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 1 Still 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 1 Let 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 1 Constant 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 2 Given , 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.
Given a quasi-isometry , there exists an unique harmonic map within bounded distance to in the sense that
Remark 3 Before these conjectures were formulated, Pansu had already established them for , and .
Remark 4 More 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:
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
In this context, the existence of in Theorem 3 can be reduced to an uniform estimate on the distances between and :
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.
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
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
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.
for all .
The proof of this theorem has two components:
- Boundary estimates: one estimates for near ;
- Interior estimates: one estimates for far from .
Remark 5 For 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:
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 ).
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
2.4.3 Lower bounds on
We start our quest for a point with far away from zero by estimating the Lebesgue measure of .
Proof: We write where
(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 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
. 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,
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 .
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.
We close this post with the following comment about the arguments of Benoist-Hulin.
Remark 6 As 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 .