Last September 28, Yves Benoist gave a beautiful talk on the occasion of the workshop Geometry and Dynamics on Moduli Spaces (that is, one of the four 2015 Clay Research Workshops) about his joint work with Dominique Hulin on a generalized version of the so-called *Schoen-Li-Wang conjecture* (on harmonic maps within bounded distance to given quasi-isometries of symmetric spaces of rank one).

Remark 1Still concerning beautiful talks delivered in this conference, I strongly recommend taking a look at this video of Peter Scholze’s talk on cohomology of algebraic varieties: indeed, I think that he was extremely sucessful in communicating his results to a broad audience of non-experts in Algebraic Geometry (such as myself).

This post is a transcription of my notes for Yves’ lecture and, as usual, all errors/mistakes below are my entire responsibility.

**1. Harmonic maps and quasi-isometries **

Definition 1Let and be Riemannian manifolds. An harmonic map is a critical point of Dirichlet energy

Equivalently, is harmonic whenever it satisfies the Euler-Lagrange equation

associated to Dirichlet energy.

Example 1Constant maps, geodesics ( interval) and, more generally, isometries with totally geodesic images are harmonic maps.

The literature dedicated to the questions of existence, uniqueness and regularity of harmonic maps is vast: see for instance these references here. In particular, we know that if is a simply connected non-positively curved Riemannian manifold, then any harmonic map is smooth () and attains the minimum of Dirichlet energy among all maps coinciding with outside a compact subset of .

Definition 2Given , a map between two metric spaces and is a –quasi-isometry whenever

for all .

A conjecture of Schoen-Li-Wang predicts the existence and uniqueness of an harmonic map within bounded distance to any given quasi-isometric self-map from a symmetric space of rank one.

**Conjecture (Schoen-Li-Wang).** Let be a non-compact symmetric space of rank one, i.e., is a hyperbolic space , , or (where is Hamilton quaternion algebra), or (where is Cayley octonion algebra).

Given a quasi-isometry , there exists an unique harmonic map within bounded distance to in the sense that

Remark 2Schoen made this conjecture for . Subsequently, Li-Wang proposed the generalized version of Schoen’s conjecture described above and they proved the uniqueness part of this conjecture.

Remark 3Before these conjectures were formulated, Pansu had already established them for , and .

Remark 4More recently, Markovic solved the case of Li-Wang conjecture and the initial conjecture of Schoen (i.e., the case ) in these papers here and here. Also, Lemm-Markovic confirmed the Li-Wang conjecture for the case , , in this paper here.

The purpose of this post is to discuss the following theorem of Benoist-Hulin establishing a *generalized* version of Schoen-Li-Wang conjecture:

Theorem 3 (Benoist-Hulin)Let and be non-compact symmetric spaces of rank one. Given a quasi-isometry , there exists an unique harmonic map such that

This result brings (at least) two novelties in comparison with previous theorems in the literature because:

- it settles Schoen-Li-Wang conjecture for (the remaining cases of) quasi-isometries of complex hyperbolic spaces, and
- it deals with quasi-isometries between and with different dimensions such as .

**2. Sketch of proof of Benoist-Hulin theorem **

For the sake of exposition, we will sketch the proof of Theorem 3 for quasi-isometries , i.e., **from now on** we will take and .

We begin by noticing that it is sufficient to show the *existence* of an harmonic map within bounded distance from : indeed, as we already told in Remark 2, the uniqueness of follows from the work of Li-Wang.

** 2.1. Regularization of quasi-isometries **

The first step in the proof of Theorem 3 is to *regularize* : by using bump functions, Benoist-Hulin show (with a 2 pages straighforward calculation) that is within bounded distance to a quasi-isometry whose covariant derivatives are bounded on for all (cf. Proposition 3.4 in Benoist-Hulin paper).

In other words, by replacing by if necessary, we can assume that is a quasi-isometry with and for some constant .

** 2.2. Reduction to a priori -estimates **

The second step is to reduce the construction of to an *a priori* -estimate (through a standard *compactness argument*).

More precisely, let us fix an origin , and let us consider the closed balls in . Next, we take the unique harmonic map satisfying the Dirichlet boundary condition on : the map minimizes the Dirichlet energy

among all maps with on . (These facts were proved by Schoen [see here] and Schoen-Uhlenbeck here and here.)

In this context, the existence of in Theorem 3 can be reduced to an uniform estimate on the distances between and :

Proposition 4Suppose that there exists such that

for all . Then, the sequence converges uniformly on compact subsets of to an harmonic map with

Before showing this proposition, we need to recall a (particular case of a) key lemma due to Cheng (see also Lemma 3.3 in Benoist-Hulin paper) allowing to “upgrade” estimates on harmonic maps into bounds on them.

Lemma 5 (Cheng)Let and a harmonic map such that the following estimate holds:

for some . Then, we have the following estimate on at :

Let us now use this lemma to prove Proposition 4.

*Proof:* By assumption, there exists such that for all , and, furthermore, for some . It follows that, given any increasing sequence , we have

for all large enough.

By Cheng’s lemma 5, this estimate leads to the following bound

for all and sufficiently large.

By Arzela-Ascoli theorem, we can find a subsequence converging uniformly on every ball , , to a continuous map with

Moreover, the harmonic maps minimize the Dirichlet energy and, on each ball , their energies are uniformly bounded

thanks to (1). By a compactness theorem of Luckhaus, this implies that the limiting map is harmonic and minimizes the Dirichlet energy.

Finally, the convergence , , of *whole* sequence follows from the convergence along subsequences and the uniqueness theorem of Li-Wang (see Remark 2) ensuring that the harmonic map with is unique.

In summary, Proposition 4 reduces the proof of Theorem 3 to

Theorem 6 (Benoist-Hulin)There exists such that

for all .

The proof of this theorem has two components:

*Boundary estimates*: one estimates for near ;*Interior estimates*: one estimates for far from .

Remark 5For the interior estimates, Benoist-Hulin use a proof by contradiction, i.e., they rule out the situation where as . Nevertheless, a careful inspection of their arguments shows that can be quantitavely bounded in terms of : see Remark 6 below.

** 2.3. Boundary estimates **

The behavior of for near is controlled by the following result:

Proposition 7Then,Let be a smooth map with and for some . Fix and denote by the harmonic map from the closed ball to with on .

for all .

*Proof:* Given , take closest to it, i.e., . Since and , one has

In order to estimate , we will use a barrier function. More precisely, we consider the geodesic passing through and , and we select a point in this geodesic with the following properties:

- is very far from , i.e., for all ;
- .

Since is harmonic, the function is subharmonic. Moreover, this function coincides with the smooth function on . Therefore, the maximum principle tells us that

for all (with equality for ), where is the smooth harmonic function coinciding with on .

In particular, our choice of implies

The barrier function can be controlled thanks to an estimate of Anderson-Schoen. More concretely, the function vanishes on and a short computation (at page 15 of Benoist-Hulin paper) reveals that its Laplacian verifies

In this context, the estimate of Anderson-Schoen (see Proposition 2.4 in Benoist-Hulin paper) asserts that

Therefore, this estimate together with (2) gives

This proves the proposition.

** 2.4. Interior estimates **

The boundary estimate in Proposition 7 says that a point with

For the sake of contradiction, let us suppose that as . Consider the polar exponential coordinates centered at . We will use these coordinates to study and on a ball of radius . More concretely, if denotes the angle between , then we will contradict the triangle inequality

by adapting an idea of Markovic to find in the set

(where is the geodesic connecting to ) such that

In fact, this will be a contradiction to (4) because we will prove that

as for all .

Pictorially, the previous paragraph is summarized in Figure 1 of Benoist-Hulin paper.

Formally, we proceed as follows:

- first, we give upper bounds for when and ;
- secondly, we give upper bounds for when and for all ;
- finally, we give lower bounds on for some .

**2.4.1 Upper bounds on **

for all with .

*Proof:* Consider as above. The triangle of vertices has sides of lengths:

- (by definition of );
- (because and is -quasi-isometric);
- (by our assumption on ).

In particular, the sum of the lengths of the sides of which are adjacent to the angle is much *bigger* than the length of the opposite side to this angle:

By “elementary hyperbolic geometry”, i.e., the fact that the angle is controlled by the Gromov product (i.e., the “excess” in the triangle inequality)

through the relation

(cf. Lemma 2.1 in Benoist-Hulin paper), we deduce from (5) that

This proves the lemma.

**2.4.2 Upper bounds on **

for all with for all (where is the geodesic from to ).

*Proof:* Note that

by our choice of the polar exponential coordinates at .

In order to estimate , we use a lemma of Gauss saying that

for all .

From the previous two estimates and our assumption that , we deduce that

Finally, we control with the aid of Cheng’s lemma. More precisely, because

for all (since and by hypothesis). Thus, Cheng’s lemma 5 ensures that

In summary, the previous estimate show that

because .

**2.4.3 Lower bounds on **

We start our quest for a point with far away from zero by estimating the Lebesgue measure of .

Lemma 10Suppose that and denote by the normalized spherical Lebesgue measure on . Then,

*Proof:* We write where

and

(where is the geodesic path between and ).

For the sake of exposition, we will discuss *only* the estimate of : indeed, the estimate of in Lemma 4.4 of Benoist-Hulin is very similar (up to a minor technicality which is treated with the aid of Cheng’s lemma) and we refer the curious reader to pages 18 and 19 of the original paper for more details.

The key tool to bound is the following Green formula:

The proof of this Green formula uses the subharmonicity of (which is a consequence of the subharmonicity of the distance function of and the harmonicity of ). More precisely, denote by the compact group of isometries of ) fixing , and let be its Haar measure.

Note that acts transitively on each sphere , . Thus, the subharmonic function takes a constant value on , so that

for all . Because , this proves (6).

Next, we observe that for all : in fact,

Therefore, by combining this estimate with (6), we see from the definition of that

i.e.,

. This completes our sketch of proof of the lemma (modulo the verification of the bound ).

Using Lemma 10, we can find with the property that is far away from zero:

*Proof:* Let us set . By Lemma 10,

(because .)

Since any subset of the Euclidean sphere with normalized Lebesgue measure has diameter , for sufficiently large, we can find such that

where is the angle between the vectors such that

Before proceeding further, we need to recall some facts about the relationship between angles of a triangle in a hyperbolic space and Gromov products (cf. Lemmas 2.1 and 2.2 in Benoist-Hulin paper). Let be a geodesic triangle in with vertices and denote by the corresponding angle at . Then,

- (a) ;
- (b) ;
- (c) whenever ;
- (d) if is a -quasi-isometry, then
where is a constant depending only on .

We use these facts as follows. Note that

On the other hand, *if* we can show that

*then* the items (c), (d), (b) and the estimate (7) ensure that

which would complete the proof of the lemma (by taking or ) in view of (8) (because and are independent of ).

In order to check (9), it is sufficient to verify that

thanks to item (d) above. For this sake, we observe that the item (a) and (7) imply that

because . It follows that the desired estimate (10) holds whenever is sufficiently large so that

This proves the lemma.

At this point, the proof of Theorem 6 is complete: indeed, it suffices to put together the estimate (4) with Lemmas 8, 9 and 11 to derive a contradiction with a scenario where as .

We close this post with the following comment about the arguments of Benoist-Hulin.

Remark 6As we already mentioned, the arguments of Benoist-Hulin provide a quantitative bound on in terms of with , . Indeed, the proofs of the lemmas above show that we get a contradiction when the quantities and satisfy the following inequalities:

- ;
- , and

where is the constant provided by item (d) in the proof of Lemma 11.Since these numerical constraints are fulfilled by any for an appropriate (“explicit”) choice of , one gets the following quantitative version of Benoist-Hulin theorem 6: for all .

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