1 Introduction↩︎

Let \(X\) and \(Y\) be complete non-compact Riemannian manifolds. A natural problem is the following, posed for instance by Gromov in [1]:

A very reasonable additional condition to impose is that \(f\) be coarse Lipschitz, meaning for some \(L\), and all \(x,z \in X\), \[\label{eqn:32coarse32Lipschitz} d_Y(f(x),f(z)) \leq L(d_X(x,z) + 1).\tag{1}\] Indeed, coarse Lipschitz maps are exactly those which preserve the relation of bounded distance in the sense that \(f \circ g \sim f \circ g'\) whenever \(g \sim g'\). In addition, if \(X\) is the universal cover of a compact manifold \(\mathcal{X}\), any locally bounded map \(f: X \to Y\) that is equivariant under an action of \(\pi_1(\mathcal{X})\) on \(Y\) is coarse Lipschitz.

It is also natural to require that the Ricci curvature of \(X\) is bounded below, \[\mathrm{Ric}(X) \geq -Kg_X.\] Besides the fact that it holds when \(X\) is the universal cover of a compact manifold, it is also necessary for any uniformity of estimates on harmonic maps.

More restrictively, we require that the sectional curvature of \(Y\) is non-positive. This still includes the important case of symmetric spaces of non-compact type, and it is also the setting that Gromov works in.

Finally, for us it is not terribly important that \(Y\) be a manifold. Using the theory of harmonic maps to Hadamard spaces of Korevaar and Schoen [2], we assume only that \(Y\) is a proper Hadamard space, i.e.a complete, locally compact CAT(0) metric space. In this context, we consider measurable maps instead of smooth ones; we will use map throughout this paper to mean a measurable map. All told, we have refined Problem [pseudoproblem] to:

Some partial answers to this problem are understood under the assumption of some negative upper bound on the sectional curvature of \(Y\) or strong assumptions on the derivative of \(f\), e.g.[1]. More recently, a sufficiently strong condition in the setting of strictly negative curvature led to the resolution by Markovic [3] of the well-known Schoen conjecture, and in a more general form by Benoist and Hulin [4], [5]. We will return to this story in Section 1.2.

In the present work, we propose a condition on \(f\), which we call stability, and prove that it is sufficient for a positive answer to Problem [problem]. Crucially, it is applicable even without any strict negative curvature assumption on \(Y\), and so we are able to apply it to produce new classes of harmonic maps into higher rank symmetric spaces.

To motivate the definition of stability, we recall three basic facts. First, since \(Y\) is a Hadamard space, the distance \(d_y\) to any point \(y\in Y\) is a convex function on \(Y\). Second, if \(h: X \to Y\) is a harmonic map, it pulls back convex functions to subharmonic functions. Third, subharmonic functions on \(X\) are those that are weakly increasing under the heat flow operator \[P_t u (x) = \int_X u(z) p_t(x,z)dz\] where \(p_t(x,z)\) is the heat kernel.

This is a variant of the Definition 6 of stability below. We find Definition 6 to be more practical, and it may be more general, but it is slightly harder to appreciate at first blush. We remark that stability is a coarse notion, in the sense that if \(f\) is stable, then so is any map at bounded distance from \(f\).

Our main theorem is

One of the key ideas which lets us avoid a strict negative curvature assumptions on \(Y\) is the Ptolemy inequality of planar geometry, which we now recall. Given any four points in the Euclidean plane, let \(D,D'\), \(E,E'\), and \(F,F'\) be the pairwise distances between them, grouped in opposite pairs. The following two inequalities always hold: \[F^2 + (F')^2 \leq D^2 + (D')^2 + E^2 + (E')^2\] and \[FF' \leq DD' + EE'.\] The first is called the parallelogram inequality, and the second the Ptolemy inequality. Equality for the first is obtained, naturally, for parallelograms, and equality for the second is obtained for quadrilaterals inscribed in the circle, where in both cases \(F\) and \(F'\) are the diagonals. Both inequalities hold also for any four points in a CAT(0) space [6].

In the regularity theory for harmonic maps to Hadamard spaces developed by Korevaar and Schoen, the parallelogram inequality plays an essential role. To prove regularity, they compare the harmonic map \(h\) with the nearly-harmonic map \(h'= h \circ \phi\) for a small diffeomorphism \(\phi\), and apply the inequality to the four points \(h(x),h(z),h'(x),h'(z)\). The symmetries of the parallelogram suggest that this is the right inequality for comparing two of the same sort of map, for example two nearly-harmonic maps. On the other hand, in the present work we need to compare a harmonic map with a coarse-Lipschitz map, and this asymmetry is reflected in the asymmetry of inscribed quadrilaterals. Indeed the quadrilateral that we study looks somewhat more like an inscribed quadrilateral than a parallelogram (see Figure 1).

Although we do not explicitly assume the presence of strict negative curvature in \(Y\), there is a weak type of non-flatness implicit in the definition of stable. Namely, if \(Y\) is isometric to \(Y_0 \times \R\), then no map to \(Y\) can be stable, which one can see by considering \(d_y\) for \(y \to \pm \infty\) in the \(\R\) direction. There is also a type of weak negative curvature condition on \(X\) implicit in stability, in the sense that unless \(\lambda_1(X) > 0\), there can be no stable maps from \(X\). Indeed, this follows from the existence of any coarse Lipschitz function \(u\) on \(X\) satisfying \(P_t u - u \geq 1\) for some \(t\).

1.1 Uniqueness, and equivariant maps↩︎

We now restrict to the case that \(X\) is connected with a cocompact isometry group, and that \(Y\) is a symmetric space of non-compact type. For the following uniqueness theorem, we emphasize that we assume no equivariance of the stable map.

Here the assumption that \(X\) admits a cocompact action by isometries is not very important, but allows us to avoid discussing metric-measure spaces with lower Ricci curvature bounds. The assumption on \(Y\) is also stronger than it needs to be, but some extra condition on \(Y\) is essential, analogous to Labourie’s condition ‘sans demies-bandes plates’ in [7].

We now turn to equivariant maps from \(X\) to \(Y\). Let \(\Gamma\) act cocompactly by isometries on \(X\), and fix an action \(\rho\) of \(\Gamma\) on \(Y\). One can construct a locally bounded \(\rho\)-equivariant map from \(X\) to \(Y\), and it is easy to see that such a map will be coarse Lipschitz and that any two \(\rho\)-equivariant locally bounded maps will be at bounded distance from each other.

Recall that the theorem of Donaldson [8] and Corlette [9] gives in this setting

1. There exists a \(\rho\)-equivariant harmonic map if and only if the Zariski closure of \(\rho\) is reductive.

2. The equivariant harmonic map is unique if and only if the centralizer of \(\rho\) is compact.

When the symmetric space is \(\SL_n(\C)/\mathrm{SU}(n)\), the centralizer condition corresponds to the representation on \(\C^n\) being irreducible. More generally, it is natural from the invariant-theoretic perspective to call this condition stable. Since stability is a coarse notion, it therefore makes sense to talk about the stability of an action \(\rho\) as the stability of such an equivariant map. Although it may appear unfortunate at first that this is weaker than our definition of stable, this issue is both minor and unavoidable, see Remark [rmk:32LK].

We note that Labourie extended the definition of reductive to actions on an arbitrary Hadamard manifold \(Y\), and proved an analog of the Donaldson-Corlette theorem in that setting [7], see also [10].

The proof of our Theorem 3 uses the Donaldson-Corlette theorem, so we are not giving a new proof of existence in the equivariant case. Instead, the main point of Theorem 3 is to give evidence that our notion of stability is a natural criterion in the context of Problem [problem].

1.2 The Schoen-Li-Wang conjecture↩︎

An asymptotic Dirichlet problem asks for a harmonic map between two spaces with prescribed boundary values ‘at infinity’. In general, it is not always clear what boundary values at infinity should mean. One way of making this precise is through the formulation in Problems [pseudoproblem] and [problem]. Of course, this is somewhat unsatisfying in the sense that the coarse map \(f\) is certainly not itself ‘at infinity’.

There is a special setting in which one can formulate the asymptotic Dirichlet problem directly. The visual boundary \(\partial_\infty Y\) of a non-positively curved space \(Y\) is the space of geodesic rays in \(Y\), in which two rays are equivalent if they lie at bounded distance. If \(X\) is also non-positively curved, a coarse Lipschitz map from \(X\) to \(Y\) may or may not induce a map from \(\partial_\infty X\) to \(\partial_\infty Y\), but in the following setting it will.

A map \(f: X \to Y\) between metric spaces is called a quasi-isometric embedding if in addition to being coarse Lipschitz 1 for some \(L\), \[d_X(x,z) \leq M(d_Y(f(x),f(z)) + 1)\] holds for some \(M\). If \(X\) and \(Y\) are hyperbolic in the sense of Gromov, then quasi-isometric embeddings between \(X\) and \(Y\) extend continuously to the visual boundary [11], and moreover are determined up to bounded distance by their boundary maps [12]. Furthermore, the class of boundary maps \(\phi\colon \partial_\infty X \to \partial_\infty Y\) that one obtains can be characterized in terms of how they distort cross-ratios; they are called quasi-symmetric maps. We will return to the notion of quasisymmetry in section 1.4.

The Schoen conjecture [13] asserts that every quasi-isometric embedding from the hyperbolic plane \(\H^2\) to itself is within bounded distance of a unique harmonic map. In other words, the conjecture is a positive answer to Problem [problem], together with a uniqueness statement, for this class of maps. By the above discussion, this can be reformulated as saying that every quasi-symmetric map from \(\H^2\) to \(\H^2\) has a unique quasi-isometric harmonic extension. Li and Wang [14] conjectured the same when \(\H^2\) is replaced by a symmetric space of rank 1.

The Schoen conjecture was one of the motivations behind Gromov’s paper [1], and it continued to provoke much work over the intervening 25 years [14]–[19] until it was finally resolved in the affirmative by Markovic in 2015 [3]. Shortly after, Benoist and Hulin proved that every quasi-isometric embedding between (possibly different) rank 1 symmetric spaces is within bounded distance of a harmonic map [4]. They later extended this further [5], replacing rank 1 symmetric spaces by pinched Hadamard manifolds, and relaxing somewhat the conditions on \(f\). The theorem for quasi-isometric embeddings was later extended in [20] to allow \(Y\) to be a locally compact Gromov hyperbolic Hadamard space. Further extensions were made by Tošić [21], [22] and Pankka-Souto [23].

Although the strict negative curvature assumptions in the works of Markovic and Benoist-Hulin allow for more techniques than we have access to in our setting, our general methods recover most of their results. In particular, using Theorem 1 we simplify the proof of the main theorem of [4].

1.3 A higher rank Schoen conjecture↩︎

The second main result of this paper, and our original inspiration, is an extension of the Schoen conjecture to the case that the target is a symmetric space of higher rank. Let us explain what we mean by this. Let \(G\) be a semi-simple Lie group, and let \(Y\) be the associated symmetric space of non-compact type. Choose a Cartan subspace \(\mathfrak{a}\), a set \(\Pi\) of simple restricted roots and let \(\mathfrak{a}^+ \subset \mathfrak{a}\) be the closed positive Weyl chamber. There is a natural vector-valued distance \(\vec{d}_Y\colon Y \times Y \to \mathfrak{a}^+\), from which we recover \(d_Y\) by taking its Euclidean norm. For any nonempty subset \(\Theta \subset \Pi\), we say a map \(f\colon X \to Y\) is a \(\Theta\)-quasi-isometric embedding if it is coarse Lipschitz and satisfies for some \(M\) \[\label{eqn:32intro32theta-qie} d_X(x,z) \leq M(\alpha \circ \vec{d}_Y(f(x),f(z)) + 1) \quad \textrm{for all } \alpha \in \Theta.\tag{2}\] It is easy to see that this is equivalent to the definition of uniformly regular quasi-isometric embeddings of Kapovich-Leeb-Porti [24]. It should also be compared with the uniformly hyperbolic bundles of Bridgeman-Labourie [25]. When \(f\) is the orbit map of a representation \(\rho\) of a finitely generated group, (2 ) is equivalent to \(\rho\) being \(\Theta\)-Anosov [24], [26].

As a consequence of their higher rank Morse lemma, Kapovich, Leeb and Porti prove that every \(\Theta\)-quasi-isometric embedding from a proper geodesic metric space \(X\) to \(Y\) has Gromov hyperbolic domain and extends to a continuous boundary map from \(\partial_\infty X\) to the partial flag variety \(\mathcal{F}_\Theta\) corresponding to \(\Theta\) [24].

We give a criterion for stability in terms of the boundary map \(\partial f\) induced by a \(\Theta\)-quasi-isometric embedding \(f\). We call a limit of \(f\) a locally uniform limit of \(f\) under recentering, and we say that a boundary map is finitely non-transverse if it transverse to any given flag except at finitely many points. As an example of the criterion, when \(\Theta = \Pi\) is the set of all simple roots, we obtain

Our criterion is fairly flexible, but is not satisfied by all boundary maps of \(\Theta\)-quasi-isometric embeddings. It would be interesting to study in full generality when it holds. In this paper, we focus on a special case in which it does hold: when \(Y = Y_d\), the symmetric space for \(\mathrm{PGL}_d(\R)\), \(\Theta = \Pi\), the full set of roots, and the boundary map \(\RP^1 \to \mathrm{Flag}(\R^d)\) is positive in the sense Fock-Goncharov [27]. This means that every triple of flags in its image is equivalent, up to the action of \(\mathrm{PGL}_d(\R)\) to the triple \(\{\sigma_0, n \cdot \sigma_0, \sigma_\infty\}\) where \(\sigma_0\) is the standard descending flag, \(\sigma_\infty\) the standard ascending flag, and \(n\) is an upper triangular unipotent matrix which is totally positive [28]–[30] i.e.all of the interesting minors of \(n\) are positive. It would be natural to try to extend the theorem to the more general setting of \(\Theta\)-positive maps as introduced by Guichard-Wienhard [31].

This allows us to conclude our second main theorem

When \(d = 2\), a \(\Pi\)-quasi-isometric embedding is the same as a quasi-isometric embedding, and the positivity condition is vacuous. Hence, this is exactly the statement of the Schoen conjecture when \(d=2\). For \(d > 2\), our main interest in this theorem is that it shows that a certain universal higher Teichmüller space can be described as a space of harmonic maps. We describe this further in the following section.

1.4 A universal higher Teichmüller space↩︎

The universal Teichmüller space was originally defined as the space of boundary values of quasi-conformal maps of the disk [32]. It is universal in the sense that it contains the Teichmüller space of any closed surface.

By work of many people including [33]–[36], these are the same as the boundary maps of quasi-isometric embeddings from \(\H^2\) to itself, i.e.the quasi-symmetric maps. We recall that the cross ratio of four points in \(\RP^1\) is \[CR(\lambda_1,\lambda_2,\lambda_3,\lambda_4) = \frac{(\lambda_4 - \lambda_3)(\lambda_2 - \lambda_1)}{(\lambda_3 - \lambda_2)(\lambda_1 - \lambda_4)}\] and a map \(\phi\) from \(\partial_\infty \H^2 \cong \RP^1\) to itself is quasisymmetric if \[CR(\phi(\lambda_1),\phi(\lambda_2),\phi(\lambda_3),\phi(\lambda_4))\] can be bounded uniformly in terms of \(CR(\lambda_1,\lambda_2,\lambda_3,\lambda_4)\). Another rephrasing of the Schoen conjecture is that the universal Teichmüller space admits a description as a space of harmonic maps.

A different sort of generalization of the classical Teichmüller space is the notion of a higher Teichmüller space as defined in [37]. This is a union of components of the space of representations of a closed surface group consisting entirely of discrete and faithful representations. The archetype of a higher Teichmüller space is, for each \(d \geq 2\), the space of Hitchin representations of a closed surface group into \(\mathrm{PGL}_d(\R)\). When \(d = 2\), these are exactly the Fuchsian representations. For \(d > 2\), these components were first described by Hitchin [38], and proved to be higher Teichmüller spaces in this sense by Labourie [39] and Fock-Goncharov [27]. Of relevance to this paper, their first description by Hitchin used equivariant harmonic maps, through the theory of Higgs bundles.

Associated to each Hitchin representation is an equivariant boundary map from \(\RP^1\) to the space \(\mathrm{Flag}(\R^d)\) of full flags of \(\R^d\). Such a map is a hyperconvex Frenet curve in the language of Labourie, or equivalently, a positive map in the language of Fock-Goncharov. This property also characterizes Hitchin representations [27], [40]. With this in mind, we define

In these terms our Theorem 6 says that the universal Hitchin component is naturally a space of harmonic maps.

By the discussion preceding the definition, for a fixed \(d\), the \(d\)-Hitchin component of any closed surface group embeds into to \(\mathrm{UHit}_d\). To complete the analogy between \(\mathrm{UHit}_d\) and universal Teichmuller space, and to recast our existence Theorem 6 as an asymptotic Dirichlet problem, it remains to characterize \(\mathrm{UHit}_d\) as a space of positive maps from \(\RP^1\) to \(\mathrm{Flag}(\R^d)\) which are moreover quasi-symmetric in some sense.

Both Fock-Goncharov [27] and Labourie [39] have suggested defining a universal Hitchin component in terms of positive maps of circles to \(\Flag(\R^d)\). Each paper describes functions of \(k\)-tuples of flags that generalize the cross-ratio; using the respective notions of cross ratio one obtains a priori different notion of quasi-symmetric maps from \(\RP^1\) to \(\mathrm{Flag}(\R^d)\). We also remark that Labourie and Toulisse define a slightly different sort of universal higher Teichmüller space containing the spaces of maximal representations into \(\SO(2,n+1)\) [41], still in terms of cross ratios. Whereas ours consists of quasisymmetric maps, theirs consists of quasi-circles. Using results of [42], they show that their universal higher Teichmüller space parametrizes a space of quasi-isometrically embedded maximal surfaces in the pseudo-Riemannian symmetric space \(\mathbb{H}^{2,n}\).

We define a cross ratio which is similar to, but different than, the edge invariant considered by Fock-Goncharov. Namely, if \(n\) and \(m^{-1}\) are totally positive unipotent upper triangular matrices, we define \(d-1\) cross-ratios of a positive four-tuple of flags in standard position by \[CR_i(m \cdot \sigma_0, \sigma_0, n \cdot \sigma_0, \sigma_\infty) = m_{i,i+1}/n_{i,i+1}\] for \(1 \leq i \leq d-1\). We say a positive map \(\phi: \RP^1 \to \mathrm{Flag}(\R^d)\) is quasi-symmetric if each \(CR_i\) is controlled by the cross-ratio in the domain. We prove

This allows us to express our Theorem 6 as a solution of an asymptotic Dirichlet problem:

We conclude with a few remarks about this theorem.

First, we note that in contrast to the case of strict negative curvature, the visual boundary of \(Y\) does not give the right target with respect to which to pose the asymptotic Dirichlet problem. This is already clear from the case of \(\H^2 \times \H^2\), for which the product structure shows that the right target for the boundary map is \(\RP^1 \times \RP^1\), which is the space of full flags.

Second, although the existence theorem is much easier in the case that the \(\Pi\)-quasi-isometric embedding arises from a Hitchin representation, we hope that the method of bounding the distance to the harmonic map will have interesting applications in the equivariant case, for instance to the asympotic geometry of the Kähler metric on the Hitchin component. It would still take some work however to make our constants explicit.

Finally, there has also been recent progress on harmonic maps from open Riemann surfaces, such as \(\H^2\), to symmetric spaces from the perspective of Higgs bundles. In [43], Li and Mochizuki construct a candidate for a universal Hitchin component, parametrized by the space of bounded holomorphic differentials \((q_2, \ldots, q_n)\) on \(\H^2\). If one could show that their family of harmonic maps coincides with ours, this would give a non-Abelian Hodge correspondence for the universal Hitchin component.

1.5 Acknowledgments↩︎

Both authors would like to thank Anna Wienhard for support throughout the project, and Michael Gekhtman for pointing out the references [44], [45]. The second author thanks Vladimir Marković for encouragement and Richard Schoen for pointing out the reference [46].

2 Preliminaries↩︎

2.1 Hadamard spaces↩︎

If \(T\) is a geodesic triangle in a metric space \(Y\), a planar comparison triangle is a triangle \(\hat{T}\) in \(\R^2\) with a map to \(T\) that is an isometry along each geodesic. A CAT(0) space \(Y\) is a metric space in which any two points can be joined by a geodesic segment, such that for any geodesic triangle \(T\) in \(Y\), the canonical map from a planar comparison triangle to \(T\) is 1-Lipschitz. A Hadamard space is a complete CAT(0) space. \(Y\) is called proper if metric balls in \(Y\) are compact. A standard reference for CAT(0) spaces is [47].

The visual boundary \(\partial_\infty Y\) of a Hadamard space \(Y\) is the set of geodesic rays \(\eta: [0,\infty) \to Y\) up to equivalence, where two rays at bounded distance are considered equivalent. The space \(\overline{Y} = Y \cup \partial_\infty Y\) is given a topology, called the visual topology, from its inclusion into the inverse limit of closed balls in \(Y\) [47]. If \(Y\) is proper, then \(\overline{Y}\) is compact.

If \(\eta \in \partial_\infty Y\), the Busemann cocycle associated to \(\eta\) is the function \(b_\eta \colon Y\times Y \to \R\) defined by \[b_\eta(y_1, y_2) = \lim_{t \to \infty} d_Y(\eta(t), y_2) - d_Y(\eta(t), y_1)\] (one checks using the triangle comparison property that this does not depend on the choice of ray in the equivalence class). We will usually de-emphasize the point \(y_1\), and speak of the Busemann function \(b_\eta(y_2)\), which is only well defined up to a constant. The CAT(0) condition implies that the distance is convex on \(Y \times Y\). In particular each distance function \(d_y \colon Y \to \R\), \(d_y(y') = d_Y(y,y')\) is convex as well as each Busemann function.

2.1.1 Quadrilateral comparison↩︎

A useful property of CAT(0) spaces is Reshetnyak’s sub-embedding theorem. For any geodesic quadrilateral \(Q\) in a metric space \(Y\), let \(D,D',E,E'\) be its edges, grouped in opposite pairs, and \(F,F'\) be its diagonals. A planar subembedding of \(Q\) is a quadrilateral \(\hat{Q}\) in \(\R^2\) with a map to \(Q\) that is an isometry along each edge, and distance non-increasing along its diagonals.

As one of the key ingredients in [2], Reshetnyak’s subembedding theorem is used to transport various versions of the parallelogram inequality from planar quadrilaterals to quadrilaterals in CAT(0) spaces. In the present work, the key ingredient turns out to be instead the Ptolemy inequality. For planar quadrilaterals with pairwise distances \(D,D'\), \(E,E'\), and \(F,F'\), this reads \[FF' \leq DD' + EE'.\] In this case equality holds if and only if the quadrilateral is inscribed in a circle. It is interesting to note that the Ptolemy inequality and the parallelogram inequality are simply the triangle inequality, applied to two generators of the space of symmetric polynomials on two letters. By the subembedding theorem, as above, we have

It is interesting to note that a metric space is CAT(0) if and only if it satisfies the Ptolemy inequality and the distance is a convex function on \(Y \times Y\) [48].

2.2 Lower Ricci curvature bounds↩︎

Recall that \(X\) is a complete Riemannian \(n\)-manifold with Ricci curvature bounded below by \(-K\).

2.2.1 Green’s functions and mean value inequalities↩︎

If \(\Omega \subset X\) is a bounded domain, we write \(G^\Omega(x,z)\) for the Green’s function of \(\Omega\). The Laplacian comparison theorem and the maximum principle gives the following well-known sharp lower bound for the Green’s function of a ball with respect to a point \(x\):

When \(X\) is non-collapsed in the sense that the volume of balls of radius 1 in \(X\) is uniformly bounded below, the following proposition follows from Proposition [prop:32noncollapsed32greens32function32bound]. But it is also true more generally.

Finally, we will use the following mean value inequality.

2.2.2 Mollification↩︎

As a consequence of the Bishop-Gromov volume comparison, we can mollify coarse Lipschitz maps to produce Lipschitz maps in a uniform way, by averaging over balls. Compare [4].

2.3 Harmonic maps↩︎

The basic theory of harmonic maps into Hadamard spaces is developed by Korevaar and Schoen in [2], and also by Jost in [53]. Sturm [54] also develops this theory in even more generality. We summarize only the relevant facts for us. Let \(Y\) be a Hadamard space, \((X,g)\) a Riemannian manifold, and \(h: X \to Y\) harmonic in the sense of Korevaar-Schoen.

Whether or not \(Y\) is a manifold, we will write \(|\nabla h|\) for \(\sqrt{2e}\).

A crude gradient bound for harmonic maps follows from Proposition [prop:32K-S], together with the mean value inequality 9 and the Caccioppoli trick.

We remark that this also follows from our Lemma 1.

For \(Y\) a Hadamard manifold, the solution of the Dirichlet problem is due to Hamilton [56]. For Hadamard spaces, it is proved by Korevaar and Schoen.

Note that properness of \(Y\) is not assumed. When \(Y\) is proper, an easy consequence of this proposition is the precompactness of bounded families of harmonic maps (see [2] Theorem 1.13). We give a slight reformulation suitable to asymptotic Dirichlet problems.

3 Stability↩︎

3.1 Coarsely strictly subharmonic functions↩︎

If \(u\) is harmonic, then applying the inequality to \(\pm u\) shows that in this case we have equality. Applying the equality to the function \(u=1\), one sees that every harmonic measure is a probability measure.

One example of a harmonic measure is the heat kernel measure \(p_t(x,z)dz\). Another is the hitting measure of a bounded domain \(\Omega\) containing \(x\). This is the unique measure that is harmonic with respect to \(x\) and is supported on \(\partial \Omega\). We will write it as \(\mu_x^\Omega\). If the boundary of \(\Omega\) is sufficiently regular, Green’s formula gives \[\mu_x^\Omega= -\frac{dG}{dn} dz,\] where \(dz\) is the area measure on the boundary of \(\Omega\) and \(dG/dn\) is the normal derivative of the Green’s function \(G=G^\Omega(x, \cdot)\) of \(\Omega\) whose pole lies at \(x\). The hitting measure can also be characterized in probabilistic terms as describing, for a subset \(U \subset \partial \Omega\), the chance that Brownian motion starting at \(x\) will first exit \(\Omega\) in the set \(U\).

One can construct more harmonic measures by convolving families of harmonic measures or using the optional stopping theorem. One that we use below is the collected heat kernel measure \(\mu_{x,t}^r\) which is defined by stopping Brownian motion at the minimimum of \(t\) and the exit time from the ball of radius \(r\) around \(x\). By Green’s formula, if \(B = B(x,r)\) and \(p^B_t\) the heat kernel of \(B\) with Dirichlet boundary conditions, \[\int_X v(z) \mu^r_{x,t}(z) = \int_{B} v(z) p^{B}_t(x,z)dz - \int_{\partial B} v(z) \left(\int_0^t \frac{d p_s^B(x,z)}{dn} ds \right) dz.\]

In other words, \(u\) is coarsely strictly subharmonic at scale \(r\) if there is an operator \(P\) such that

1. \(P\) is given by convolution with a family of measures,

2. \(P\) has propogation at most \(r\),

3. \(P w - w \geq 0\) for all subharmonic functions \(w\), and

4. \(P u - u \geq 1\).

In this case, we will say \(u\) is coarsely strictly subharmonic with respect to the operator \(P\), or to the family of measures \(\mu_x\).

The following proposition says that coarse strict subharmonicity is a coarse property.

A natural measure of the degree of coarse strict subharmonicity is given by the drift.

It would be nice to know if the converse of Proposition [prop:32drift32implies32stable] holds as well.

3.2 Stability of maps↩︎

We recall that \(X\) is a complete Riemannian manifold with Ricci curvature bounded below by \(-K\), and \(Y\) is a proper Hadamard space.

We note that the implicit family of measures \(\{\mu_x\}\), or equivalently the operator \(P\), is allowed to depend on \(y\). By Proposition [prop:32subharmonicity32is32coarse], stability is a coarse property.

We also define, but do not use in an essential way;

As in Proposition [prop:32drift32implies32stable], a coarse Lipschitz map of positive drift is stable at some scale. This shows that Definition 1 in the introduction is at least as strong as Definition 6.

Definition 7 bears a strong resemblance to the conditions of the multiplicative ergodic theorem of [57].

4 Existence↩︎

In this section we prove Theorem 1. We will construct \(h\) by taking a limit of a sequences of solutions to Dirichlet problems. Since \(f\) itself is not assumed to be of locally finite energy, as a first step we mollify \(f\), using Proposition [prop:32Lipchitz], to obtain a Lipschitz map \(f^{(1)}\) that is at distance at most \(L\) from \(f\).

Then, we construct a suitable exhaustion of \(X\) by Lipschitz domains \(\Omega_i\), and let \(h_i \colon \Omega \to Y\) be the solution to the Dirichlet problem \[\label{eqn:32Dirichlet32problem} \begin{cases} \Delta h_i = 0 \\ h_i |_{\partial \Omega_i} = f^{(1)} |_{\partial \Omega_i} \end{cases}\tag{5}\] which exists and is unique by Proposition [prop:32dirichlet32problem]. According to Proposition [prop:32asymptotic32Dirichlet], as long as we can find a distance \(R\) such that the uniform \(C^0\) bound \[\label{eqn:32uniform32bound} d_Y(h_i(x),f(x)) \leq R\tag{6}\] holds for all \(i\) and all \(x \in \Omega_i\), the maps \(h_i\) will sub-converge to a harmonic map \(h\) from \(X\) to \(Y\), lying at distance at most \(R\) from \(f\). The key lemma will be the following interior \(C^0\) estimate. Recall that for \(y \in Y\) we let \(d_y \colon Y \to \R\) denote the distance function \(d_y(y')=d_Y(y,y')\). Also recall that we define a harmonic measure on a domain \(B\) with respect to a point \(x\) to be any measure with the property that \(\int_B u(z) \mu(z) \geq u(x)\) for any subharmonic function \(u\). The assumption of the following Proposition could be rephrased as saying that \(f\) is stable at scale \(r\) at the point \(x\).

According to this proposition, each map \(h_i:\Omega_i \to Y\) will satisfy \[d_Y(h(x),f(x)) \leq R\] provided that there is a point \(x_i\) in \(\Omega\) maximizing \(d_Y(h_i(z),f(z))\) which lies at distance at least \(r+2\) from the boundary. Since the distance is at most \(L\) on the boundary, it will be maximized at some point of \(\Omega\); to deal with the possibility that this point lies close to \(\partial \Omega_i\) we need to choose the exhaustion with a little bit of care. This will be done in the next section, following the proof of the interior estimate.

4.1 Interior \(C^0\) estimate↩︎

We begin with a variant of the gradient estimate [prop:32gradient32estimate], which gives a better estimate when the harmonic map sends a point very close to the boundary of the ball. We refer to it as a Harnack estimate.

The following lemma is a simple consequence of the Ptolemy inequality for CAT(0) spaces. It may be helpful to refer to Figure 1 below.

We now proceed to the proof of the interior estimate.

4.2 Boundary \(C^0\) estimate↩︎

In this section we construct an exhaustion \(\Omega_i\) of \(X\) for which we can control the distance between \(f\) and the solutions \(h_i\) to the Dirichlet problems 5 in a fixed neighborhood of the boundary of \(\Omega_i\).

It is easy to construct domains satisfying the exterior \(\rho\)-ball condition.

The last ingredient in the proof of the main theorem is the following boundary estimate.

The proof will involve a lemma to the effect that if \(\Omega\) satisfies the exterior ball condition then a Brownian motion that starts near the boundary has a positive probability of hitting the boundary before it travels too far. This is rephrased in the following lemma. For the following lemma, refer to Figure 2.

We now prove the boundary estimate.

We note that stability of \(f\) does not play a role in the boundary estimate.

4.3 Conclusion of the proof↩︎

We have now completed all the steps of the proof of the main existence theorem 1, which we now restate in a more quantitative form:

We have already explained every element of the proof, but for the reader’s convenience we put everything together in the following.

5 Uniqueness and equivariant maps↩︎

In this section, we prove Theorems 2 and 3. We assume throughout this section that \(Y\) is a manifold, and for the theorems, we assume further that \(Y\) is a symmetric space. Some more generality is possible, but some version of the Jacobi field extension property of Proposition [prop:32Y32has32condition32E] for \(Y\) is a necessary ingredient.

We also assume that \(X\) is connected and that the isometry group of \(X\) acts cocompactly on \(X\). We believe that cocompactness is not an essential assumption, and could be removed by using the theory of metric-measure spaces with lower Ricci curvature bounds, or more quantitative estimates on \(Y\).

5.1 Variations through geodesics↩︎

Let \(I\) be an interval of \(\R\). We call \(H: I \times \Omega \to Y\) a variation through geodesics if \(t \mapsto H(t,x)\) is a geodesic for every \(x \in \Omega\). If \(H\) is a variation through geodesics, the vector fields obtained by differentiating \(H\) in directions tangent to \(\Omega\) are Jacobi fields. We call \(H\) a parallel variation through geodesics if every such Jacobi field is parallel along the geodesic. Note that a Jacobi field is parallel if and only if its norm is constant in \(t\). Furthermore, since \(Y\) is a Hadamard manifold, the norm of a Jacobi field is a convex function of \(t\).

Parallel variations through geodesics arise in the following two situations.

Both propositions are well-known and follow from the first and second variation formulas for the induced metric along the family. For the first, one uses the negative gradient flow of \(\tfrac{1}{2}d^2\) on \(Y \times Y\).

This is a natural extension of Labourie’s condition ‘sans demies-bandes plates’ in [7].

Not every manifold has extendable parallel Jacobi fields; for instance, a non-flat surface with a flat open sub-surface will not. However, since the coefficients of the Jacobi equation of a real analytic manifold are real analytic, we have for instance:

5.2 Uniqueness↩︎

5.3 Stability of equivariant maps↩︎

In this subsection we prove Theorem 3.

6 \(\Theta\)-quasi-isometric embeddings and stability↩︎

In this section we prove a more general version of the criterion for stability in terms of boundary maps, Theorem 5. We first give the definition of \(\Theta\)-quasi-isometric embeddings from a metric space to a symmetric space \(Y\), and recall the higher rank Morse lemma of [24], and the existence of continuous extensions to the flag variety. We then prove a slightly stronger version of the extension theorem than we could find in [24] (Proposition [prop:32uniform32convergence]), and show that stability of \(\Theta\)-quasi-isometric embeddings can be deduced as a consequence of stability in the sense of [59] of certain measures on the visual boundary of the symmetric space. In the case that \(Y=Y_d\) is the symmetric space of \(\PGL_d(\R)\), \(\Theta = \Pi\), the set of all simple roots, and \(f\) has positive boundary map (Definition 16), this will allow us to conclude that \(f\) is stable.

6.1 Symmetric space reminders↩︎

Let \(G\) be a connected semisimple Lie group with finite center and let \(Y\) be the associated symmetric space of non-compact type. There is a unique \(G\)-invariant Riemannian metric \(g\) on \(Y\) up to rescaling on each de Rham factor. It will be convenient to fix the metric so that the minimium sectional curvature of any tangent \(2\)-plane is \(-1\) in each de Rham factor. Fix a point \(o\in Y\), and let \(K\) be the stabilizer of \(o\) in \(G\). The Lie algebra \(\mathfrak{g}\) of \(G\) admits a Cartan involution \(\theta\) whose fixed point set is precisely \(\mathfrak{k}\), the Lie algebra of \(K\). Let \(\mathfrak{p}\) denote the \(-1\)-eigenspace of \(\theta\). For any point \(y \in Y\) there exists a unique \(v \in \mathfrak{p}\) so that \(e^v o = y\).

Let \(\mathfrak{a}\) denote a Cartan subspace of \(\mathfrak{p}\). The adjoint action of \(\mathfrak{a}\) on \(\mathfrak{g}\) induces the restricted root space decomposition of \(\mathfrak{g}\) and we let \(\Sigma \subset \mathfrak{a}\) denote the set of restricted roots. Choose a subset \(\Pi \subset \Sigma\) of simple roots and a corresponding positive closed Weyl chamber \(\mathfrak{a}^+ \subset \mathfrak{a}\). For any \(v \in \mathfrak{p}\) there exists a unique \(a \in \mathfrak{a}^+\) and a (typically non-unique) element \(k \in K\) so that \(\Ad(k)(v)=a\).

In particular, for any pair of points \(y_1, y_2\) in \(Y\), there exists a unique element \(a \in \mathfrak{a}^+\) and an element \(g \in G\) so that \[g(y_1,y_2) = (o,e^a o) ,\] the element \(a \in \mathfrak{a}^+\) is called the vector-valued distance of \((y_1,y_2)\). For a pair of points \(y_1,y_2 \in Y\) we will often write \(\vec{d}_Y(y_1,y_2)\) for the vector-valued distance.

6.2 \(\Theta\)-quasi-isometric embeddings↩︎

\(\Theta\)-quasi-isometric embeddings are equivalent to \(\tau_{mod}\)-uniformly regular quasi-isometric embeddings as introduced by Kapovich-Leeb-Porti [24]. They are also strongly related to the uniformly hyperbolic bundles of [25]. When \(Y\) is negatively curved (i.e.has rank \(1\)), there is a unique simple restricted root \(\Pi=\{\alpha\}\). In this case, \(\{\alpha\}\)-quasi-isometric embeddings are the same as quasi-isometric embeddings. In higher rank, every \(\Pi\)-quasi-isometric embedding is a quasi-isometric embedding but the converse does not hold in general.

Note that the vector-valued distance is not symmetric in general, but instead satisfies \(-w_0 \vec{d}_Y(y_1,y_2) = \vec{d}_Y(y_2,y_1)\) where \(w_0\) is the longest element of the Weyl group. The isometry \(-w_0 \colon \mathfrak{a} \to \mathfrak{a}\) is called the opposition involution, and it is non-trivial exactly when the root system of \(\mathfrak{a}\) is of type \(A_n,D_{2n+1}\) or \(E_6\). In any case, interchanging the roles of \(z_1\) and \(z_2\) in Definition 10 shows that \((M,\Theta)\)-quasi-isometric embeddings are also \((M,\Theta \cup (-w_0 \Theta))\)-quasi-isometric embeddings. So we may, and do from now on, assume that \(\Theta\) is closed under the opposition involution.

6.2.1 Flag topology↩︎

Let \(\mathcal{F}_\Theta\) denote the flag manifold associated to \(\Theta\). We need to recall the definition of the flag topology on \(Y \cup \mathcal{F}_\Theta\), making it into a suitable bordification of \(Y\).

There are several possible descriptions of the flag topology. We will give a description as a quotient of a subspace of the visual compactification. It can also be described in terms of a Satake compactification or a suitable horofunction compactifications [60].

The visual boundary of \(Y\) has a simplicial structure as a spherical building, inherited from the Coxeter complex structure on the visual boundary of each maximal flat. In particular, there is a type map from \(\partial_\infty Y\) to the model spherical Weyl chamber \(\sigma_{mod} = \mathbb{P}(\mathfrak{a}^+)\). The fibers of the type map are precisely the \(G\)-orbits of \(\partial_{\infty} Y\). There is a distinguished face \(\tau_\Theta\) of \(\sigma_{mod}\) corresponding to \(\Theta\): it is the intersection of the walls \(\ker \alpha\) for \(\alpha \in \Pi \setminus \Theta\) with \(\sigma_{mod}\). The flag manifold \(\mathcal{F}_\Theta\) is naturally identified with the set of simplices of \(\partial_\infty Y\) of type \(\tau_\Theta\).

The \(\Theta\)-regular part of \(\sigma_{mod}\) is the subset where \(\alpha\) is positive for each \(\alpha \in \Theta\). The \(\Theta\)-regular part of \(\partial_\infty Y\), denoted \(\partial^{\Theta-reg}_\infty Y\), is the subset of \(\partial_\infty Y\) with types in the \(\Theta\)-regular part of \(\sigma_{mod}\).

Fixing \(o \in Y\), let \[o^{\Theta-sing} = \{ y \in Y : \exists \alpha \in \Theta \text{ s.t. } \alpha \circ \vec{d}_Y(o,y) =0 \} .\] There is a natural continuous projection \[\overline{\pi_\Theta} = \pi_\Theta \cup \partial \pi_\Theta \colon (Y \setminus o^{\Theta-sing}) \cup \partial_\infty^{\Theta-reg}Y \to \mathcal{F}_\Theta\] taking \(y \in Y \setminus o^{\Theta-sing}\) to the unique simplex of type \(\tau_\Theta\) in any chamber containing \(\vec{oy}\). If a segment \(y_1y_2\) is congruent to \(oy\) for \(y \in Y\setminus o^{\Theta-sing}\) then it is called \(\Theta\)-regular. For \(\tau \in \mathcal{F}_\Theta\) (respectively \(y \in Y \setminus o^\mathrm{\Theta-sing}\)), the fiber \(\pi_\Theta^{-1}(\tau)\) is called a Weyl cone, and denoted \(V_\Theta(o,\tau)\) (respectively, \(V_\Theta(o,y)\)).

Then the flag topology on \(Y \cup \mathcal{F}_\Theta\) is the quotient topology under the map \[\mathrm{Id} \cup \partial \pi_\Theta \colon Y \cup \partial_\infty^{\Theta-reg}Y \to Y \cup \mathcal{F}_\Theta\] which is independent of the basepoint \(o \in Y\).

6.2.2 The higher rank Morse lemma↩︎

As a consequence of Gromov hyperbolicity, quasigeodesics in rank 1 symmetric spaces enjoy the Morse lemma: they lie uniformly close to geodesics. Kapovich-Leeb-Porti have proven a generalization of this statement for \(\Theta\)-quasigeodesics into symmetric spaces of higher rank.

When \(\Theta = \Pi\), a Weyl cone \(V_\Pi(y,\sigma)\) is an open Euclidean Weyl chamber with tip at \(y\) asymptotic to the ideal Weyl chamber \(\sigma \subset \partial_\infty Y\). That case already covers the material we need in Section 7 and for Theorem 6, and we encourage the reader to keep that special case in mind.

Kapovich-Leeb-Porti also obtain the following consequence of Theorem 13. A pair \(\tau,\tau' \in \mathcal{F}_\Theta\) is called transverse if there exists a geodesic \(c \colon \R \to Y\) so that \(c(+\infty)\) is in the relative interior of \(\tau\) and \(c(-\infty)\) is in the relative interior of \(\tau'\).

Our next goal is to prove Proposition [prop:32uniform32convergence], which states that \(L\)-Lipschitz \((M,\Theta)\)-quasi-isometric embeddings taking some point \(x\) to some point \(y\) which converge converge locally uniformly on \(Z\) actually converge uniformly as maps \(Z \cup \partial_\infty Z\) to \(Y \cup \mathcal{F}_\Theta\). This is more-or-less a direct consequence of the higher rank Morse lemma, but we could not find the precise result we need in the literature.

Recall that \(\mathfrak{a}^+\) is a closed Euclidean Weyl chamber. Define \(m_\Theta \colon \mathfrak{a}^+ \to \R^{\geq 0}\) by \[m_\Theta(v) \mathrel{\vcenter{:}}= \min_{\alpha \in \Theta} \alpha(v),\] and for brevity write also for \(y \in Y\), \[m_\Theta(y) = m_\Theta(\vec{d}_Y(o,y)).\] The flag variety \(\mathcal{F}_\Theta\) inherits a Riemannian metric \(g_\mathcal{F}\) as a quotient of \(K\), the stabilizer of \(o\) in \(G\). Let \(d_\mathcal{F}\) be the corresponding distance function. Using the projection \(\pi_\Theta\) from \(Y \setminus o^{\Theta-\mathrm{sing}}\) to \(\mathcal{F}_\Theta\), we can also evaluate \(d_\mathcal{F}\) on points of \(Y \setminus o^{\Theta-\mathrm{sing}}\).

6.3 Transversality↩︎

In this section we define some notions that we will need in order to state our criterion for stability, Theorem 15 below. Specifically, we give definitions of almost-everywhere transversality for measures on a flag variety \(\mathcal{F}_\theta\) and for maps to \(\mathcal{F}_\theta\). We first recall the definition of Kapovich-Leeb-Millson [59] of stability of measures on \(\partial_\infty Y\).

For a convex Lipschitz function \(b \colon Y \to \R\), the asymptotic slope of \(b\) along a geodesic ray \(c \colon [0,\infty) \to Y\) is defined by \[\slope(b,c) \mathrel{\vcenter{:}}= \lim_{t \to \infty} \frac{b \circ c(t)}{t} .\] A pair of asymptotic geodesic rays have the same asymptotic slopes, so the definition descends to a pairing of convex Lipschitz functions with the visual boundary of \(Y\). The slope is closely related to the angular metric on \(\partial_\infty Y\), defined by \[\angle(\xi,\eta) \mathrel{\vcenter{:}}= \sup_{y \in Y} \angle_y(\xi,\eta)\] where \(\angle_y(\xi, \eta)\) is the Riemannian angle at \(y\) between the geodesic rays emanating from \(y\) asymptotic to \(\xi,\eta\). In a symmetric space, the angular metric is achieved at \(y\) when \(y, \xi,\eta\) all belong to a common (maximal) flat, and there always exists a maximal flat containing both \(\xi\) and \(\eta\) in its visual boundary. As a consequence, \[\label{eqn:32slope32vs32angle} - \cos \angle(\xi,\eta) = \slope(b_\xi,\eta)\tag{13}\] holds for all \(\xi,\eta \in \partial_\infty Y\). When the rank of \(Y\) is at least \(2\), the angular metric coincides with its associated length metric, which is called the Tits metric.

Following Kapovich-Leeb-Millson [59], we recall

Recall that \(\Theta\) is closed under the opposition involution and that a pair \(\tau,\tau' \in \mathcal{F}_\Theta\) is called transverse if there exists a geodesic \(c \colon \R \to Y\) so that \(c(+\infty)\) is in the relative interior of \(\tau\) and \(c(-\infty)\) is in the relative interior of \(\tau'\). The star of a simplex \(\tau\) is the union of all closed ideal Weyl chambers \(\sigma\) containing \(\tau\).

We will also use the following invariant of the root system \((\mathfrak{a},\Pi)\) and the subset \(\Theta\).

In the case of interest, we aim to deduce stability of a \(\Theta\)-quasi-isometric embedding purely from properties of its boundary map. To this end, the following definition will be useful.

If \(\nu\) is a non-atomic finite measure on \(\Lambda\) and \(\xi\) is finitely non-transverse, then \(\xi_*\nu\) is generically transverse.

In particular, finitely-non transverse maps are finite-to-one. For us, \(\Lambda\) will arise as the visual boundary of a pinched Hadamard space, and \(\xi\) will be the boundary map of an \((L,M,\Theta)\)-quasi-isometric embedding.

In the next section, we will extract measures on \(\partial_\infty Y\) from \(\Theta\)-quasi-isometric embeddings. We have the following criterion for those measures to be stable, even quantifiably. Recall the continuous map \(\pi_\Theta \colon \partial^{\Theta-reg}_\infty Y \to \mathcal{F}_\Theta\) from the \(\Theta\)-regular part of the ideal boundary to the flag manifold.

6.4 A criterion for stability↩︎

Let \(f: X \to Y\) be a continuous map between symmetric spaces, and suppose \(f(x) = y\) for some \(x\in X\) and \(y \in Y\). For the following corollary, we call a continuous map \(g: X \to Y\) a limit of \(f\) if there is a sequence of isometries \(\gamma_i \in \mathrm{Isom}(X)\) and \(\gamma'_i \in \mathrm{Isom}(Y)\) such that \(\gamma'_i \circ f \circ \gamma_i (x) = y\) and \(\gamma'_i \circ f \circ \gamma_i\) converges locally uniformly to \(g\).

In our main existence theorem 10, the bound on the distance between an \(L\)-Lipschitz map \(f\) that is stable at scale \(r\) and the harmonic map depends only on \(L\), \(r\), \(X\), and \(Y\). Hence from Corollary 3 we deduce:

If \(Y\) is Gromov hyperbolic, then the issue of transversality between points in the visual boundary of \(Y\) simplifies considerably; any two distinct points are transverse. Using Theorem 15, we recover in this case:

7 The universal Hitchin component of \(\PGL_d(\R)\)↩︎

In this section we study boundary maps of \(\Pi\)-quasi-isometric embeddings from \(\H^2\) to the symmetric space \(Y_d\) of \(\PGL_d(\R)\).

In the first part of this section, we discuss positivity of configurations of flags, and the finite non-transversality of positive circles in the flag variety. A recent theorem of Saldanha-Shapiro-Shapiro in fact gives an explicit upper bound on the number.

In the remainder of this section, we discuss cross ratios and quasisymmetry of positive configurations of flags, and prove Theorem 7 that the universal Hitchin component can be described purely in terms of boundary maps.

7.1 Positivity in \(\Flag(\R^d)\)↩︎

Let \(N\) be the subgroup of \(\SL_d(\R)\) of strictly upper triangular matrices: \[N \mathrel{\vcenter{:}}= \left\{ \begin{bmatrix} 1 & \ast & \cdots & \ast \\ & \ddots & & \vdots \\ & & 1 & \ast \\ 0 & & & 1 \end{bmatrix} \right\} .\] An element \(n \in N\) is called totally positive if all if its minors which can be positive are positive. In particular, all the “right-justified" minors, i.e.those involving any \(k\) rows and the last \(k\) columns, are positive.

We let \[\sigma_\infty = \langle e_1 \rangle \subset \langle e_1,e_{2} \rangle \subset \cdots \subset \langle e_1,e_{2}, \dots, e_{d-1} \rangle\] denote the standard ascending flag, and let \[\sigma_0 = \langle e_d \rangle \subset \langle e_d,e_{d-1} \rangle \subset \cdots \subset \langle e_d,e_{d-1}, \dots, e_{2} \rangle\] denote the standard descending flag. Note that \(N\) is the unipotent radical of the stabilizer of \(\sigma_\infty\) in \(\SL_d(\R)\). An arbitrary triple of flags \((\sigma_1,\sigma_2,\sigma_3)\) is called positive if there exists \(g \in \PGL_d(\R)\) and a totally positive \(n \in N\) such that \(g(\sigma_1,\sigma_2,\sigma_3) = (\sigma_0, n \sigma_0, \sigma_\infty)\).

The following fact is standard, and the proof works for any flag manifold \(\mathcal{F}_\Theta = G/P_\Theta\) supporting a positive structure in the sense of Guichard-Wienhard [31]. Recall that a continuous map \(\xi \colon \RP^1 \to \Flag(\R^d)\) is called transverse when it takes distinct pairs to transverse flags.

The fact that positive maps yield generically transverse measures is a direct consequence of a theorem of Saldanha-Shapiro-Shapiro. They prove that, if \(\Lambda\) is the image of a positive map, then each vertex \(v\) of \(\partial_\infty Y\) is non-transverse to \(\Lambda\) at finitely many points. In fact, they give an explicit upper bound.

7.2 Proof of Theorem 6↩︎

Let \(Y_d\) be the symmetric space of \(\PGL_d(\R)\).

Recall (Definition 2) that we define the universal Hitchin component \(\mathrm{UHit}_d\) to be the space of \(\Pi\)-quasi-isometric embeddings from \(\H^2\) to \(Y_d\) with positive boundary map from \(\RP^1\) to \(\mathrm{Flag}(\R^d)\), up to bounded distance. With this definition, Corollary 6 says:

Using heat kernel estimates on \(\H^2\), we also obtain a positive lower bound on the drift (Definition 7) of an \((M,\Pi)\)-quasi-isometric embedding.

Extending our techniques to other higher Teichmüller spaces hinges on the following question.

For example, when \(G=\SO_0(2,n+1)\), for \(n \ge 0\), there is a subset \(\Theta\) of \(\Pi\) so that \((G,\Theta)\) admits a positive structure and moreover \(\mathcal{F}_\Theta = \Ein^{1,n}\), the Einstein universe. When \(n \ge 2\), the restricted root system has type \(B_2\) and \(\Theta\) is the singleton consisting of the long simple root. Using the fact that positive maps \(\xi \colon S^1 \to \Ein^{1,n}\) are the same as spacelike maps in the conformal Lorentzian structure, it is not hard to verify that they satisfy the finite non-transversality property. Then, by following the proof of Corollary 6 with the previous sentences replacing Theorem 16, any \(\Theta\)-quasi-isometric embedding \(f \colon \H^2 \to \SO_0(2,n+1)/\SO(2)\times \SO(n+1)\) with positive boundary map \(\partial f \colon \partial_\infty \H^2 \to \Ein^{1,n}\) is at bounded distance from a unique harmonic map.

7.3 Positive quasi-symmetric maps↩︎

We always consider \(\partial_\infty \H^2 = \RP^1\).

7.3.1 \(\PGL_2(\R)\)↩︎

The cross ratio of four points in \(\RP^1\) is \[\label{eqn:32cross32ratio} CR(\lambda_1,\lambda_2,\lambda_3,\lambda_4) = \frac{(\lambda_4 - \lambda_3)(\lambda_2 - \lambda_1)}{(\lambda_3 - \lambda_2)(\lambda_1 - \lambda_4)}\tag{18}\] With this convention \[CR(x,0,1,\infty) = x.\] In particular the cross ratio of four dihedrally ordered points is negative. Let \((\RP^1)^{(k)}\) denote the space of pairwise distinct \(k\)-tuples in \(\RP^1\), and \((\RP^1)^{(k)+}\) the space of dihedrally ordered pairwise distinct \(k\)-tuples. If \(\phi\) is a homeomorphism from \(\RP^1\) to \(\RP^1\), we write \(\phi^k\) for the induced self-homeomorphism of \((\RP^1)^{(k)}\).

A homeomorphism \(\phi: \RP^1 \to \RP^1\) is called \(K\)-quasisymmetric if \[\label{eqn:32K32quasi-symmetric32to32RP1} \frac{1}{K} \leq -CR(\phi^4(\tau)) \leq K\tag{19}\] for all \(\tau \in (\RP^1)^{(4)}\) such that \(-CR(\tau) = 1\).

On the other hand a homeomorphism \(\phi: \R \to \R\) is called \(K\)-quasisymmetric if \[\label{eqn:32K32quasi-symmetric32to32R} \frac{1}{K} \le \frac{\phi(x+t) - \phi(x)}{\phi(x)-\phi(x-t)} \le K\tag{20}\] Since \(CR(x-t,x,x+t,\infty)=-1\), the restriction of a \(K\)-quasisymmetric self-homeomorphism of \(\RP^1\) fixing \(\infty\) to a self-homeomorphism of \(\R\) is still \(K\)-quasisymmetric. On the other hand, it is a standard lemma in the theory of quasi-symmetric maps that if \(\phi\colon \R \to \R\) is \(K\)-quasisymmetric then the extension to a homeomorphism of \(\RP^1\) is \(K'\)-quasisymmetric for some explicit \(K'(K)\).

We also define, for \(\tau \in (\RP^1)^{(4)+}\) the log-cross-ratio \[cr(\tau) \mathrel{\vcenter{:}}= \log(-CR(\tau)).\]

From this lemma follows both the Hölder continuity of quasisymmetric maps, and the compactness of the family of normalized \(K\)-quasisymmetric maps, i.e.those satisfying \(\phi^3(0,1,\infty) = (0,1,\infty)\).

7.3.2 \(\PGL_d(\R)\)↩︎

In order to define quasi-symmetric maps to the flag variety \(\mathrm{Flag}(\R^d)\), we first generalize the cross-ratio and log-cross-ratio to four-tuples of flags.

A dihedrally ordered \(k\)-tuple of flags is called positive if it can be extended to a positive map of the circle. Any pairwise-transverse four-tuple of flags is equivalent, up to the action of \(\mathrm{PGL}_d(\R)\), to one of the form \[\label{eqn:32standard324-tuple} (m \cdot \sigma_0, \sigma_0, n \cdot \sigma_0, \sigma_\infty).\tag{21}\] where \(m\) and \(n\) are unipotent upper triangular matrices. The four-tuple is positive if \(m\) and \(n\) can be chosen such that \(n\) and \(m^{-1}\) are totally positive.

Noting that \(CR_i\) is always negative, we define:

We give another interpretation of this cross ratio in terms of projections from triples of flags to the symmetric space. Let \(H\) be the Cartan subgroup of diagonal matrices. Then \(H\) acts on the space of positive triples \((\sigma_0,n\cdot \sigma_0, \sigma_\infty)\) by conjugation on the matrix \(n\). Since \(H\) acts simply transitively on the superdiagonal of \(n\), there is always a unique representative of the triple such that \(n_{i,i+1} = 1\) for all \(i\).

Since the stabilizer in \(\mathrm{PGL}_d(\R)\) of a normalized triple of flags is trivial, any positive triple of flags determines a unique element of the group \(G\), and so we may choose any \(o\) in the flat \(F(\sigma_0,\sigma_\infty)\) and define:

Note that \(p_d(\sigma,\sigma', \sigma'')\) always lies in the flat determined by \(\sigma\) and \(\sigma''\). Since the connected component \(H_0 = \exp(\mathfrak{a})\) of \(H\) acts simply transitively on the flat determined by \(\sigma_0\) and \(\sigma_\infty\), for any pair of positive triples \[(\sigma_0, n \cdot \sigma_0, \sigma_\infty), (\sigma_0, m \cdot \sigma_0, \sigma_\infty)\] there is always a unique element \(v \in \mathfrak{a}\) such that \[\mathrm{exp}(v) \cdot p(\sigma_0, n \cdot \sigma_0, \sigma_\infty) = p(\sigma_0, m \cdot \sigma_0, \sigma_\infty).\] Moreover, when the quadruple \[(m \cdot \sigma_0, \sigma_0, n \cdot \sigma_0, \sigma_\infty)\] is positive, the relationship with the cross-ratio is given by:

With our generalized cross ratios in hand, the obvious definition of quasisymmetric is:

Let \(\phi\colon \RP^1 \to \mathrm{Flag}(\R^d)\) be a normalized positive map. Then for all \(t\in\R\), \[\phi(t) = n(t)\phi(0)\] where \(n(t) \in N\), the unipotent radical of the stabilizer of \(\sigma_\infty\). Writing \[n(t) = \begin{bmatrix} 1 & n_{1,2}(t) & * & * \\ & \ddots & \ddots & * \\ & & 1 & n_{d-1,d}(t) \\ & & & 1 \end{bmatrix},\] we denote by \(n^1(t)\) the superdiagonal, i.e.the vector of entries of \(n\) one above the diagonal. From positivity, one sees that each component \(n_{i,i+1}\) of \(n^1(t)\) is a monotonic function of \(t\), and from continuity, one deduces that in fact each component of \(n^1(t)\) is a homeomorphism from \(\R\) to \(\R\). The map \(\phi\) is normalized if and only if each homeomorphism \(n_{i,i+1}\) is normalized in the sense that it sends 0 to 0 and 1 to 1.

From the definition of our generalized cross-ratios, one sees immediately that if \(\phi\) is \(K\)-quasisymmetric, then each \(n_{i,i+1}\) is a \(K\)-quasisymmetric homeomorphism from \(\R\) to \(\R\).

In fact, one can recover a normalized positive map from the \(d-1\) homeomorphisms \(n_{i,i+1}\). This is the content of the following proposition.

For differentiable curves, Fock-Goncharov observed this in [27].

7.4 Quasiperiodicity↩︎

By Corollary 2, the family of positive maps arising as boundary maps of \(L\)-Lipschitz \((M,\Pi)\)-quasi-isometric embeddings is quasiperiodic.

From the continuity of the cross-ratio on the space of positive 4-tuples, one sees immediately that any quasi-periodic family is \(K\)-quasisymmetric for some \(K\). In particular,

Our other example of a quasi-periodic family is the family of \(K\)-quasisymmetric positive maps.

We will use the following lemma twice. The proof is obvious, just like the proof of Proposition [prop:32piqie32implies32qs].

7.5 Extending quasisymmetric maps↩︎

Recall that \(Y_d\) is the symmetric space for \(\PGL_d(\R)\). The following two theorems allow us to establish a correspondence between equivalence classes of positive \(\Pi\)-quasi-isometric embeddings from \(\H^2\) to \(Y_d\) and their boundary maps.

Such maps are unique up to bounded distance:

7.5.1 Uniqueness↩︎

We begin with uniqueness, Theorem 18. We prove two lemmas first.

Let \(\rho\) be the half-sum of all positive roots: \[\rho = \frac{1}{2} \sum_{\alpha \in \Pi} \alpha ,\] sometimes called the Weyl vector, which lies in the interior of \(\mathfrak{a}^+\) and is invariant by the opposition involution (these are the only two properties of \(\rho\) we will use).

For \(\sigma \in \Flag(\R^d)\) we write \(\rho(\sigma)\) for the element of \(\partial_\infty Y\) with type \(\rho\) in the chamber corresponding to \(\sigma\). For \(\xi \in \partial_\infty Y\) we write \(\delta_\xi\) for the Dirac measure of \(\xi\) on \(\partial_\infty Y\). We may sometimes suppress \(\rho\), and write \(b_\sigma = b_\rho(\sigma)\) for the Busemann function associated to \(\rho(\sigma) \in \partial_\infty Y\).

The second lemma is quite general.

7.5.2 Existence↩︎

Let \(\phi: \RP^1 \to \mathrm{Flag}(\R^d)\) be a positive map. The \(\PGL_d(\R)\)-equivariant projection \(p_d\) (Definition 20) induces a map \[p_d \circ \phi^3: (\RP^1)^{(3)} \to \mathrm{Flag}(\R^d)^{(3)+} \to Y.\]

On the other hand, \(p_2: (\RP^1)^{(3)} \to \H^2\) naturally identifies \((\RP^1)^{(3)}\) with the unit tangent bundle of \(\H^2\), and we metrize \((\RP^1)^{(3)}\) correspondingly.

Since the fibers of \(p_2\) are bounded diameter, \(p_2\) is a quasi-isometric embedding. Therefore, any section \(s: \H^2 \to (\RP^1)^{(3)}\) gives a coarse inverse, and the composition \(p_d \circ \phi^3 \circ s\) is still \(L\)-coarse Lipschitz for a (possibly new) \(L\) not depending on \(s\). Fix any \(s\).

This completes the proof of existence.

References↩︎