and 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 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 .

Domination results in n$n$‐Fuchsian fibers in the moduli space of Higgs bundles

“…This is motivated from the result by Deroin-Tholozan [9] that any 𝑆𝐿(2, ℂ)representation can be dominated by some Fuchsian representation using harmonic map method (for 𝑆𝐿(2, ℝ)-representations, the result is also proved independently by Guéritaud-Kassel-Wolff [12] with a different method). Recently, such domination results are generalized to complete surfaces by Sagman [32] and surfaces with boundary by Gupta-Su [13]. One can view the domination results in this paper as a generalization to the 𝑆𝐿(𝑛, ℂ)-case as much as possible.…”

Stability conditions and Teichmüller space

“…On the other hand, it follows from Lemma 4.6 that φ has at worst a simple pole at any p i ∈ M. Thus the differential φ has vanishing leading coefficient at p i and its principal part is zero. If m i = 2 then by Proposition 5.5 of [40], the Hopf differential φ has a pole of order two at p i with leading coefficient…”

“…In particular, we see that dist(F, F ′ ) is bounded on U i \ {p i } whenever m i = 0. In the case m i = 2, it was shown in Lemma 5.9 of [40] that dist(F, F ′ ) is also bounded on U i \ {p i }. For m i ≥ 3, this was shown in Proposition 3.9 of [25].…”

“…If m i = 0 then Lemma 5 of [37] says that the diameter of F (∂U l i ) tends to zero as l → ∞. If m i = 2 then Lemma 5.4 of [40] says there exists a hyperbolic isometry R stabilizing the corresponding boundary component of C such that dist(F, R • f i )(z) → 0 as z → p i . Finally, if m i ≥ 3, then by considering a polygonal exhaustion as in Proposition 2.29 of [25], one sees that F surjects onto the hyperbolic crown corresponding to p i .…”

“…Gupta [25] proved the case where the Hopf differentials have only poles of order ≥ 3, and Sagman [40] further studied the case where the Hopf differentials have double poles. From the perspective of the theory of Higgs bundles, Theorem 1.2 is also closely related to the work of Simpson [45],…”