Riestenberg and Smillie (arXiv:2511.11469) proved a higher-rank extension of the Schoen conjecture: every Π\Pi-quasi-isometric embedding of the hyperbolic plane into the symmetric space of PGLd(R)\mathrm{PGL}_d(\mathbb{R}) whose boundary map is positive in the sense of Fock–Goncharov lies within bounded distance of a unique harmonic map. The proof runs through a stability criterion that requires the boundary maps of all recentered limits of the embedding to be finitely non-transverse, a property supplied in the PGLd\mathrm{PGL}_d case by a theorem of Saldanha–Shapiro–Shapiro. The paper poses, as the hinge for extending its techniques to all other higher Teichmüller spaces, Question 78 (Question 77 in v1): for a pair (G,Θ)(G,\Theta) admitting a Guichard–Wienhard positive structure, is every continuous positive map from RP1\mathbb{RP}^1 to the partial flag variety FΘF_\Theta finitely non-transverse? The question is open — settled only for split PGLd(R)\mathrm{PGL}_d(\mathbb{R}) and, informally, for SO0(2,n+1)\mathrm{SO}_0(2,n+1) — and no follow-up literature resolves any remaining family.

Background

A Hadamard space is a complete metric space of nonpositive curvature in the sense of Alexandrov (a CAT(0) space); symmetric spaces of noncompact type are the main finite-dimensional examples. For maps from a Riemannian domain into a metric target, harmonic maps are defined by minimizing the Korevaar–Schoen energy, which generalizes the classical Dirichlet energy. Schoen conjectured in the early 1990s that every quasiconformal homeomorphism of the sphere at infinity of real hyperbolic nn-space extends to a harmonic quasi-isometry of hyperbolic space; Markovic proved this in 2017. Riestenberg and Smillie extend the conjecture to higher-rank targets. Let GG be a semisimple Lie group with symmetric space Y=G/KY = G/K of noncompact type. The vector-valued distance d⃗Y\vec d_Y takes values in the positive Weyl chamber, and for a nonempty subset Θ\Theta of the simple restricted roots Π\Pi, a map ff from a metric space XX to YY is a Θ\Theta-quasi-isometric embedding if it is coarsely Lipschitz and each root α\alpha in Θ\Theta applied to d⃗Y(f(x),f(z))\vec d_Y(f(x),f(z)) coarsely dominates dX(x,z)d_X(x,z); this coincides with the uniformly regular quasi-isometric embeddings of Kapovich–Leeb–Porti, and for orbit maps of group representations it corresponds to Θ\Theta-Anosov representations — a robust discreteness and properness condition for representations into GG defined by dynamics on suitable flag-manifold flows, which generalizes convex cocompactness. By the higher-rank Morse lemma, such embeddings of a rank-one symmetric space XX have Gromov-hyperbolic domain and extend continuously to a boundary map from ∂∞X\partial_\infty X to the partial flag variety FΘ=G/PΘF_\Theta = G/P_\Theta. Two flags are transverse when they are in general position; a point η\eta of the visual boundary ∂∞Y\partial_\infty Y is transverse to a flag τ∈FΘ\tau \in F_\Theta if η\eta lies in the star of some flag τ′∈FΘ\tau' \in F_\Theta transverse to τ\tau (the star of a simplex being the union of the closed ideal Weyl chambers containing it), and a map ξ:Λ→FΘ\xi : \Lambda \to F_\Theta is finitely non-transverse if for every η∈∂∞Y\eta \in \partial_\infty Y the set of λ∈Λ\lambda \in \Lambda with ξ(λ)\xi(\lambda) not transverse to η\eta is finite. Guichard and Wienhard introduced positive structures (G,Θ)(G,\Theta), generalizing Lusztig total positivity and the positivity of Fock–Goncharov: a positive structure distinguishes a class of positive triples of flags, and a continuous map from the circle RP1\mathbb{RP}^1 to FΘF_\Theta is positive if it takes every pairwise distinct triple to a positive triple. Their classification shows that positive structures exist in exactly four families: split real forms with Θ=Π\Theta = \Pi; Hermitian groups of tube type with Θ\Theta the long simple root (so FΘF_\Theta is the Shilov boundary, the distinguished minimal closed group orbit on the geometric boundary of the symmetric space); SO(p+1,p+k)\mathrm{SO}(p+1,p+k) with p>1p > 1, k>1k > 1 and Θ\Theta the long simple roots; and real forms with restricted root system F4F_4 with Θ={α1,α2}\Theta = \{\alpha_1,\alpha_2\}. Higher Teichmüller spaces, such as Hitchin components, are components of character varieties of surface groups consisting entirely of discrete and faithful representations, and Θ\Theta-positivity organizes the known examples. Theorem 9 of Riestenberg–Smillie states that for d≥2d \geq 2 every Π\Pi-quasi-isometric embedding H2→YdH^2 \to Y_d (YdY_d the symmetric space of PGLd(R)\mathrm{PGL}_d(\mathbb{R})) with positive boundary map is within bounded distance of a unique harmonic map, identifying their universal Hitchin component — the space of such embeddings up to bounded distance — with a space of harmonic maps. The engine of the proof is a stability criterion: if every recentered limit of ff has finitely non-transverse boundary map, then ff is stable and the harmonic approximation and uniqueness follow. In the PGLd\mathrm{PGL}_d case, finite non-transversality of positive boundary maps is a theorem of Saldanha–Shapiro–Shapiro, who bound the number of non-transverse incidences by an explicit constant C(k,d)C(k,d); the sharp value k(d−k)k(d-k) is predicted by the Grassmann convexity conjecture of Shapiro–Shapiro and remains open. Remark 64 of the source paper shows that finite non-transversality is not automatic for general Π\Pi-quasi-isometric embeddings (an H2×RH^2 \times \mathbb{R} example), so positivity must be used in an essential way. Beyond PGLd\mathrm{PGL}_d, the paper verifies informally, via the identification of positive maps into the Einstein universe Ein1,n\mathrm{Ein}_{1,n} (the conformal compactification of (1,n)(1,n)-Minkowski space) with spacelike curves (curves whose tangent directions have positive Minkowski norm), that the property also holds for G=SO0(2,n+1)G = \mathrm{SO}_0(2,n+1) and derives the corresponding harmonic-extension corollary. The authors state that extending their techniques to other higher Teichmüller spaces hinges on the question posed below.

Problem Statement

Let GG be a semisimple Lie group, Y=G/KY = G/K its symmetric space of noncompact type, Π\Pi the simple restricted roots, and Θ⊂Π\Theta \subset \Pi nonempty. Suppose (G,Θ)(G,\Theta) admits a positive structure in the sense of Guichard–Wienhard, and let FΘ=G/PΘF_\Theta = G/P_\Theta be the corresponding partial flag variety. This is Question 78 of Riestenberg–Smillie (Question 77 in v1), which asks, for a general pair (G,Θ)(G,\Theta) admitting a positive structure, whether every continuous positive map ξ:RP1→FΘ\xi : \mathbb{RP}^1 \to F_\Theta is finitely non-transverse, that is, for every η∈∂∞Y\eta \in \partial_\infty Y (the visual boundary of YY, whose simplices range over all flag types), ξ(λ)\xi(\lambda) is non-transverse to η\eta for only finitely many λ∈RP1\lambda \in \mathbb{RP}^1. Because the positive structures are classified into the four families described in Background, the question ranges over every (G,Θ)(G,\Theta) admitting a positive structure. The property is established for a family by an argument covering every continuous positive map in it, and refuted for a family by a single continuous positive map that is non-transverse to some η∈∂∞Y\eta \in \partial_\infty Y at infinitely many λ\lambda. For any class of positive structures on which the property is established, the paper's stability criterion (Theorem 8 / Corollary 67) then yields the corresponding extension of the higher-rank Schoen conjecture: every Θ\Theta-quasi-isometric embedding H2→G/KH^2 \to G/K with positive boundary map is within bounded distance of a unique harmonic map. The property is currently known for split PGLd(R)\mathrm{PGL}_d(\mathbb{R}) (via Saldanha–Shapiro–Shapiro) and, by an informal verification in the source paper, for SO0(2,n+1)\mathrm{SO}_0(2,n+1), and it is open for the remaining positive structures.

References

  1. J. Maxwell Riestenberg and Peter Smillie, Harmonic maps to Hadamard spaces and a universal higher Teichmüller space, arXiv:2511.11469 (2025, revised 2026), DOI 10.48550/arXiv.2511.11469, https://arxiv.org/abs/2511.11469
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