Posted by: matheuscmss | November 18, 2015

Harmonic quasi-isometries (after Benoist and Hulin)

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 ${M}$ and ${N}$ be Riemannian manifolds. An harmonic map ${h:M\rightarrow N}$ is a critical point of Dirichlet energy

$\displaystyle E(h):=\frac{1}{2}\int_M \|dh\|^2 d vol_M$

Equivalently, ${h:M\rightarrow N}$ is harmonic whenever it satisfies the Euler-Lagrange equation

$\displaystyle \textrm{tr}(D^2h)=0$

associated to Dirichlet energy.

Example 1 Constant maps, geodesics ${\gamma:I\rightarrow N}$ (${I\subset \mathbb{R}}$ 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 ${N}$ is a simply connected non-positively curved Riemannian manifold, then any harmonic map ${h:M\rightarrow N}$ is smooth (${C^{\infty}}$) and ${h}$ attains the minimum of Dirichlet energy among all maps coinciding with ${h}$ outside a compact subset of ${M}$.

Definition 2 Given ${c\geq 1}$, a map ${f:X\rightarrow Y}$ between two metric spaces ${X}$ and ${Y}$ is a ${c}$quasi-isometry whenever

$\displaystyle \frac{1}{c}d_X(x,x')-c\leq d_Y(f(x),f(x'))\leq c d_X(x,x')+c$

for all ${x,x'\in X}$.

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 ${X}$ be a non-compact symmetric space of rank one, i.e., ${X}$ is a hyperbolic space ${\mathbb{H}^p_{\mathbb{K}}}$, ${p\geq 2}$, ${\mathbb{K} = \mathbb{R}, \mathbb{C}}$ or ${\textbf{H}}$ (where ${\textbf{H}}$ is Hamilton quaternion algebra), or ${\mathbb{H}^2_{\textbf{O}}}$ (where ${\textbf{O}}$ is Cayley octonion algebra).

Given a quasi-isometry ${f:X\rightarrow X}$, there exists an unique harmonic map ${h:X\rightarrow X}$ within bounded distance to ${f}$ in the sense that

$\displaystyle \sup\limits_{x\in X} d(h(x),f(x))<\infty$

Remark 2 Schoen made this conjecture for ${X=\mathbb{H}^2_{\mathbb{R}}}$. Subsequently, Li-Wang proposed the generalized version of Schoen’s conjecture described above and they proved the uniqueness part of this conjecture.

Remark 3 Before these conjectures were formulated, Pansu had already established them for ${X=\mathbb{H}^p_{\textbf{H}}}$, ${p\geq 2}$ and ${\mathbb{H}^2_{\textbf{O}}}$.

Remark 4 More recently, Markovic solved the case ${X=\mathbb{H}^3_{\mathbb{R}}}$ of Li-Wang conjecture and the initial conjecture of Schoen (i.e., the case ${X=\mathbb{H}^2_{\mathbb{R}}}$) in these papers here and here. Also, Lemm-Markovic confirmed the Li-Wang conjecture for the case ${\mathbb{H}^p_{\mathbb{R}}}$, ${p\geq 3}$, 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 ${X}$ and ${Y}$ be non-compact symmetric spaces of rank one. Given a quasi-isometry ${f:X\rightarrow Y}$, there exists an unique harmonic map ${h:X\rightarrow Y}$ such that

$\displaystyle \sup\limits_{x\in X} d(h(x),f(x))<\infty$

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 ${f:\mathbb{H}^p_{\mathbb{C}}\rightarrow\mathbb{H}^p_{\mathbb{C}}}$ of complex hyperbolic spaces, and
• it deals with quasi-isometries between ${X}$ and ${Y}$ with different dimensions such as ${f:\mathbb{H}^2_{\mathbb{R}}\rightarrow \mathbb{H}^3_{\mathbb{R}}}$.

2. Sketch of proof of Benoist-Hulin theorem

For the sake of exposition, we will sketch the proof of Theorem 3 for quasi-isometries ${f:\mathbb{H}^2_{\mathbb{R}}\rightarrow \mathbb{H}^3_{\mathbb{R}}}$, i.e., from now on we will take ${X=\mathbb{H}^2_{\mathbb{R}}}$ and ${Y=\mathbb{H}^3_{\mathbb{R}}}$.

We begin by noticing that it is sufficient to show the existence of an harmonic map ${h:X\rightarrow Y}$ within bounded distance from ${f:X\rightarrow Y}$: indeed, as we already told in Remark 2, the uniqueness of ${h}$ 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 ${f}$: by using bump functions, Benoist-Hulin show (with a 2 pages straighforward calculation) that ${f:X\rightarrow Y}$ is within bounded distance to a ${C^{\infty}}$ quasi-isometry ${\widetilde{f}}$ whose covariant derivatives ${D^p\widetilde{f}}$ are bounded on ${X}$ for all ${p\geq 1}$ (cf. Proposition 3.4 in Benoist-Hulin paper).

In other words, by replacing ${f}$ by ${\widetilde{f}}$ if necessary, we can assume that ${f:X\rightarrow Y}$ is a ${C^{\infty}}$ quasi-isometry with ${\|Df\|\leq c}$ and ${\|D^2 f\|\leq c}$ for some constant ${c\geq 1}$.

2.2. Reduction to a priori ${C^0}$-estimates

The second step is to reduce the construction of ${h}$ to an a priori ${C^0}$-estimate (through a standard compactness argument).

More precisely, let us fix an origin ${O\in X}$, and let us consider the closed balls ${B_R:=B(O,R)}$ in ${X}$. Next, we take ${h_R:B_R\rightarrow Y}$ the unique harmonic map satisfying the Dirichlet boundary condition ${h_R = f}$ on ${\partial B_R}$: the map ${h_R\in C^{\infty}(B_R)}$ minimizes the Dirichlet energy

$\displaystyle E_R(g)=\int_{B_R}\|dg(x)\|^2 d vol_X(x)$

among all ${C^1}$ maps ${g:B_R\rightarrow Y}$ with ${g=f}$ on ${\partial B_R}$. (These facts were proved by Schoen [see here] and Schoen-Uhlenbeck here and here.)

In this context, the existence of ${h}$ in Theorem 3 can be reduced to an uniform ${C^0}$ estimate on the distances between ${h_R}$ and ${f}$:

Proposition 4 Suppose that there exists ${M\geq 1}$ such that

$\displaystyle d(h_R,f):=\sup\limits_{x\in B_R} d(h_R(x),f(x))\leq M$

for all ${R\geq 1}$. Then, the sequence ${(h_R)_{R\geq 1}}$ converges ${C^0}$ uniformly on compact subsets of ${X}$ to an harmonic map ${h:X\rightarrow Y}$ with

$\displaystyle \sup\limits_{x\in X} d(h(x),f(x))\leq M$

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” ${C^0}$ estimates on harmonic maps into ${C^1}$ bounds on them.

Lemma 5 (Cheng) Let ${x_0\in X=\mathbb{H}_{\mathbb{R}}^2}$ and ${g:B(x_0,r_0)\rightarrow Y=\mathbb{H}_{\mathbb{R}}^3}$ a ${C^{\infty}}$ harmonic map such that the following ${C^0}$ estimate holds:

$\displaystyle g(B(x_0,r_0))\subset B(y_0,R_0)$

for some ${R_0>0}$. Then, we have the following ${C^1}$ estimate on ${g}$ at ${x_0}$:

$\displaystyle \|Dg(x_0)\|\leq 2^6 \frac{1+r_0}{r_0} R_0$

Let us now use this lemma to prove Proposition 4.

Proof: By assumption, there exists ${M\geq 1}$ such that ${d(h_R,f)\leq M}$ for all ${R\geq 1}$, and, furthermore, ${\|Df\|\leq c}$ for some ${c\geq 1}$. It follows that, given any increasing sequence ${(R_n)_{n\in\mathbb{N}}\rightarrow\infty}$, we have

$\displaystyle h_{R_n}(B_{2S})\subset B(f(O), 2 c S+M)$

for all ${n\in\mathbb{N}}$ large enough.

By Cheng’s lemma 5, this ${C^0}$ estimate leads to the following ${C^1}$ bound

$\displaystyle \|Dh_{R_n}(x)\|\leq 2^7 (2 c S + M) \ \ \ \ \ (1)$

for all ${x\in B_S}$ and ${n\in\mathbb{N}}$ sufficiently large.

By Arzela-Ascoli theorem, we can find a subsequence ${(h_{R_{n_k}})_{n_k\in\mathbb{N}}}$ converging ${C^0}$ uniformly on every ball ${B_S}$, ${S\geq 1}$, to a continuous map ${h:X\rightarrow Y}$ with

$\displaystyle \sup\limits_{x\in X} d(h(x),f(x))\leq M$

Moreover, the harmonic maps ${h_{R_n}}$ minimize the Dirichlet energy and, on each ball ${B_S}$, their energies are uniformly bounded

$\displaystyle \limsup\limits_{n\rightarrow\infty} E_S(h_{R_n})<\infty$

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

Finally, the convergence ${h_R\rightarrow h}$, ${R\rightarrow\infty}$, of whole sequence follows from the convergence along subsequences and the uniqueness theorem of Li-Wang (see Remark 2) ensuring that the harmonic map ${h}$ with ${\sup\limits_{x\in X}d(h(x),f(x))<\infty}$ is unique. $\Box$

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

Theorem 6 (Benoist-Hulin) There exists ${M\geq 1}$ such that

$\displaystyle \rho_R:=d(h_R,f):=\sup\limits_{x\in B_R} d(h_R(x), f(x))\leq M$

for all ${R\geq 1}$.

The proof of this theorem has two components:

• Boundary estimates: one estimates ${d(h_R(x),f(x))}$ for ${x}$ near ${\partial B_R}$;
• Interior estimates: one estimates ${d(h_R(x),f(x))}$ for ${x}$ far from ${\partial B_R}$.

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

2.3. Boundary estimates

The behavior of ${d(h_R(x), f(x))}$ for ${x}$ near ${\partial B_R}$ is controlled by the following result:

Proposition 7 Let ${f:\mathbb{H}_{\mathbb{R}}^2\rightarrow \mathbb{H}_{\mathbb{R}}^3}$ be a smooth map with ${\|Df\|\leq c}$ and ${\|D^2f\|\leq c}$ for some ${c\geq 1}$. Fix ${O\in\mathbb{H}_{\mathbb{R}}^2=X}$ and denote by ${h_R:B_R\rightarrow\mathbb{H}_{\mathbb{R}}^3}$ the harmonic map from the closed ball ${B_R=B(O,R)}$ to ${\mathbb{H}_{\mathbb{R}}^3=Y}$ with ${h_R=f}$ on ${\partial B_R}$.Then,

$\displaystyle d(h_R(x),f(x))\leq 8c^2 d(x,\partial B_R)$

for all ${x\in B_R}$.

Proof: Given ${x\in B_R}$, take ${w\in\partial B_R}$ closest to it, i.e., ${d(x,w)=d(x,\partial B_R)}$. Since ${h_R(w)=f(w)}$ and ${\|Df\|\leq c}$, one has

$\displaystyle \begin{array}{rcl} d(f(x),h_R(x))&\leq& d(f(x),f(w)) + d(h_R(w),h_R(x)) \\ &\leq& c d(x,\partial B_R) + d(h_R(w),h_R(x)) \end{array}$

In order to estimate ${d(h_R(w), h_R(x))}$, we will use a barrier function. More precisely, we consider the geodesic passing through ${h_R(w)}$ and ${h_R(x)}$, and we select a point ${y_0}$ in this geodesic with the following properties:

• ${y_0}$ is very far from ${f(B_R)}$, i.e., ${F(z):=d(f(z),y_0)\geq 1}$ for all ${z\in B_R}$;
• ${d(h_R(w),h_R(x)) = d(h_R(x),y_0) - d(h_R(w),y_0)}$.

Since ${h_R}$ is harmonic, the function ${z\mapsto d(h_R(z),y_0)}$ is subharmonic. Moreover, this function coincides with the smooth function ${F}$ on ${\partial B_R}$. Therefore, the maximum principle tells us that

$\displaystyle d(h_R(z),y_0)\leq H(z)$

for all ${z\in B_R}$ (with equality for ${z\in\partial B_R}$), where ${H:B_R\rightarrow\mathbb{R}}$ is the smooth harmonic function coinciding with ${F}$ on ${\partial B_R}$.

In particular, our choice of ${y_0}$ implies

$\displaystyle d(h_R(w),h_R(x))\leq H(x)-H(w),$

so that

$\displaystyle d(f(x),h_R(x))\leq c d(x,\partial B_R) + H(x)-H(w) \ \ \ \ \ (2)$

The barrier function ${H}$ can be controlled thanks to an estimate of Anderson-Schoen. More concretely, the function ${G:=F-H}$ vanishes on ${\partial B_R}$ and a short computation (at page 15 of Benoist-Hulin paper) reveals that its Laplacian verifies

$\displaystyle |\Delta G|=|\Delta (d(.,y_0)\circ f)|\leq 6c^2$

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

$\displaystyle |G(x)|\leq 6c^2 d(x,\partial B_R)$

Therefore, this estimate together with (2) gives

$\displaystyle \begin{array}{rcl} d(f(x),h_R(x))&\leq & c d(x,\partial B_R) + H(x)-H(w) \\ &\leq& c d(x,\partial B_R) + |G(x)| + |F(x)-F(w)| \\ &\leq& c d(x,\partial B_R) + 6c^2 d(x,\partial B_R) + d(f(x),f(w)) \\ &\leq & c d(x,\partial B_R) + 6c^2 d(x,\partial B_R) + c d(x,\partial B_R) \\ &\leq & 8c^2 d(x,\partial B_R) \end{array}$

This proves the proposition. $\Box$

2.4. Interior estimates

The boundary estimate in Proposition 7 says that a point ${x_R\in B_R}$ with

$\displaystyle d(f(x_R), h_R(x_R)) = \sup\limits_{x\in B_R} d(f(x),h_R(x)):=\rho_R$

is not close to ${\partial B_R}$, i.e.,

$\displaystyle d(x_R,\partial B_R)\geq \frac{1}{8c^2}\rho_R \ \ \ \ \ (3)$

For the sake of contradiction, let us suppose that ${\rho_R\rightarrow\infty}$ as ${R\rightarrow\infty}$. Consider the polar exponential coordinates ${(\rho(y),v(y))\in (0,\infty)\times T^1_{y_R}Y}$ centered at ${y_R:=f(x_R)\in Y=\mathbb{H}_{\mathbb{R}}^3}$. We will use these coordinates to study ${f}$ and ${h_R}$ on a ball ${B(x_R, r_R)}$ of radius ${r_R=\rho_R^{1/3}(\ll \rho_R^{1/2})}$. More concretely, if ${\theta(v_1,v_2)}$ denotes the angle between ${v_1, v_2\in T^1_{y_R}Y}$, then we will contradict the triangle inequality

$\displaystyle \theta(v(f(z_0)), v(h_R(x_R)))\leq \theta(v(f(z_0)), v(h_R(z_0))) + \theta(v(h_R(z_0)), v(h_R(x_R))) \ \ \ \ \ (4)$

by adapting an idea of Markovic to find ${z_0=z_0(R)}$ in the set

$\displaystyle W_R:=\{z\in\partial B(x_R,r_R): \rho(h_R(z_0))\geq \rho_R - \frac{r_R}{2c} \textrm{ and } \rho(h_R(z_t))\geq\frac{\rho_R}{2} \, \forall 0\leq t\leq r_R\}$

(where ${(z_t)_{0\leq t\leq r_R}}$ is the geodesic connecting ${x_R}$ to ${z}$) such that

$\displaystyle \liminf\limits_{R\rightarrow\infty} \theta (v(f(z_0)), v(h_R(x_R))) > \frac{1}{4c^2}$

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

$\displaystyle \theta(v(f(z)), v(h_R(z)))\rightarrow 0 \textrm{ and } \theta(v(h_R(z)), v(h_R(x_R)))\rightarrow 0$

as ${R\rightarrow\infty}$ for all ${z\in W_R}$.

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

Formally, we proceed as follows:

• first, we give upper bounds for ${\theta_1:=\theta(v(f(z)), v(h_R(z)))}$ when ${z\in\partial B(x_R, r_R)}$ and ${\rho(h_R(z))\geq \rho_R - r_R/2c}$;
• secondly, we give upper bounds for ${\theta_2:=\theta(v(h_R(z)), v(h_R(x_R)))}$ when ${z\in\partial B(x_R, r_R)}$ and ${\rho(h_R(z_t))\geq \rho_R/2}$ for all ${0\leq t\leq r_R}$;
• finally, we give lower bounds on ${\theta_0:=\theta (v(f(z_0)), v(h_R(x_R)))}$ for some ${z_0\in W_R}$.

2.4.1 Upper bounds on ${\theta_1}$

Lemma 8 Suppose that ${1\leq r_R\leq \frac{1}{16c^2}\rho_R}$. Then,

$\displaystyle \theta_1=\theta(v(f(z)), v(h_R(z)))\leq 4 e^{c/2} e^{-r_R/4c}$

for all ${z\in\partial B(x_R, r_R)}$ with ${\rho(h_R(z))\geq \rho_R - r_R/2c}$.

Proof: Consider ${z}$ as above. The triangle ${T}$ of vertices ${y_R=f(x_R), f(z), h_R(z)}$ has sides of lengths:

• ${\ell_0:=d(h_R(z),f(z))\leq \rho_R}$ (by definition of ${\rho_R}$);
• ${\ell_1:=d(f(z),f(x_R))\geq \frac{1}{c}d(z,x_R)-c = r_R/c -c}$ (because ${z\in\partial B(x_R,r_R)}$ and ${f}$ is ${c}$-quasi-isometric);
• ${\ell_2:=d(f(x_R),h_R(z)):=\rho(h_R(z))\geq \rho_R - r_R/2c}$ (by our assumption on ${z}$).

In particular, the sum ${\ell_1+\ell_2}$ of the lengths of the sides of ${T}$ which are adjacent to the angle ${\theta_1:=\theta(v(f(z)), v(h_R(z)))}$ is much bigger than the length ${\ell_0}$ of the opposite side to this angle:

$\displaystyle \ell_1+\ell_2-\ell_0\geq \rho_R-\frac{r_R}{2c} + \frac{r_R}{c} - c - \rho_R = \frac{r_R}{2c} - c \ \ \ \ \ (5)$

By “elementary hyperbolic geometry”, i.e., the fact that the angle ${\theta(v(f(z)), v(h_R(z)))}$ is controlled by the Gromov product (i.e., the “excess” in the triangle inequality)

$\displaystyle \begin{array}{rcl} (f(z),h_R(z))_{f(x_R)} &:=& \frac{1}{2}(d(f(x_R),f(z))+d(f(x_R),h_R(z))-d(f(z),h_R(z)))\\ &=&\frac{1}{2}(\ell_1+\ell_2-\ell_0) \end{array}$

through the relation

$\displaystyle \theta_1=\theta(v(f(z)), v(h_R(z)))\leq 4 e^{-(f(z),h_R(z))_{f(x_R)}}$

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

$\displaystyle \theta_1\leq 4 e^{-(\ell_1+\ell_2-\ell_0)/2}\leq 4 e^{c/2} e^{-r_R/4c}$

This proves the lemma. $\Box$

2.4.2 Upper bounds on ${\theta_2}$

Lemma 9 Suppose that ${1\leq r_R\leq \frac{1}{16c^2}\rho_R}$. Then,

$\displaystyle \theta_2:=\theta(v(h_R(z)), v(h_R(x_R)))\leq \frac{8\rho_R^2}{\sinh(\rho_R/4)}$

for all ${z\in\partial B(x_R,r_R)}$ with ${\rho(h_R(z_t))\geq \rho_R/2}$ for all ${0\leq t\leq r_R}$ (where ${(z_t)_{0\leq t\leq r_R}}$ is the geodesic from ${x_R}$ to ${z}$).

Proof: Note that

$\displaystyle \begin{array}{rcl} \theta(v(h_R(z)), v(h_R(x_R))) &=&\int_0^{r_R} \frac{d}{dt} \theta(v(h_R(z_t)), v(h_R(x_R))) dt \\ &\leq& r_R\cdot \sup\limits_{z_t}\|D(v\circ h_R)(z_t)\| \end{array}$

by our choice of the polar exponential coordinates ${(\rho, v)}$ at ${y_R=f(x_R)}$.

In order to estimate ${\|D(v\circ h_R)(z_t)\|}$, we use a lemma of Gauss saying that

$\displaystyle 2\sinh(\rho(y)/2)\|Dv(y)\|\leq 1$

for all ${y=\exp_{f(x_R)}(\rho(y)v(y))\in \mathbb{H}_{\mathbb{R}}^3-\{y_R\}:=Y-\{y_R\}}$.

From the previous two estimates and our assumption that ${\rho(h_R(z_t))\geq\rho_R/2}$, we deduce that

$\displaystyle \theta_2:=\theta(v(h_R(z)), v(h_R(x_R)))\leq \frac{r_R}{2\sinh(\rho_R/4)}\cdot \sup\limits_{z_t} \|D h_R(z_t)\|$

Finally, we control ${\|Dh_R(z_t)\|}$ with the aid of Cheng’s lemma. More precisely, ${h_R(B(x_R,r_R))\subset B(h(x_R),4\rho_R)}$ because

$\displaystyle \begin{array}{rcl} d(h_R(w),h_R(x_R)) &\leq& d(h_R(w),f(w))+d(f(w),f(x_R))+d(f(x_R),h_R(x_R)) \\ &\leq& \rho_R+c d(w,x_R) + \rho_R\leq 2\rho_R + c r_R \\ &\leq& 4\rho_R \end{array}$

for all ${w\in B(x_R,r_R)}$ (since ${\|Df\|\leq c}$ and ${r_R\leq\rho_R/16c^2}$ by hypothesis). Thus, Cheng’s lemma 5 ensures that

$\displaystyle \sup\limits_{z_t}\|Dh_R(z_t)\|\leq 2^8\rho_R$

In summary, the previous estimate show that

$\displaystyle \theta_2:=\theta(v(h_R(z)), v(h_R(x_R)))\leq \frac{2^7 r_R\rho_R}{\sinh(\rho_R/4)}\leq \frac{2^3\rho_R^2}{\sinh(\rho_R/4)}$

because ${r_R\leq \rho_R/16c^2}$. $\Box$

2.4.3 Lower bounds on ${\theta_0}$

We start our quest for a point ${z_0\in W_R}$ with ${\theta_0:=\theta (v(f(z_0)), v(h_R(x_R)))}$ far away from zero by estimating the Lebesgue measure of ${W_R}$.

Lemma 10 Suppose that ${1\leq r_R\leq \frac{1}{16c^2}\rho_R}$ and denote by ${\sigma_R}$ the normalized spherical Lebesgue measure on ${\partial B(x_R,r_R)}$. Then,

$\displaystyle \sigma_R(W_R)\geq \frac{1}{3c^2} - 2^{13}c\frac{r_R^2}{\rho_R}$

Proof: We write ${W_R=U_R\cap V_R}$ where

$\displaystyle U_R:=\{z\in\partial B(x_R,r_R): \rho(h_R(z))\geq \rho_R - \frac{r_R}{2c}\}$

and

$\displaystyle V_R:=\{z\in\partial B(x_R,r_R): \rho(h_R(z_t))\geq\rho_R/2 \,\forall 0\leq t\leq r_R\}$

(where ${z_t}$ is the geodesic path between ${x_R}$ and ${z}$).

For the sake of exposition, we will discuss only the estimate of ${\sigma_R(U_R)}$: indeed, the estimate of ${\sigma_R(\partial B(x_R,r_R)-V_R)}$ 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 ${\sigma_R(U_R)}$ is the following Green formula:

$\displaystyle \int_{\partial B(x_R,r_R)} (\rho(h_R(z))-\rho_R) d\sigma_R(z)\geq 0 \ \ \ \ \ (6)$

The proof of this Green formula uses the subharmonicity of ${\rho(h_R(z)) := d(h_R(z), f(x_R))}$ (which is a consequence of the subharmonicity of the distance function of ${Y=\mathbb{H}_{\mathbb{R}}^3}$ and the harmonicity of ${h_R}$). More precisely, denote by ${\Gamma(\simeq SO(2,\mathbb{R})}$ the compact group of isometries of ${X(\simeq\mathbb{H}_{\mathbb{R}}^2}$) fixing ${x_R}$, and let ${d\gamma}$ be its Haar measure.

Note that ${\Gamma}$ acts transitively on each sphere ${\partial B(x_R,t)}$, ${t\leq r_R}$. Thus, the subharmonic function ${F(w):=\int_{\Gamma} \rho(h_R(\gamma(w))) d\gamma}$ takes a constant value ${F_t\in\mathbb{R}}$ on ${\partial B(x_R,t)}$, so that

$\displaystyle \rho_R=\rho(h_R(x_R))=F_0\leq F_t$

for all ${0\leq t\leq r_R}$. Because ${F_{r_R}= \int_{\partial B(x_R,r_R)}\rho(h_R(z)) d\sigma_R(z)}$, this proves (6).

Next, we observe that ${\rho(h_R(z))-\rho_R\leq c r_R}$ for all ${z\in\partial B(x_R,r_R)}$: in fact,

$\displaystyle \rho(h_R(z)):=d(h_R(z),f(x_R))\leq d(h_R(z),f(z)) + d(f(z),f(x))\leq\rho_R+cr_R$

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

$\displaystyle c r_R\sigma_R(U_R) - \frac{r_R}{2c}(1-\sigma_R(U_R))\geq \int_{\partial B(x_R,r_R)}(\rho(h_R(z))-\rho_R) d\sigma_R(z)\geq 0,$

i.e.,

$\displaystyle \sigma_R(U_R)\geq \frac{1}{1+2c^2}\geq \frac{1}{3c^2}$

. This completes our sketch of proof of the lemma (modulo the verification of the bound ${\sigma_R(\partial B(x_R,r_R)-V_R)\leq 2^{13}c r_R^2/\rho_R}$). $\Box$

Using Lemma 10, we can find ${z_0\in W_R}$ with the property that ${\theta_0:=\theta (v(f(z_0)), v(h_R(x_R)))}$ is far away from zero:

Lemma 11 Suppose that ${\rho_R\rightarrow\infty}$ as ${R\rightarrow\infty}$ and let ${r_R=\rho_R^{1/3}}$. Then,

$\displaystyle \liminf\limits_{R\rightarrow\infty} \sup\limits_{z_0\in W_R}\theta (v(f(z_0)), v(h_R(x_R))) > 0$

Proof: Let us set ${\sigma_0:=(4c)^{-2}}$. By Lemma 10,

$\displaystyle \liminf\limits_{R\rightarrow\infty}\sigma_R(W_R)>\sigma_0>0$

(because ${r_R\ll \rho_R^{1/2}}$.)

Since any subset ${W}$ of the Euclidean sphere ${S^1\simeq T^1_{x_R}\mathbb{H}_{\mathbb{R}}^2}$ with normalized Lebesgue measure ${>\sigma_0}$ has diameter ${>\varepsilon_0=\sigma_0/2}$, for ${R}$ sufficiently large, we can find ${z_1, z_2\in W_R}$ such that

$\displaystyle \theta_{x_R}(z_1, z_2)\geq \varepsilon_0 = \sigma_0/2 \ \ \ \ \ (7)$

where ${\theta_{x_R}(z_1,z_2)}$ is the angle between the vectors ${v(z_1), v(z_2)}$ such that

$\displaystyle z_j=\exp_{x_R}(r_R v(z_j)), j=1, 2.$

Before proceeding further, we need to recall some facts about the relationship between angles of a triangle ${T}$ in a hyperbolic space ${\mathbb{H}_{\mathbb{R}}^p}$ and Gromov products ${(x_1,x_2)_{x_0}:=\frac{1}{2}(d(x_1,x_0)+d(x_2,x_0)-d(x_1,x_2))}$ (cf. Lemmas 2.1 and 2.2 in Benoist-Hulin paper). Let ${T}$ be a geodesic triangle in ${\mathbb{H}_{\mathbb{R}}^p}$ with vertices ${x_0, x_1, x_2}$ and denote by ${\theta(x_0)}$ the corresponding angle at ${x_0}$. Then,

• (a) ${(x_0,x_2)_{x_1}\geq d(x_0,x_1)\sin^2(\theta_0/2)}$;
• (b) ${\theta(x_0)\leq 4e^{-(x_1,x_2)_{x_0}}}$;
• (c) ${\theta(x_0)\geq e^{-(x_1,x_2)_{x_0}}}$ whenever ${\min\{(x_0,x_1)_{x_2}, (x_0,x_2)_{x_1}\}\geq 1}$;
• (d) if ${f:\mathbb{H}_{\mathbb{R}}^p\rightarrow \mathbb{H}_{\mathbb{R}}^q}$ is a ${c}$-quasi-isometry, then

$\displaystyle \frac{1}{c}(x_1,x_2)_{x_0}-A\leq (f(x_1),f(x_2))_{f(x_0)}\leq c (x_1,x_2)_{x_0}+A$

where ${A=A(c)>0}$ is a constant depending only on ${c\geq 1}$.

We use these facts as follows. Note that

$\displaystyle \max\{\theta (v(f(z_1)), v(h_R(x_R))),\theta (v(f(z_2)), v(h_R(x_R)))\}\geq \frac{1}{2} \theta(v(f(z_1)), v(f(z_2))) \ \ \ \ \ (8)$

by the triangle inequality.

On the other hand, if we can show that

$\displaystyle \min\{(f(x_R),f(z_1))_{f(z_2)}, (f(x_R),f(z_2))_{f(z_1)}\}\geq 1, \ \ \ \ \ (9)$

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

$\displaystyle \begin{array}{rcl} \theta(v(f(z_1)), v(f(z_2)))&\geq& e^{-(f(z_1),f(z_2))_{f(x_R)}} \\ &\geq& e^{-A} e^{-(z_1,z_2)_{x_R}} \\ &\geq& e^{-A}(\theta_{x_R}(z_1,z_2)/4)^c \\ &\geq& e^{-A} (\varepsilon_0/4)^c \end{array}$

which would complete the proof of the lemma (by taking ${z_0=z_1}$ or ${z_2}$) in view of (8) (because ${A=A(c)>0}$ and ${\varepsilon_0=\sigma_0/2=1/8c^2}$ are independent of ${R}$).

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

$\displaystyle \min\{(x_R,z_1)_{z_2}, (x_R,z_2)_{z_1}\}\geq (A+1)c \ \ \ \ \ (10)$

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

$\displaystyle \min\{(x_R,z_1)_{z_2}, (x_R,z_2)_{z_1}\}\geq r_R \sin^2(\theta_{x_R}(z_1, z_2)/2)\geq r_R\sin^2(\varepsilon_0/2)$

because ${d(z_1,x_R)=r_R=d(z_2,x_R)}$. It follows that the desired estimate (10) holds whenever ${R}$ is sufficiently large so that

$\displaystyle r_R=\rho_R^{1/3}\geq\frac{(A+1)c}{\sin^2(\varepsilon_0/2)}.$

This proves the lemma. $\Box$

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 ${\rho_R\rightarrow\infty}$ as ${R\rightarrow\infty}$.

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 ${\rho_R}$ in terms of ${c\geq 1}$ with ${\|Df\|\leq c}$, ${\|D^2f\|\leq c}$. Indeed, the proofs of the lemmas above show that we get a contradiction when the quantities ${r_R=\rho_R^{1/3}}$ and ${\rho_R}$ satisfy the following inequalities:

• ${\sigma_R(W_R)\geq \frac{1}{3c^2} - 2^{13}c\frac{r_R^2}{\rho_R}>\sigma_0:=\frac{1}{4c^2}}$;
• ${r_R\geq (A+1)c/\sin^2(1/2^4c^2)}$, and
• ${\theta_1+\theta_2\leq 4e^{c/2}e^{-r_R/4c}+8\frac{\rho_R^2}{\sinh(\rho_R/4)}<\frac{1}{2}e^{-A}(\frac{1}{2^5c^2})^c\leq\theta_0}$

where ${A=A(c)>0}$ is the constant provided by item (d) in the proof of Lemma 11. Since these numerical constraints are fulfilled by any ${\rho_R\geq M=M(c)}$ for an appropriate (“explicit”) choice of ${M=M(c)\gg 1}$, one gets the following quantitative version of Benoist-Hulin theorem 6: ${d(h_R,f)\leq M(c)}$ for all ${R\geq 1}$.

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