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Quasiisometric harmonic maps between rank one symmetric spaces

We prove that a quasiisometric map between rank one symmetric spaces is within bounded distance from a unique harmonic map. In particular, this completes the proof of the Schoen-Li-Wang conjecture.


Main result
We first explain the title.
A symmetric space is a Riemannian manifold X such that, for every point x in X, there exists a symmetry centered at x, i.e. an isometry s x of X fixing the point x and whose differential at x is minus the identity.
In this article we will call rank one symmetric space a symmetric space X whose sectional curvature is everywhere negative : K X < 0. The list of rank one symmetric spaces is well known. They are the real hyperbolic spaces H p R , the complex hyperbolic spaces H p C , the quaternion hyperbolic spaces H p Q , with p ≥ 2, and the Cayley hyperbolic plane H 2 Ca . A map f : X → Y between two metric spaces X and Y is said to be quasiisometric if there exists a constant c ≥ 1 such that f is c-quasiisometric i.e. such that, for all x, x ′ in X, one has (1.1) A map h : X → Y between two Riemannian manifolds X and Y is said to be harmonic if its tension field is zero i.e. if it satisfies the elliptic nonlinear partial differential equation tr(D 2 h) = 0 where D 2 h is the second covariant derivative of h. For instance, an isometric map with totally geodesic image is always harmonic. The problem of the existence, regularity and uniqueness of harmonic maps under various boundary conditions is a very classical topic (see [8], [15], [7], [34], [33] or [21]). In particular, when Y is simply connected and has non positive curvature, a harmonic map is always C ∞ i.e. it is indefinitely differentiable, and is a minimum of the energy functional -see Formula (3.3)-among maps that agree with h outside a compact subset of X.
The aim of this article is to prove the following Theorem 1.1. Let f : X → Y be a quasiisometric map between rank one symmetric spaces X and Y . Then there exists a unique harmonic map h : X → Y which stays within bounded distance from f i.e. such that The uniqueness of h is due to Li and Wang in [20,Theorem 2.3]. In this article we prove the existence of h.

Previous results and conjectures
When X is equal to Y , Theorem 1.1 was conjectured by Li and Wang in [20,Introduction] extending a conjecture of Schoen in [30] for the case X = Y = H 2 R . When these conjectures were formulated, the case where X = Y is either equal to H p Q or H 2 Ca was already known by a previous result of Pansu in [26]. In that case the harmonic map h is an onto isometry.
The uniqueness part of the Schoen conjecture was quickly settled by Li and Tan in [19], and the proof was extended by Li and Wang to rank one symmetric spaces in their paper [20]. Partial results towards the existence statement in the Schoen conjecture were obtained in [35], [14], [28], [22], [4]. A major breakthrough was then achieved by Markovic who proved successively the Li-Wang conjecture for the case X = Y = H 3 R in [24], for the case X = Y = H 2 R in [23] thus solving the initial Schoen conjecture, and very recently with Lemm for the case X = Y = H p R with p ≥ 3 in [18]. As a corollary of Theorem 1.1, we complete the proof of the Li-Wan conjecture. In particular, we obtain the following. Corollary 1.2. For p ≥ 1, any quasiisometric map f : H p C → H p C is within bounded distance from a unique harmonic map h : H p C → H p C . Another new feature in Theorem 1.1 is that one does not assume that X and Y have the same dimension. Even the following special case is new.
We finally recall that, according to a well-known result of Kleiner and Leeb in [17], every quasiisometric map f : X → Y between irreducible higher rank symmetric spaces stays within bounded distance of an isometric map, after a suitable scalar rescaling of the metrics. Another proof of this result has also been given by Eskin and Farb in [9].

Motivation
We now briefly recall a few definitions and facts that are useful to understand the context and the motivation of Theorem 1.1. None of them will be used in the other sections of this article.
Let X and Y be rank one symmetric spaces. Recall first that X is diffeomorphic to R k and has a visual compactification X ∪ ∂X. The visual boundary ∂X is homeomorphic to a topological sphere S k−1 . Choosing a base point O in X, this boundary is endowed with the Gromov quasidistance d ′ defined by d ′ (ξ, η) := e −(ξ|η) O for ξ, η in ∂X where (ξ|η) O denotes the Gromov product (see [12,Sec. 7.3]). A non-constant continuous map F : ∂X → ∂Y between the boundaries is called quasisymmetric if there exists The following nice fact, which is also true for a wider class of geodesic Gromov hyperbolic spaces, gives another point of view on quasiisometric maps : Two quasiisometric maps f, g : X → Y have the same boundary map ∂f = ∂g if and only if f and g are within bounded distance from one another. c) Any quasisymmetric map F : ∂X → ∂Y is the boundary map F = ∂f of a quasiisometric map f : X → Y .
This fact has a long history. Point a is in the paper of Mostow [25] and was extended later by Gromov in [13,Sec. 7] (see also [12,Sec. 7]). Point b is in the paper of Pansu [26,Sec. 9]. Point c is in the paper of Bourdon and Pajot [5, Section 2.2] extending previous results of Tukia in [36], of Paulin in [27], and of Bonk, Heinonen, Koskela in [3].
Recall that a diffeomorphism f of X is said to be quasiconformal if the function x → Df (x) Df (x) −1 is uniformly bounded on X. The original formulation of the Schoen conjecture involved quasiconformal diffeomorphisms instead of quasiisometries : Every quasisymmetric homeomorphism F of S 1 is the boundary map of a unique quasiconformal harmonic diffeomorphism of H 2 R . Relying on a previous result of Wan in [37, Theorem 13], Li and Wang pointed out in [20,Theorem 1.8] that a harmonic map between the hyperbolic plane H 2 R and itself is a quasiconformal diffeomorphism if and only if it is a quasiisometric map. This is why Li and Wang formulated in [20] the higher dimensional generalisation of the Schoen conjecture using quasiisometries instead of quasiconformal diffeomorphisms.
Note that in dimension k ≥ 6 there exist harmonic maps between compact manifolds of negative curvature which are homotopic to a diffeomor-phism but which are not diffeomorphisms (see [10]).

Strategy
To prove our Theorem 1.1, we start with a c-quasiisometric map f : X → Y between rank one symmetric spaces. We want to exhibit a harmonic map h : X → Y within bounded distance from f .
We will first gather in Chapter 2 a few properties of Hadamard manifolds : images of triangles under quasiisometric maps, Hessian of the distance function, gradient estimate for functions with bounded Laplacian.
The first key point in our proof is the simple remark that, thanks to a smoothing process, we may assume without loss of generality that the cquasiisometric map f is C ∞ and that its first and second covariant derivatives Df and D 2 f are uniformly bounded on X (Proposition 3.4). We fix a point O in X. For R > 0, we denote by B R := B(O, R) the closed ball in X with center O and radius R and by ∂B R the sphere that bounds B R . We introduce the unique harmonic map h R : B R → Y whose restriction to the sphere ∂B R is equal to f . This map h R is C ∞ on the closed ball B R . The harmonic map h will be constructed as the limit of the maps h R when R goes to infinity. In order to prove the existence of this limit h, using a classical compactness argument that we will recall in Section 3.3, we just have to check that on the balls B R the distances are uniformly bounded in R. We will argue by contradiction and assume that we can find radii R with ρ R arbitrarily large.
The second key point in our proof is what we call the boundary estimate (Proposition 3.8). It tells us that the ratio is uniformly bounded for R ≥ 1 and x in B R . In particular, when ρ R is large, the ball B(O, R − 1) contains a ball B(x R , r R ) whose center x R satisfies d(h R (x R ), f (x R )) = ρ R and whose radius r R ≥ 1 is quite large. A good choice for the radius r R will be r R = ρ 1/3 R . We will focus on the restriction of the maps f and h R to this ball B(x R , r R ). Let The third key point in our proof is to write for each point z on the sphere S(x R , r R ) the triangle inequality and, adapting an idea of Markovic in [23], to focus on the set The contradiction will come from the fact that when both R and ρ R go to infinity, the two angles θ 1 := θ(v f (z), v h (z)) and θ 2 := θ(v h (z), v R ) converge to 0 uniformly for z in W R , while one can find z = z R in W R such that the other angle θ 0 = θ(v f (z), v R ) stays away from 0. Here is a rough sketch of the arguments used to estimate these three angles.
To get the upper bound for the angle θ 1 (Lemma 4.5), we use the relation between angles and Gromov products (Lemma 2.1) and we notice that the set W R has been chosen so that the Gromov product (f (z)|h R (z)) y R is large.
To get the upper bound for the angle θ 2 (Lemma 4.6), we check that the gradient Dv h is uniformly small on the geodesic segment between x R and z. This follows from the comparison inequality 2 sinh(ρ h /2) Dv h ≤ Dh R , from the bound for Dh R that is due to Cheng (Lemma 3.3), and from the definition of W R that ensures that the factor sinh(ρ h /2) stays very large on this geodesic.
To find a point z = z R in W R such that the angle θ 0 is not small (Lemma 4.7), we use the almost invariance of the Gromov products -and hence of the angles-under a quasiisometric map (Lemma 2.2). We also use a uniform lower bound on the measure of W R (Lemma 4.4). This lower bound is a consequence of the subharmonicity of the function ρ h (Lemma 3.2) and of Cheng's estimate.
We would now like to point out the difference between our approach and those of the previous papers. The starting point of Markovic's method in [23] is the fact that any K-quasisymmetric homeomorphism of the circle is a uniform limit of K-quasisymmetric diffeomorphisms. This fact has no known analog in high dimension. The starting point of the methods in both [24] and [18] is the fact that a quasisymmetric homeomorphism of the sphere S k−1 is almost surely differentiable. This fact is not true on S 1 . Since our strategy avoids the use of quasisymmetric maps, it gives a unified approach for all H p R with p ≥ 2, and also works when X and Y have different dimensions.
Both authors thank the MSRI for its hospitality, the Simons Foundation and the GEAR Network for their support, in Spring 2015 at the beginning of this project.

Hadamard manifolds
In this preliminary chapter, we recall various estimates on a Hadamard manifold : for the angles of a geodesic triangle, for the Hessian of the distance function, and also for functions with bounded Laplacian.

Triangles and quasiisometric maps
We first recall basic estimates for triangles in Hadamard manifolds and explain how one controls the angles of the image of a triangle under a quasiisometric map.
All the Riemannian manifolds will be assumed to be connected and to have dimension at least two. We will denote by d their distance function.
A Hadamard manifold is a complete simply connected Riemannian manifold X of non positive curvature K X ≤ 0. For instance, the Euclidean space R k is a Hadamard manifold with zero curvature K X = 0, while the rank one symmetric spaces are Hadamard manifolds with negative curvature K X < 0. We will always assume without loss of generality that the metric on a rank one symmetric space X is normalized so that −1 ≤ K X ≤ −1/4. Let x 0 , x 1 , x 2 be three points on a Hadamard manifold X. The Gromov product of the points x 1 and x 2 seen from x 0 is defined as We recall the basic comparison lemma which is one of the motivations for introducing the Gromov product.
Lemma 2.1. Let X be a Hadamard manifold with −1 ≤ K X ≤ −a 2 < 0, let T be a geodesic triangle in X with vertices x 0 , x 1 , x 2 , and let θ 0 be the angle of T at the vertex x 0 .
The hyperbolic triangle equality reads as In particular, one has sin 2 (θ 0 /2) ≤ ℓ 1 −m ℓ 1 . This proves Point a. Points b and c follow from (2.2) and the basic inequalities : When the sectional curvature of X is pinched between −1 and −a 2 , the triangle comparison theorems of Alexandrov and Toponogov (see [16,Theorems 4.1 and 4.2]) ensure that the result also holds in X.
We now recall the effect of a quasiisometric map on the Gromov product.
Proof. This is a general property of quasiisometric maps between Gromov δ-hyperbolic spaces which is due to M. Burger. See [12,Prop. 5.15].

Hessian of the distance function
We now recall basic estimates for the Hessian of the distance function and of its square on a Hadamard manifold.
When x 0 is a point in a Riemannian manifold X, we denote by d x 0 the distance function defined by d x 0 (x) = d(x 0 , x) for x in X. We denote by d 2 x 0 the square of this function. When F is a C 2 function on X, we denote by DF the differential of F and by D 2 F the Hessian of F , which is by definition the second covariant derivative of F . Lemma 2.3. Let X be a Hadamard manifold and x 0 ∈ X. a) The Hessian of the square d 2 x 0 satisfies on X The general statement follows from this model case and the Alexandrov triangle comparison theorem. b) Assume first that X = H 2 R is the real hyperbolic plane with curvature −1. Using the expression cosh(ℓ t ) = cosh(ℓ 0 ) cosh(t) for the length ℓ t of the hypothenuse of a right triangle with side lengths ℓ 0 and t, one infers that The general statement follows by the same argument combined again with the Alexandrov and Toponogov triangle comparison theorems.

Functions with bounded Laplacian
We give a bound for functions defined on balls of a Hadamard manifold, when their Laplacian is bounded and their boundary value is equal to 0.
The Laplace-Beltrami operator ∆ on a Riemannian manifold X is defined as the trace of the Hessian. In local coordinates, the Laplacian of a function F reads as Then, for all x in B R , one has the upper bound Remark.
The assumption in Proposition 2.4 that the curvature is negative is essential. Indeed, the function G := R 2 − d 2 O on the ball B R of the Euclidean space X = R k satisfies (2.6) with M = 2k but does not satisfy (2.7) because the gradient of G at a point x on the sphere ∂B R has norm 2 R.
The proof of Proposition 2.4 relies on the following. Lemma 2.5. Let X be a Hadamard manifold with K X ≤ −a 2 and x 0 be a point of X. Then, the function d x 0 is subharmonic. More precisely, the distribution ∆ d x 0 −a is non-negative.
Proof of Lemma 2.5. Since X is a Hadamard manifold, Lemma 2.3 ensures that the function d x 0 is C ∞ on X \{x 0 } and satisfies ∆ d x 0 (x) ≥ a for x = x 0 . It remains to check that the distribution ∆ d x 0 −a is non-negative on X. The function d x 0 is the uniform limit when ε converges to 0 of the C ∞ functions d x 0 ,ε := (ε 2 + d 2 x 0 ) 1/2 . One computes their Laplacian on X \ {x 0 } : x 0 ) 1/2 . Since both sides are continuous functions on X, this inequality also holds on X. Since a limit of non negative distributions is non negative, one gets the inequality ∆d x 0 ≥ a on X by letting ε go to 0.
Proof of Proposition 2.4. According to Lemma 2.5, both functions one has ∆G ± ≤ 0. Since they vanish on the boundary ∂B R , the maximum principle ensures that these functions G ± are non-negative on the ball B R .

Harmonic maps
In this chapter we begin the proof of Theorem 1.1. We first recall basic facts satisfied by harmonic maps. We then explain why we can assume our c-quasiisometric map f to be C ∞ with bounded covariant derivatives. We also explain why an upper bound on d(h R , f ) implies the existence of the harmonic map h. Finally we provide this upper bound near the boundary ∂B R .

Harmonic maps and the distance function
In this section, we recall two useful facts satisfied by a harmonic map h : the subharmonicity of the functions d y 0 • h, and Cheng's estimate for the differential Dh.  Proof. The proof is similar to the proof of Lemma 2.5. We first recall the formula for the Laplacian of a composed function. Let f : X → Y be a C ∞ map and F ∈ C ∞ (Y ) be a C ∞ function on Y . Then one has where (e i ) 1≤i≤k is an orthonormal basis of the tangent space to X. Since Y is a Hadamard manifold, the continuous function ρ h = d y 0 • h is C ∞ outside h −1 (y 0 ). Using Formula (3.1), the harmonicity of h and Lemma 2.3, we compute the Laplacian on X \ h −1 (y 0 ) : The function ρ h is the uniform limit when ε go to 0 of the C ∞ functions ρ h,ε := (ε 2 + ρ 2 h ) 1/2 . We compute their Laplacian on X \ h −1 (y 0 ) : It follows that the inequality ∆ρ h,ε ≥ 0 also holds on the whole X.
One finally gets ∆ρ h ≥ 0 as a distribution on X by letting ε go to 0.
Another crucial property of harmonic maps is the following bound for their differential due to Cheng. Lemma 3.3. Let X, Y be two Hadamard manifolds with −b 2 ≤ K X ≤ 0. Let k = dim X, x 0 be a point of X, r 0 > 0 and let h : B(x 0 , r 0 ) → Y be a harmonic C ∞ map such that the image h(B(x 0 , r 0 )) lies in a ball of radius R 0 . Then one has the bound In the applications, we will use this inequality with b = 1 and r 0 = 1.
Proof. This is a simplified version of [6, Formula 2.9] obtained by keeping track of the constants in the proof.

Smoothing quasiisometric maps
The following proposition will allow us to assume in Theorem 1.1 that the quasiisometric map f is C ∞ with bounded covariant derivatives.
Proposition 3.4. Let X, Y be two symmetric spaces of non positive curvature and f : X → Y be a quasiisometric map. Then there exists a C ∞ quasiisometric map f : X → Y within bounded distance from f and whose covariant derivatives D p f are bounded on X for all p ≥ 1.
This regularized map f will be constructed as follows. Let α : R → R be a non-negative C ∞ function with support [−1, 1]. For x in X, we introduce the positive finite measure on X Let µ x := f * α x denote the image measure on Y . It is defined, for any positive function ϕ on Y , by We choose α so that each µ x is a probability measure. By homogeneity of X, this fact does not depend on the point x. We will define f (x) ∈ Y to be the center of mass of the measure µ x . To be more precise, we will need the following Lemma 3.5 which is an immediate consequence of Lemma 2.3.
Lemma 3.5. For x in X, let Q x be the function on Y defined for y in Y by For x in X, the functions Q x are proper and uniformly strictly convex. More precisely, for all x in X and y in Y , the Hessian admits the lower bound Proof of Proposition 3.4. For x in X, we define the point f (x) ∈ Y to be the center of mass of µ x , i.e. to be the unique point where the function Q x reaches its infimum. Equivalently, the point y = f (x) ∈ Y is the unique critical point of the function Q x , i.e. it is defined by the implicit equation Since the map f is c-quasiisometric, the support of the measure µ x lies in the ball B(f (x), 2 c). Since Y is a Hadamard manifold, the balls of Y are convex (see [2]) so that the center of mass f (x) also belongs to the ball B(f (x), 2 c). In particular, one has We now check that the map f : X → Y is C ∞ . Since the Hessians D 2 y Q x are nondegenerate, this follows from the implicit function theorem applied to the C ∞ map To prove that the first derivative of f is bounded on X, we first notice that Lemma 3.5 ensures that the covariant derivative is an invertible linear map with (D y Ψ(x, y)) −1 ≤ 1/2 .
We also notice that, since the point y = f (x) is at distance at most 4c of all the points f (z) with z in the ball B(x, 1), the norm is also uniformly bounded on X. Hence the norm D f of the derivative of f is uniformly bounded on X.
For the same reason, since Y is homogeneous, the norm of each covariant derivative D p x D q y Ψ(x, f (x)) is uniformly bounded on X for p, q ≥ 0. Hence the norm of each covariant derivative D p f is uniformly bounded on X for p ≥ 1.

Existence of harmonic maps
In this section we prove Theorem 1.1, taking for granted Proposition 3.6 below.
Let X, Y be rank one symmetric spaces and f : X → Y be a c-quasiisometric C ∞ map whose first two covariant derivatives are bounded.
We fix a point O in X. For R > 0, we denote by B R := B(O, R) the closed ball in X with center O and radius R and by ∂B R the sphere that bounds B R . Since the manifold Y is a Hadamard manifold, there exists a unique harmonic map h R : B R → Y satisfying the Dirichlet condition h R = f on the sphere ∂B R . Moreover, this harmonic map h R is energy minimizing. This means that the map h R achieves the minimum of the energy functional among all C 1 maps g on the ball which agree with f on the sphere ∂B R , i.e. one has These facts are due to Schoen (see [29] or [8,Thm 12.11]). Thanks to Schoen and Uhlenbeck in [31] and [32], the harmonic map h R is known to be C ∞ on the closed ball B R . We denote by the distance between these two maps.
The main point of this article is to prove the following uniform estimate.
Proposition 3.6. There exists a constant M ≥ 1 such that, for any R ≥ 1, Even though the argument is very classical, we first explain how to deduce our main theorem from this Proposition.
Proof of Theorem 1.1. As explained in Proposition 3.4, we may also assume that the c-quasiisometric map f is C ∞ with bounded covariant derivatives. Pick an increasing sequence of radii R n converging to ∞ and let h Rn : B Rn → Y be the harmonic C ∞ map which agrees with f on the sphere ∂B Rn . Proposition 3.6 ensures that the sequence of maps h Rn is locally uniformly bounded. More precisely there exists M ≥ 1 such that, for all S ≥ 1, for n large enough, one has Using the Cheng Lemma 3.3 with b = 1 and r 0 = 1, it follows that the derivatives are also uniformly bounded on the balls B S . More precisely one has, for all S ≥ 1, for n large enough, The Ascoli-Arzela theorem implies that, after extraction, the sequence h Rn converges uniformly on every ball B S towards a continuous map h : X → Y .
By construction this limit map h stays within bounded distance from the quasiisometric map f . We claim that the limit map h is harmonic. Indeed, the harmonic maps h Rn are energy minimizing and, on each ball B S , the energies of h Rn are uniformly bounded : Hence the Luckhaus compactness theorem for energy minimizing harmonic maps (see [34,Section 2.9]) tells us that the limit map h is also harmonic and energy minimizing.
Remark 3.7. By Li-Wang uniqueness theorem in [21], the harmonic map h which stays within bounded distance from f is unique. Hence the above argument also proves that the whole family of harmonic maps h R converges to h uniformly on the compact subsets of X when R goes to infinity.

Boundary estimate
In this section we begin the proof of Proposition 3.6 : we bound the distance between h R and f near the sphere ∂B R . Let h R : B R → Y be the harmonic C ∞ map whose restriction to the sphere ∂B R is equal to f . Then, for all x in B R , one has An important feature of this upper bound is that it does not depend on the radius R, provided the distance d(x, ∂B R ) remains bounded. This is why we call (3.4) the boundary estimate. The proof relies on an idea of Jost in [15,Section 4] Proof. Let x be a point in B R and w in ∂B R such that d(x, w) = d(x, ∂B R ). Since h R (w) = f (w), the triangle inequality reads as (3.5) The assumption on f ensures that To estimate the other term, we choose a point y 0 on the geodesic ray starting from h R (x) and passing by h R (w). This choice of y 0 ensures that one has the equality We also choose y 0 far enough so that This function F is then C ∞ on the ball B R . Let H : B R → R be the harmonic C ∞ function whose restriction to the sphere ∂B R is equal to F . By Lemma 3.2, since h R is a harmonic map, the function z → d(h R (z), y 0 ) is subharmonic on B R . Since this function is equal to H on the sphere ∂B R , the maximum principle ensures that with equality for z in ∂B R . Combining (3.7) and (3.9), one gets To estimate the right-hand side of (3.10), we observe that the function G := F − H vanishes on ∂B R and has bounded Laplacian : Indeed, using Formulas (3.1), (2.5) and (3.8), one computes Using Proposition 2.4, one deduces that and therefore, combining with (3.5), (3.6) and (3.10), one concludes that This proves (3.4).

Interior estimate
In this chapter we complete the proof of Proposition 3.6. We follow the strategy explained in the introduction (Section 1.4).

Notation
We first explain more precisely the notation and the assumptions that we will use in the whole chapter.
Let X and Y be rank one symmetric spaces and k = dim X. We start with a C ∞ quasiisometric map f : X → Y all of whose covariant derivatives are bounded. We fix a constant c ≥ 1 such that, for all x, x ′ in X, one has Note that the additive constant c on the right-hand side term of (1.1) has been removed since the derivative of f is bounded by c.
We fix a point O in X. For R > 0, we introduce the harmonic C ∞ map h R : B(O, R) → Y whose restriction to the sphere ∂B(O, R) is equal to f . We let We denote by x R a point of B(O, R) where the supremum is achieved : According to the boundary estimate in Proposition 3.8, one has, d(x R , ∂B(O, R)) ≥ 1 8kc 2 ρ R . When ρ R is large enough, we introduce a ball B(x R , r R ) with center x R , and whose radius r R is a function of R satisfying Note that this condition ensures the inclusion B(x R , r R ) ⊂ B(O, R−1). Later on, in section 4.5, we will assume that r R := ρ 1/3 R . We will focus on the restrictions of the maps f and h R to this ball B(x R , r R ). We will express the maps f and h R through the polar exponential coordinates (ρ, v) in Y centered at the point y R := f (x R ). For z in B(x R , r R ), we will thus write Note that ρ h and v h are shorthands for ρ h R and v h R . For simplicity, we do not write the dependance on R.
We denote by [x R , z] the geodesic segment between x R and z.
Definition 4.1. We introduce the following subsets of the sphere S(x R , r R ) :

Measure estimate
We first notice that one can control the size of ρ h (z) and of Dh R (z) on the ball B(x R , r R ). We will then give a lower bound for the measure of W R .
Proof. The triangle inequality and (4.2) give, for z in B(x R , r R ), Proof. For all z, z ′ in B(O, R) with d(z, z ′ ) ≤ 1, the triangle inequality and (4.2) yield Applying Cheng's lemma 3.3 with b = 1 and r 0 = 1, one gets for all z in B(O, R−1) the bound Dh R (z) ≤ 2 8 k ρ R .
We now give a lower bound for the measure of W R . We will denote by the same letter σ the probability measure on each sphere S(x R , r R ) that is invariant under all the isometries of X that fix the point x R .
Proof. The proof relies on the subharmonicity of the function ρ h on the ball B(x R , r R ) (see Lemma 3.2). We claim that Let us give a short proof of this special case of the Green formula. Since X is a symmetric space, the group Γ of isometries of X that fix the point x R is a compact group which acts transitively on the spheres S(x R , t). Let dγ be the Haar probability measure on Γ. The function F := Γ ρ h • γ dγ, defined as the average of the translates of the function ρ h under Γ, is equal to a constant F t on each sphere S(x R , t) of radius t ≤ r R . By the maximum principle applied to this subharmonic function F , one gets F 0 ≤ F t for all t ≤ r R . Since F 0 = ρ h (x R ) = ρ R , this proves (4.5).
First step. We prove (4.6) By Lemma 4.2, the function ρ h is bounded by ρ R + c r R , hence Equation For z in the complementary subset V c R ⊂ S(x R , r R ), we define We claim that, for each z in V c R , one has Indeed, the length of the curve t → h R (z t ) between t = s z and t = t z is at least ρ R 4 . Hence, using Lemma 4.3, one gets which prove (4.8).
The Green formula also gives the following variation of (4.5) By Lemma 4.2, the function ρ h is bounded by ρ R + c r R , hence Equation (4.9) implies Using the bound ρ h (z t ) ≤ 3 4 ρ R , for all t in the interval [s z , t z ], one deduces from (4.8) and (4.10) that This proves (4.7). Since W R = U R ∩ V R , the bound (4.4) follows from (4.6) and (4.7).

Upper bound for
For all v in U R , we give an upper bound for the angle between v f (z) and v h (z).
For two vectors v 1 , v 2 of the unit sphere T 1 y R Y of the tangent space T y R Y , we denote by θ(v 1 , v 2 ) the angle between these two vectors. Proof. For z in U R , we consider the triangle with vertices y R , f (z) and h R (z). Its side lengths satisfy Since K Y ≤ −1/4, applying Lemma 2.1 with a = 1 2 , one gets θ(v f (z)), v h (z)) ≤ 4 e − 1 4 (ℓ 1 +ℓ 2 −ℓ 0 ) ≤ 4 e

Upper bound for θ(v h (z), v R )
For all v in V R , we give an upper bound for the angle between v h (z) and v R . Proof. Let us first sketch the proof. We recall that the curve t → z t , for 0 ≤ t ≤ r R , is the geodesic segment between x R and z. By definition, for each z in V R , the curve t → h R (z t ) lies outside of the ball B(y R , ρ R /2) and by Cheng's bound on Dh R (z t ) one controls the length of this curve. We now detail the argument. For z in V R , we have the inequality Since Hence since the point z belongs to V R one deduces Hence using Lemma 4.3 one gets θ(v h (z), v R ) ≤ 2 8 k ρ R r R 2 sinh(ρ R /4) .

Lower bound for θ(v f (z), v R )
When ρ R is large enough, we find a point z = z R in W R for which the angle between v f (z) and v R is bounded below.  Assume that there exists a sequence of radii R going to infinity such that ρ R goes to infinity. Then, choosing r R := ρ 1/3 R , the diameters diam({v f (z) | z ∈ W R }) do not converge to 0 along this sequence.
There exists ε 0 > 0 such that every subset W of the Euclidean sphere S k−1 whose normalized measure is at least σ 0 contains two points whose angle is at least ε 0 .
Hence if R and ρ R are large enough one can find z 1 , z 2 in W R such that θ x R (z 1 , z 2 ) ≥ ε 0 and r R ≥ (A + 1)c sin 2 (ε 0 /2) , (4.13) where θ x R (z 1 , z 2 ) is the angle between z 1 and z 2 seen from x R , and where A is the constant given by Lemma 2.2. According to Lemma 2.1.a, we infer that min((x R |z 1 ) z 2 , (x R |z 2 ) z 1 )) ≥ (A + 1)c.
This proves our claim.
End of the proof of Proposition 3.6. Assume that there exists a sequence of radii R going to infinity such that ρ R goes also to infinity. We set r R = ρ 1/3 R . Using Lemmas 4.5 and 4.6 and the triangle inequality, one gets This contradicts Lemma 4.7.