Conformal equivalence of visual metrics in pseudoconvex domains

Abstract

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between bounded, smooth strongly pseudoconvex domains in ${\mathbb {C}}^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth extensions of biholomorphic mappings between bounded smooth pseudoconvex domains. The proofs are inspired by Mostow’s proof of his rigidity theorem and are based on the asymptotic hyperbolic character of the Kobayashi or Bergman metrics and on the Bonk-Schramm hyperbolic fillings.

1 Introduction

Let $D \subset {\mathbb {C}}^n (n\ge 2)$ be a bounded, strongly pseudo-convex domain with $C^\infty$-smooth boundary. Denote by $d_K$ the distance function corresponding to a Finsler structure K satisfying suitable estimates, see (2.8). For example, one may consider the Bergman metric or the Kobayashi metric or the inner Carathéodory metric.

In [2, 3], Balogh and Bonk have proved that the metric space $(D,d_K)$ is hyperbolic in the sense of Gromov and its visual boundary coincides with the topological boundary $\partial D$. They also show that the Carnot–Carathéodory metric $d_{CC}$ corresponding to the Levi form on $\partial D$, determines the canonical class of snowflake equivalent visual metrics on $\partial D$. As a consequence, results from the theory of Gromov hyperbolic spaces can be immediately applied in this setting. Among these we recall that every quasi-isometry between such spaces extends to a quasi-conformal map between the visual boundaries, endowed with their families of visual metrics, see for instance [6, 17] and references therein.

Our main contribution is to show that extensions of isometries are actually diffeomorphisms that are conformal with respect to the Carnot–Carathéodory metric. We only need to show that the extension is 1-quasi-conformal, as the smoothness then follows from the recent results in [12].

As in [3], our strategy involves the Bonk-Schramm hyperbolic filling metric g defined in (1.2). This metric provides a stepping stone to connect the Carnot–Carathéodory distance, defined on the boundary by the Levi form (see Sect. 2.2), with the invariant metric defined in the domain.

Theorem 1.1

Let $D_1,D_2\subset {\mathbb {C}}^n$ be bounded strongly pseudoconvex $C^\infty$-smooth domains and denote by $d_{K}$ the distance function corresponding to a Finsler structure K satisfying (2.8), and by $d_{CC}$ the Carnot–Carathéodory distance on the boundaries induced by the Levi form. If $f:(D_1, d_{K}) \rightarrow (D_2,d_{K})$ is an isometry then the induced boundary map $F: (\partial D_1, d_{CC})\rightarrow (\partial D_2, d_{CC})$ is a diffeomorphism, conformal with respect to the metric $d_{CC}$.

We emphasize that the result holds when $d_K$ is the Bergman, the Kobayashi, or the inner Carathéodory metrics. Indeed, these distances satisfy (2.8) in view of the work in [2, 3, 21].

As we noted above, the proof of Theorem 1.1 is based on the study of the relation between the visual distances associated to $d_K$ and the visual distance of an ad-hoc hyperbolic filing metric, built through the Carnot–Carathéodory distance: For $x\in D$ denote by $h(x):=\sqrt{d_E(x,\partial D)}$ and by $\pi (x) \in \partial D$ a closest point in $\partial D$ with respect to the Euclidean distance $d_E(\cdot , \cdot )$, noting it is uniquely defined in a neighborhood of $\partial D$. Set

$$\begin{aligned} g(x,y):=2\log \bigg ( \frac{d_{\mathrm{CC}}(\pi (x), \pi (y))+ \max (h(x),h(y) ) }{\sqrt{h(x) h(y)} } \bigg ). \end{aligned}$$

(1.2)

This is an hyperbolic filling metric built from the metric space $(\partial D, d_{\mathrm{CC}})$ (see Bonk and Schramm [10]). Balogh and Bonk [3, Corollary 1.3], showed that g is a metric in a neighborhood of $\partial D$ and that g and the invariant distance function $d_K$ are (1, C)-quasi-isometric. As a consequence, they give rise to quasi-conformally equivalent visual metrics.

The main technical point of our work is to refine this result in a quantitative fashion. We show that a particular visual quasi-distance $\rho ^K_o$ associated to the invariant metric $d_K$ is in fact pointwise and asymptotically $(1+\epsilon )$-quasi-conformally equivalent to the Carnot–Carathéodory $d_{CC}$ metric. By pointwise and asymptotically we mean that for every point $x\in \partial D$ in the boundary, and for every $\epsilon >0$, one can choose a base point o for the definition of the visual distances so that the identity map has distortion less than $1+\epsilon$ at x. Following ideas in CAT$(-1)$ spaces, given a pointed metric space (X, d, o) we consider the visual function

$$\begin{aligned} \rho ^{d}_o ( x,y) = \exp (- \langle x, y \rangle _o) , \end{aligned}$$

(1.3)

where $\langle x, y \rangle _o$ denotes the Gromov product in (X, d), see Sect. 2. Usually, $\rho ^{d}_o$ is called Bourdon distance since for CAT($-1$) spaces it satisfies the triangle inequality. In our setting, $\rho ^{d}_o$ may not be a distance.

Moreover, Bourdon showed in [9] that on a CAT($-1$) space X the visual boundaries $(\partial _{\infty } X, \rho ^d_o)$ corresponding to different base points $o,o'\in X$ are conformally equivalent, thus implying immediately that any isometry of X extends to a conformal maps of its visual boundaries. Since pseudoconvex domains may not have negative curvature (see [19]) and may not be simply connected, they are not CAT$(-1)$ spaces and so one cannot apply Bourdon’s result.

Theorem 1.1 is achieved in two steps: First one shows that the Carnot–Carathéodory distance is conformally equivalent¹ to the function $\rho ^g_o$ associated to the hyperbolic filling metric g.

Proposition 1.4

For any $o\in D$, the functions $d_{CC}$ and $\rho ^g_o$ are conformally equivalent.

In other words, the identity map $(\partial D, d_{CC}) \rightarrow (\partial D,\rho ^g_o)$ has distortion that is identically equal to one. See (2.1) for the definition of distortion.

Next, we show that at every boundary point, and for any $\epsilon >0$, one can find a base point $o\in D$ such that the corresponding visual functions $\rho ^K_o$ and $\rho ^g_o$ are $(1+\epsilon )$-biLipschitz equivalent in a neighborhood of that point. In the following we denote Euclidean balls in ${\mathbb {C}}^n$ with the notation B(x, r).

Proposition 1.5

For any ${\bar{p}}\in \partial D$ and ${\bar{\epsilon }}>0$ there exists $r>0$ such that for all $\omega \in \partial D\cap B({\bar{p}},r){\setminus } \{{\bar{p}}\}$ there exists $r'>0$ such that for all $o\in D\cap B(\omega ,r')$ the two functions $\rho ^g_o$ and $\rho ^K_o$ are $(1+{\bar{\epsilon }})$-biLipschitz on $\partial D \cap B({\bar{p}},r')$.

The proof of Proposition 1.5 and Theorem 1.1 are in Sect. 5. Theorem 1.1 follows rather directly from Propositions 1.4 and 1.5 and from the following diagram

At the center of this chain of compositions there is an isometry, the rest of the links are either $(1+\epsilon )$-biLipschitz maps or conformal maps, so that the total distortion is at most $\epsilon$ away from being equal to 1 everywhere.

From the conformal equivalence theorem above and the results in [12], one can immediately infer a result about boundary extensions for biholomorphisms between strictly pseudoconvex domains in ${\mathbb {C}}^n$, originally established by Fefferman [15].

Corollary 1.6

Let $D_1,D_2\in {\mathbb {C}}^n (n\ge 2)$ be bounded strongly pseudo-convex domains with $C^\infty$-smooth boundaries. If $f:D_1\rightarrow D_2$ is a biholomorphism then it extends to a smooth map $F: \partial D_1 \rightarrow \partial D_2$ that is conformal with respect to the corresponding subRiemannian contact structure. In particular, at every boundary point, its differential is a similarity between the maximally complex tangent planes.

Since the publication of [15] there have been several significative extensions and simplifications of the result. A small sample of this extensive line of inquiry can be found in the references [1, 4, 5, 8, 13, 20, 22].

The contribution of the present paper does not lie so much in an innovation on a technical level, but rather in two new insights: namely, that one can deduce Fefferman’s result from the conformality of the boundary extension and that one can prove this with relative ease from a combination of the general theory of Gromov hyperbolic spaces in combination with careful estimates in Theorem 4.1.

We conclude by observing that our work can be seen as an instance of a dictionary, introduced by Bonk, Heinonen, and Koskela in [7], translating back and forth problems in domains in Euclidean spaces by means of ad hoc hyperbolic or quasi-hyperbolic metrics, that endow such domains with an hyperbolic structure in the sense of Gromov. For more results along this line, see also the recent, interesting work of Zimmer in [24].

2 Preliminaries

In this section we recall some basic definitions and results. We start by discussing distortion and conformal maps on subRiemannian manifolds. Then we discuss pseudoconvex domains and their metrics. Finally we review hyperbolicity in the sense of Gromov.

2.1 Distorsion in subRiemannian geometry

By a previous work of the authors together with Ottazzi, we know that several definitions of conformal maps are equivalent in the setting of contact subRiemannian manifolds. We now recall the two definitions that we shall need in this paper.

For a homeomorphism $F:X\rightarrow Y$ between general metric spaces, we consider the following quantities

$$\begin{aligned} {\text {L}}_F(x) := \limsup _{x'\rightarrow x} \dfrac{d(F(x), F(x'))}{d(x,x')} \qquad \text { and }\qquad \ell _F(x):= \liminf _{x'\rightarrow x} \dfrac{d(F(x), F(x'))}{d(x,x')}. \end{aligned}$$

The quantity ${\text {L}}_F(x)$ is sometimes denoted by $\mathrm{Lip}_{ F}(x)$ and is called the pointwise Lipschitz constant. Within this paper, we define the distortion of f at a point $x\in X$ as

$$\begin{aligned} H^*(x,F, d_{X}, d_{Y}) := \dfrac{{\text {L}}_F(x) }{\ell _F(x)}. \end{aligned}$$

(2.1)

The homeomorphism f is said to be quasi-conformal if there exists K such that for all $x\in X$ one has

$$\begin{aligned} \limsup _{r \rightarrow 0}\frac{ \sup \{d_Y(F(p),F(q)): {d_X(p,q)\le r} \}}{ \inf \{d_Y(F(p),F(q)) : {d_X(p,q)\ge r}\}} \le K. \end{aligned}$$

It is well-known that in the literature there are several other equivalent definitions of quasi-conformality in ‘geometrically nice’ spaces, see [23]. However, the equivalence is not quantitative, in the sense that each definition has an associated constant (like the K above) and the value of of these constants can be different from definition to definition. Thus we need to clarify what is a conformal map. To do this we invoke some recent results due to Citti, Ottazzi and the authors of the present paper.

Lemma 2.2

[12, Theorem 1.3 and Theorem 1.19] Let $F:X\rightarrow Y$ be a quasi-conformal homeomorphism between two equiregular subRiemannian manifolds.

• (i) The requirement $H^*(\cdot ,F, d_{X}, d_{Y})\equiv 1$ is equivalent to other notions of 1-quasi-conformality.

• (ii) If X and Y are contact manifolds, then 1-quasi-conformality of F is equivalent to F being conformal (i.e., smooth and with horizontal differential that is a homothety).

One of the advantages to work with (2.1) is that it immediately yields a chain rule:

$$\begin{aligned} H^*(x,F_1\circ F_2) \le H^*(x, F_2 ) H^*(F_2 (x),F_1 ). \end{aligned}$$

(2.3)

The last equation follows from the fact that $\limsup a_nb_n \le \limsup a_n \limsup b_n$ whenever $a_n, b_n\ge 0$. Moreover, we trivially have that if F is an L-biLipschitz homeomorphism, then

$$\begin{aligned} H^*(x,F ) \le L^2. \end{aligned}$$

(2.4)

2.2 Pseudoconvex domains and hermitian metrics

We recall some of the basic definitions about pseudoconvex domains and hermitian metrics, as well as some key results proved by Balogh and Bonk in [3].

Let $D\subset {\mathbb {C}}^n$, $n\ge 2$ be a smooth, bounded open set. Let ${\varphi }: {\mathbb {C}}^n \rightarrow {\mathbb {R}}$ denote the signed distance function from $\partial D$, negative in D and positive in its complement. Set $N_\delta =\{x\in D\ |\ d_E(x, \partial D)<\delta \}$.

Lemma 2.5

(Tubular Neighborhood Theorem) Let $D\subset {\mathbb {C}}^n$, $n\ge 2$ be a bounded domain with smooth boundary. There exists $\delta _0>0$ such that the projection $\pi :N_{\delta _o}\rightarrow \partial D$ is a smooth, well defined map and the distance function $d_E(\cdot ,\partial D)$ is smooth on $N_{\delta _0}$.

We will denote by n(x) the outer unit normal at $x\in \partial D$, so that the fiber $\pi ^{-1}(x)\cap N_{\delta _0}=\{ x+sn(x) | s\in (0,\delta _0)\}.$

For $p\in \partial D$, one can define the tangent space $T_p\partial D=\{ Z\in {\mathbb {C}}^n | Re\langle {\bar{\partial }} {\varphi }(p), Z\rangle =0 \}$ and its maximal complex subspace $H_p\partial D=\{ Z\in {\mathbb {C}}^n| \langle \bar{\partial } {\varphi }(p),Z\rangle =0\}$, where $\langle Z,Z'\rangle =\sum _{i=1}^n Z_i {\bar{Z}}'_i$ is the hermitian product. By definition, the domain D is strictly pseudoconvex if for every $p\in \partial D$, the Levi form

$$\begin{aligned} L_{\varphi }(p,Z) := \sum _{\alpha ,\beta =1}^n \partial ^2_{z_{\alpha }{\bar{z}}_{\beta }} {\varphi }(p) Z_\alpha {\bar{Z}}_{\beta } \end{aligned}$$

(2.6)

is positive definite on $H_p\partial D$.

For each $p\in \partial D$ one has a splitting ${\mathbb {C}}^n=H_p\partial D\oplus N_p \partial D$, where $N_p \partial D$ is the complex one-dimensional subspace orthogonal to $H_p \partial D$. This splitting at p induces a decomposition $Z=Z_H+Z_N$ for all $Z\in {\mathbb {C}}^n$.

Metrics that are invariant under the action of biholomorphisms play a key role in several complex variables. Important examples are the Bergman metric, the Kobayashi metric, and the inner Carathéodory metric (see [19]). In all cases, for $x\in D$ the length of a complex vector $Z\in T_x D={\mathbb {C}}^n$ is given by a Finsler structure K(x, Z). We will rely on the following result, which can be found (along with more references) in [2] and also [11, 14, 16], and [3, Proposition 1.2].

Proposition 2.7

(Balogh-Bonk) Let $D\subset {\mathbb {C}}^n$, $n\ge 2$ be a bounded, strictly pseudoconvex domain with smooth boundary and let K(x, Z) be the Finsler structure associated to the Bergman metric or the Kobayashi metric or the inner Carathéodory metric. For every ${\bar{\epsilon }}>0$ there exists $\delta _0,C>0$ such that for all $x\in D$ with $d_E(x,\partial D)\le \delta _0$ and $Z\in {\mathbb {C}}^n$ one has

$$\begin{aligned}&(1-C \sqrt{d_E(x,\partial D) }) \Bigg ( \frac{|Z_N|^2}{4d_E^2(x,\partial D)} + (1-{\bar{\epsilon }}) \frac{L_{\varphi }(\pi (x), Z_H)}{d_E(x,\partial D)} \Bigg )^{\frac{1}{2}} \le K(x,Z) \nonumber \\&\quad \le (1 +C \sqrt{d_E(x,\partial D) }) \Bigg ( \frac{|Z_N|^2}{4d_E^2(x,\partial D)} + (1+{\bar{\epsilon }}) \frac{L_{\varphi }(\pi (x), Z_H)}{d_E(x,\partial D)} \Bigg )^{\frac{1}{2}}, \end{aligned}$$

(2.8)

where $Z=Z_H+Z_N$ is the splitting at $\pi (x)$.

The subbundle $H \partial D$ is a contact distribution on $\partial D$ and the triplet $(\partial D, H\partial D, L_{\varphi })$ yields a contact subRiemannian manifold. In this structure, the horizontal curves are those arcs in $\partial D$ that are tangent to the contact distribution, and the Carnot–Carathéodory distance $d_{CC}(p,q)$ between $p,q\in \partial D$ is defined as the minimum time it takes to reach one point from the other traveling along horizontal curves at unit speed with respect to the Levi form, see [18].

As in [3], we will need to use a family of Riemannian metrics on $\partial D$ that approximate the sub-Riemannian metric associated to the Levi form, and that in fact have corresponding distance functions that converge in the sense of Gromov-Hausdorff to the Carnot–Carathéodory distance. For every $k>0$ we define a Riemannian metric $g_k$ on $T\partial D$ as

$$\begin{aligned} g_k^2(p, Z):= L_{\varphi }(p,Z_H)+ k^2 |Z_N|^2, \end{aligned}$$

(2.9)

for every $p\in \partial D$ and every $Z=Z_H+Z_N\in T_p \partial D$. Here we just recall a basic comparison result (see for instance [3, Lemma 3.2]) relating the distance function $d_k$ associated to $g_k$ to the Carnot–Carathéodory distance $d_{CC}$.

Lemma 2.10

There exists a constant $C>0$ such that for all $k>0$, and for all points $p,q\in \partial D$, with $d_{CC}(p,q)\ge k^{-1}$ one has

$$\begin{aligned} C^{-1} d_k(p,q)\le d_{CC}(p,q) \le C d_k (p,q). \end{aligned}$$

(2.11)

2.3 Gromov hyperbolicity

Let x, y, o be three points in a metric space (X, d). Then the Gromov product of x and y at o, denoted $\langle x,y \rangle _o$, is defined by

$$\begin{aligned} \langle x,y \rangle _{o} = \frac{1}{2} \big ( d(x, o) + d(y, o) - d(x, y) \big ). \end{aligned}$$

Then X is called Gromov hyperbolic if there exists $\delta \ge 0$ such that

$$\begin{aligned} \langle x,y \rangle _o\ge \min \{ \langle x, z\rangle _o , \langle z, y\rangle _o\} -\delta , \quad \text { for all }x, y, z,o \in X. \end{aligned}$$

For a Gromov hyperbolic space X one can define a boundary set $\partial _\infty X$ as follows, see [6, p.431-2]. Fix a base point $o\in X$. A sequence $(x_i)$ in X is said to converge at infinity if $\lim _{i,j\rightarrow \infty }\langle x_i, x_j\rangle _o = \infty$. Two sequences $(x_i)$ and $(y_i)$ converging at infinity are called equivalent if $\lim \langle x_i, y_i\rangle _o =\infty .$ These notions do not depend on the choice of the base point o. The set $\partial _\infty X$ is now defined as the set of equivalence classes of sequences converging at infinity.

For $p,q \in \partial _\infty X$ and $o\in X$ we define

$$\begin{aligned} \langle p, q\rangle _o = \sup \liminf _{i\rightarrow \infty } \langle x_i, y_i\rangle _o , \end{aligned}$$

where the supremum is taken over all sequences $(x_i)$ and $(y_i)$ representing the boundary points p and q, respectively. Actually, there exists such sequences $(x_i)$ and $(y_i)$ for which $\langle p, q\rangle _o = \lim _{i\rightarrow \infty } \langle x_i, y_i\rangle _o$, see [6, Remark 3.17].

Balogh and Bonk have proved that if $D\subset {\mathbb {C}}^n$, $n\ge 2$ is a bounded, strictly pseudoconvex domain with smooth boundary, and K(x, Z) is a norm satisfying (2.8), then the corresponding metric space $(D,d_K)$ is Gromov hyperbolic and its visual boundary coincides with the topological boundary. See [3, Theorem 1.4].

3 Conformal equivalence of boundary metrics

3.1 Boundary distances of hyperbolic fillings

An important contribution of Bonk and Schramm [10], is that the functor $X\rightarrow \partial _\infty X$ has an inverse functor, in the form of hyperbolic filling spaces Con(Z). To be more precise, one defines $Con(Z) = Z \times (0,D)$, endowed with the metric given by

$$\begin{aligned} g((x,u),(y,v))=2\log \bigg ( \frac{d_1(x, y)+ \max (u,v ) }{\sqrt{u v} } \bigg ). \end{aligned}$$

(3.1)

The space (Con(Z), g) is Gromov hyperbolic, and its visual boundary is Z, with the canonical class of snowflake equivalent metrics given by $d_1$. In this section we prove that a particular visual metric, i.e. the visual metric generated by g through the formula (1.3), is actually conformal to $d_1$. Choose a generic base point $o=(z,s)$, with $z\in Z$ and $s\in (0,D)$. For any two points $x,y\in Z$ so that $d_1(x,y) < s$. consider $u,v \in (0, d_1(x,y) )$. Following (1.3), we let $d_2(x,y)$ be defined as follows

$$\begin{aligned} d_2(x,y)=\lim _{u,v\rightarrow 0} e^{-\langle ( x ,u) , (y , v) \rangle _o}. \end{aligned}$$

Notice that in general, functions defined as in (1.3), associated to the hyperbolic fillings are a quasi-distance. By quasi-distance we intend that the triangle inequality is satisfied modulo a multiplicative constant.

Proposition 3.2

Let $d_1$ a distance on a bounded space Z. If $d_2$ is defined as in (1.3), associated to the hyperbolic filling for $d_1$, then $d_1$ and $d_2$ are conformally equivalent.

Proof

In order to show that $d_1, d_2$ are conformally equivalent it suffices to prove that the limit $\lim _{y\rightarrow x}d_1 (x,y) / d_2(x,y)$ exists for every $x\in Z$. Fix any $z\in Z$ and $s\in (0,D)$. Let $o=(z,s)$. Take two points $x,y\in Z$ so that $d_1(x,y) < s$. Take $u,v \in (0, d_1(x,y) )$.

The rest of the proof follows from

$$\begin{aligned} \langle ( x ,u) , (y , v) \rangle _o= & {} {1 \over 2}( d_2(( x ,u),o) + d_2((y , v),o) - d_2(( x ,u),(y , v)) ) \\= & {} \log \bigg ( \frac{d_1(x, z)+ \max (u,s ) }{\sqrt{u s} } \bigg ) +\log \bigg ( \frac{d_1(y, z)+ \max (v,s ) }{\sqrt{v s} } \bigg ) \\&\quad -\log \bigg ( \frac{d_1(x,y)+ \max (u,v) }{\sqrt{uv} } \bigg ) \\= & {} \log \frac{ ( d_1(x, z)+ s ) ( d_1(y, z)+ s )}{ s ( d_1(x,y)+ \max (u,v) )} \\= & {} \log \bigg ( \frac{ ( d_1(x, z)+ s ) ( d_1(y, z)+ s )}{ s ( d_1(x,y)+ \max (u,v) )} \frac{ d_1(x,y) }{ d_1(x,y)} \bigg ) \\= & {} - \log ( d_1(x,y) ) + \log \bigg ( \frac{ d_1(x,y) ( d_1(x, z)+ s ) ( d_1(y, z)+ s )}{ s ( d_1(x,y)+ \max (u,v) )} \bigg ) \end{aligned}$$

We calculate $\lim _{y\rightarrow x}d_1 (x,y) / d_2(x,y)$. Consider the quotient

$$\begin{aligned} d_2 (x,y) / d_1(x,y)= & {} \lim _{u,v\rightarrow 0} \frac{e^{-\langle ( x ,u) , (y , v) \rangle _o }}{ d_1(x,y)} \\= & {} \lim _{u,v\rightarrow 0} \frac{ e^{ \log \big ( d_1(x,y) \big ) } e^{- \log \bigg ( \dfrac{ d_1(x,y) ( d_1(x, z)+ s ) ( d_1(y, z)+ s )}{ s ( d_1(x,y)+ \max (u,v) )} \bigg ) } }{ d_1(x,y)} \\= & {} \lim _{u,v\rightarrow 0} \bigg ( \frac{ d_1(x,y) ( d_1(x, z)+ s ) ( d_1(y, z)+ s )}{ s ( d_1(x,y)+ \max (u,v) )} \bigg )^{-1}\\= & {} \frac{ s d_1(x,y) }{ d_1(x,y) ( d_1(x, z)+ s ) ( d_1(y, z)+ s ) } \\= & {} \frac{s}{ ( d_1(x, z)+ s ) ( d_1(y, z)+ s )}. \end{aligned}$$

The latter implies that

$$\begin{aligned} \lim _{y\rightarrow x}\frac{d_2 (x,y) }{ d_1(x,y)}= \frac{s}{ ( d_1(x, z)+ s )^2} , \end{aligned}$$

which gives the conclusion. $\square$

As a corollary of the previous result, one immediately has a proof for Proposition 1.4. In fact, by setting $d_1=d_{\mathrm{CC}}$ and applying Proposition 3.2 it follows that the identity map $(\partial D, d_{CC}) \rightarrow (\partial D,\rho ^g_o)$ is 1-quasi-conformal.

For possible future reference, we conclude this section by providing an explicit formula for the “distance” function $\rho ^g_o$, where g is as defined in (1.2) and $\rho ^g_o$ is its visual metric as defined in (1.3). We represent p and q by two sequences $x_i$ and $y_i\in D$, respectively. Notice that since $x_i\rightarrow p$ in ${\mathbb {C}}^n$ then $\pi (x_i)\rightarrow p$ and $h(x_i)\rightarrow 0$. In particular, we also have that $\max (h(x_i),h(o)) = h(o)$. Similar considerations apply to $y_i$ and q. Elementary computations yield the following

$$\begin{aligned} \rho ^{g}_o ( p,q)= & {} \exp (- \langle p,q \rangle _o) \\= & {} \lim _{i\rightarrow \infty } \exp (- \langle x_i, y_i \rangle _o) \\= & {} \lim _{i\rightarrow \infty } \exp (- {1 \over 2}( g(x_i,o) + g(y_i,o) - g(x_i,y_i ))) \\= & {} \lim _{i\rightarrow \infty } \exp \bigg ( - \log \bigg ( \frac{d_{\mathrm{CC}}(\pi (x_i), \pi (o))+ \max (h(x_i),h(o) ) }{\sqrt{h(x_i) h(o)} } \bigg )\\&\quad - \log \bigg ( \frac{d_{\mathrm{CC}}(\pi (y_i), \pi (o))+ \max (h(y_i),h(o) ) }{\sqrt{h(y_i) h(o)} } \bigg ) \\&\quad + \log \bigg ( \frac{d_{\mathrm{CC}}(\pi (x_i), \pi (y_i))+ \max (h(x_i),h(y_i) ) }{\sqrt{h(x_i) h(y_i)} } ) \bigg ) \\= & {} \lim _{i\rightarrow \infty } \bigg ( \frac{d_{\mathrm{CC}} (p, \pi (o))+ h(o) }{\sqrt{h(x_i) h(o)} } \bigg )^{-1} \bigg ( \frac{d_{\mathrm{CC}}(q, \pi (o))+ h(o) }{\sqrt{h(y_i) h(o)} } \bigg )^{-1} \frac{d_{\mathrm{CC}}(p,q) }{\sqrt{h(x_i) h(y_i)} } \\= & {} \frac{ d_{\mathrm{CC}}(p,q) \,\, h(o) }{ (d_{\mathrm{CC}}(p, \pi (o))+ h(o) )( d_{\mathrm{CC}}(q, \pi (o))+ h(o) )}. \end{aligned}$$

4 Comparing d and g, after Balogh and Bonk

The quantitative bounds on the distortion of the identity map in Proposition 1.5 follow from the following result, which is a refinement of an analogue statement of Balogh and Bonk [3, Corollary 1.3]. We follow largely their arguments, but where in [3] the noise due to the rough geometry would yield an additive constant, here instead we need to exploit the fact that the geometry is asymptotically hyperbolic to show that such constants can be chosen arbitrarily small the closer one gets to the boundary.

Theorem 4.1

For every ${\bar{p}}\in \partial D$ and $\epsilon >0$ there exists $r>0$ such that for all distinct $p,q\in \partial D\cap B({\bar{p}},r)$ there exists $r'>0$ such that for all $x \in D\cap B(p,r')$ and all $y \in D\cap B(q,r')$

$$\begin{aligned} g(x, y) -\epsilon \le d_K(x, y)\le g(x, y) +\epsilon . \end{aligned}$$

(4.2)

In the rest of the paper we will refer to this result in connection with the quintuplet $({\bar{p}}, p,q,x,y)$.

4.1 Lemmata

The proof of Theorem 4.1 is based on preliminary estimates established in Lemmas 4.3, 4.8, and 4.12.

Lemma 4.3

Let $\delta _0>0$ to be the constant in Lemma 2.5. For $x_1,x_2\in D$ with $d_E(x_i, \partial D)<\delta _0$, and $h(x_1)\ge h(x_2)$, consider a piecewise $C^1$ curve $\gamma :[0,1]\rightarrow N_{\delta _0}$ with $\gamma (0)=x_1$ and $\gamma (1)=x_2$. The length $l_K(\gamma )$ of $\gamma$ with respect to the metric $d_K$ satisfies

$$\begin{aligned} l_K(\gamma )\ge \ln \frac{h(x_1)}{h(x_2)} - C\bigg (h(x_1)- h(x_2)\bigg ), \end{aligned}$$

(4.4)

where C is the same constant as in (2.8). Moreover, if the curve is a segment $\gamma (t)=x_1+t(x_2-p_1)\subset \pi ^{-1}(p)$ for some $p\in \partial D$ then one has

$$\begin{aligned} l_K(\gamma )\le \ln \frac{h(x_1)}{h(x_2)} + C\bigg (h(x_1)- h(x_2)\bigg ), \end{aligned}$$

(4.5)

Proof

Recall from [3, page 517] that

$$\begin{aligned} \bigg | \frac{d}{dt} h(\gamma (t)) \bigg | =\frac{ | Re\langle n(\pi (\gamma (t))), \gamma '(t)\rangle |}{2h(\gamma (t))} \le \frac{|\gamma _N'(t)|}{2 h(\gamma (t))}. \end{aligned}$$

The latter and (2.8) yield

$$\begin{aligned} l_K(\gamma )= & {} \int _0^1 K(\gamma (t), \gamma '(t)) dt \\\ge & {} \int _0^1 (1- C d_E^{\frac{1}{2}}(\gamma (t), \partial D)) \bigg ( \frac{ |\gamma _N'(t)|^2}{4 d_E^2(\gamma (t), \partial D)} + (1-{\bar{\epsilon }}) \frac{ L_{\varphi }(\pi (\gamma (t), \gamma _H'(t)) }{ d_E(\gamma (t), \partial D)} \bigg )^\frac{1}{2} dt \\\ge & {} \int _0^1 (1- C d_E^{\frac{1}{2}}(\gamma (t), \partial D)) \frac{ |\gamma _N'(t)|}{2 d_E(\gamma (t), \partial D)}dt \ge \int _0^1 \frac{(1- C h(\gamma (t))}{h(\gamma (t))} \frac{d}{dt} h(\gamma (t)) dt \\= & {} \ln \frac{h(x_1)}{h(x_2)} - C (h(x_1)-h(x_2)), \end{aligned}$$

which gives (4.4).

On the other hand, if $\gamma (t)=x_1+t(x_2-x_1)$, then we observe that $\gamma '$ is parallel to the unit normal at $\pi (x_i)$ and so has no tangent component, hence no horizontal component with respect to the splitting at $\pi (x_i)$. Using the fact that

$$\begin{aligned} d_E(\gamma (t),\partial D)= |\gamma (t)-p|=| t (x_2-x_1)|+|x_1-p|, \end{aligned}$$

and (2.8) one has

$$\begin{aligned} l_K(\gamma )= & {} \int _0^1 K(\gamma (t), \gamma '(t)) dt \\\le & {} \int _0^1 (1+ C d_E^{\frac{1}{2}}(\gamma (t), \partial D)) \bigg ( \frac{ |\gamma _N'(t)|^2}{4 d_E^2(\gamma (t), \partial D)} + (1+{\bar{\epsilon }}) \frac{ L_{\varphi }(\pi (\gamma (t), \gamma _H'(t)) }{ d_E(\gamma (t), \partial D)} \bigg )^{1 \over 2}dt \\= & {} \frac{1}{2} \int _0^1 \frac{ |x_2-x_1|}{ t |x_2-x_1|+ |x_1-p|} dt + \frac{C}{2} \int _0^1 \frac{1}{\sqrt{ t |x_2-x_1|+ |x_1-p| }} dt \\= & {} \ln \frac{h(x_1)}{h(x_2)} + C (h(x_1)-h(x_2)), \end{aligned}$$

which gives (4.5). $\square$

An immediate consequence of Lemma 4.3 is the following.

Corollary 4.6

Let $\delta _0>0$ to be the constant in Proposition 2.5. If $x_1,x_2\in D$, with $\delta _0>h(x_1)\ge h(x_2)$, then

$$\begin{aligned} \ln \frac{h(x_1)}{h(x_2)} - C\big (h(x_1)- h(x_2)\big ) \le d_K(x_1,x_2). \end{aligned}$$

Moreover, if $\pi (x_1)=\pi (p_2)$, then we also have

$$\begin{aligned} d_K(x_1,x_2) \le \ln \frac{h(x_1)}{h(x_2)} + C\big (h(x_1)- h(x_2)\big ) \end{aligned}$$

(4.7)

where C is the same constant as in (2.8).

The next lemma provides an upper bound for $d_K(x_1,x_2)$ in the case when both points $x_1,x_2$ are at the same distance from the boundary and equal to the Carnot–Carathéodory distance between the projections $\pi (x_1), \pi (x_2)$.

Lemma 4.8

Let $p_1,p_2\in \partial D$. If we set $x_i := p_i - d_{CC}(p_1,p_2) n(p_i)$, $i=1,2$, then

$$\begin{aligned} d_K(x_1,x_2)\rightarrow 0, \qquad \text {as } \,d_{CC}(p_1,p_2)\rightarrow 0. \end{aligned}$$

(4.9)

Proof

Let $\eta >0$ and let $\alpha :[0,1]\rightarrow \partial D$ be any horizontal curve with $\alpha (0)=p_1$ and $\alpha (1)=p_2$, such that its subRiemannian length $l_{CC}$, satisfies

$$\begin{aligned} l_{CC}(\alpha )= \int _0^1 L^{\frac{1}{2}}_{\varphi }(\alpha (t), \alpha '(t)) dt \le d_{CC}(p_1,p_2)(1+\eta ). \end{aligned}$$

Define a new curve $\gamma :[0,1]\rightarrow D$ as a lift at height $h\in (0,\delta _0)$ of $\alpha$ by the formula

$$\begin{aligned} \gamma (t):= \alpha (t) - h \ n(\alpha (t)). \end{aligned}$$

(4.10)

Arguing as in the proof of [3, Lemma 2.2] yields the following relations between $\alpha '$ and $\gamma '$,

$$\begin{aligned} L(\alpha (t), \gamma _H'(t))= & {} L(\alpha (t), \alpha '(t)) +O(h |\alpha '(t)|^2 ) \nonumber \\ |\gamma _N'(t)|= & {} O(h |\alpha '(t)|). \end{aligned}$$

(4.11)

In fact, from (4.10) one has $\gamma '(t)|_H= \alpha '(t)-h [dn|_{\alpha (t)} \alpha '(t)]_H$ which, together with the bilinearity of the Levi form, yields (4.11). Consequently we have

$$\begin{aligned} l_K(\gamma )= & {} \int _0^1 K(\gamma (t), \gamma '(t)) dt \\\le & {} \int _0^1 (1+ C d_E^{\frac{1}{2}}(\gamma (t), \partial D)) \bigg ( \frac{ |\gamma _N'(t)|^2}{4 d_E^2(\gamma (t), \partial D)} + (1+{\bar{\epsilon }}) \frac{ L_{\varphi }(\pi (\gamma (t), \gamma _H'(t)) }{ d_E(\gamma (t), \partial D)} \bigg )^{1 \over 2}dt \\= & {} \int _0^1 (1 +C h^{{1 \over 2}} ) \bigg ( \frac{ |\gamma _N'(t)|^2}{4 h^2} + (1+{\bar{\epsilon }}) \frac{ L_{\varphi }(\pi (\gamma (t), \gamma _H'(t)) }{ h} \bigg )^{1 \over 2}dt \\\le & {} \int _0^1 (1 +C h^{{1 \over 2}} ) \bigg ( C |\alpha '(t)|^2 + (1+{\bar{\epsilon }}) \frac{L_{\varphi }(\alpha (t), \alpha '(t))}{h} \bigg )^{1 \over 2}dt \\\le & {} (1+C\sqrt{h}) (1+\eta ) \bigg [C d_{CC}(p_1,p_2) + (1+{\bar{\epsilon }})^{1 \over 2}h^{-{1 \over 2}} d_{CC}(p_1,p_2)\bigg ]. \end{aligned}$$

Setting $h=d_{CC}(x_1,x_2)$ in the latter yields the conclusion. $\square$

The next lemma will be instrumental in establishing a lower bound for $d_K(x_1,x_2)$ in the case when a length minimizing arc $\gamma$ joining two points $x_1,x_2\in D$ will travel at a distance further than the Carnot–Carathéodory distance between their projections.

Lemma 4.12

Let $\delta _0>0$ be smaller than the similarly named constants in Propositions 2.5 and 2.7. Consider two points $x_1,x_2\in D$ with $d_E(x_i,\partial D) <\delta _0$. Set $p_i=\pi (x_i)\in \partial D$, and let $\gamma :[0,1]\rightarrow D$ denote an arc joining $x_1$ and $x_2$. If $\max _{z\in \gamma } h(z)\ge d_{CC}(p_1,p_2)$ then

$$\begin{aligned} l_K(\gamma ) \ge 2 \ln \bigg ( \frac{d_{CC}(p_1,p_2)}{\sqrt{h(x_1)h(x_2)} } \bigg )- C(2d_{CC}(p_1,p_2)- h(x_1)-h(x_2)), \end{aligned}$$

(4.13)

where C is the same constant as in (2.8).

Proof

Choose $t_0\in [0,1]$ such that $h(\gamma (t_0))=\max _{z\in \gamma } h(z)$. Set $\gamma _1,\gamma _2$ be the two branches of the curve $\gamma$ corresponding to the subintervals $[0,t_0]$ and $[t_0,1]$. Set also ${\bar{\gamma }}_1$ and ${\bar{\gamma }}_2$ to be the connected components on $\gamma _1$ and $\gamma _2$ joining $x_i$ to the closest points $y_i\in \gamma$ such that $h(y_i)=d_{CC}(p_1,p_2)$, for $i=1,2$. More formally, $y_1=\gamma (t_1)$, with $t_1=\inf \{t\in [0,t_1] \$such that $h(\gamma (t))\ge d_{CC}(p_1,p_2)\}$. The point $y_2$ is defined analogously.

Next we invoke Lemma 4.3 to deduce

$$\begin{aligned} l_K(\gamma )\ge & {} l_K({\bar{\gamma }}_1)+ l_K({\bar{\gamma }}_2) \\\ge & {} \ln \frac{d_{CC}(p_1,p_2)}{h(x_1)} + \ln \frac{d_{CC}(p_1,p_2)}{h(x_2)} - C(2d_{CC}(p_1,p_2)- h(x_1)-h(x_2)) \\= & {} 2 \ln \bigg ( \frac{d_{CC}(p_1,p_2)}{\sqrt{h(x_1)h(x_2)} } \bigg )- C(2d_{CC}(x_1,x_2)- h(x_1)-h(x_2)), \end{aligned}$$

which is the desired bound (4.13). $\square$

4.2 Proof of Theorem 4.1

Thanks to the previous lemmata we can now prove the main result of the section.

Proof of Theorem 4.1

We shall show that for all ${\bar{p}}\in \partial D$ and $\epsilon >0$ one can choose $r>0$ small enough so that for all distinct $p,q\in \partial D\cap B({\bar{p}},r)$ one can find $r'\in (0, r)$ such that (4.2) holds for all $x \in D\cap B(p,r')$ and all $y \in D\cap B(q,r')$. In our proof we begin with arbitrary values of r and $r'$ and then put several constrains on them.

If p and q are distinct, then the value $d_1:=d_{CC}(p,q)$ is strictly positive. We shall choose r smaller that the constants $\delta _0$ in Propositions 2.5 and 2.7 and so that $d_1$ is small enough to be determined later. Denote by ${\bar{x}}$, and ${\bar{y}}$ the projections on the boundary of x and y, respectively. Note that since the projections are the closest points in $\partial D$, then ${\bar{x}}\in B(p,2r')$ and ${\bar{y}}\in B(q,2r')$. Set $d_2:=d_{CC}({\bar{x}}, {\bar{y}})$. Notice that as $r'\rightarrow 0$ we have $d_2\rightarrow d_1$. We shall choose $r'$ sufficiently small so that $r' <d_2$ and $d_2\in (d_1/2, 2 d_1)$. In particular, if r was chosen small enough, then $d_2$ is positive and smaller than the constants $\delta _0$ in Propositions 2.5 and 2.7.

Proof of the upper bound in (4.2). Set $x':= {\bar{x}} - d_2 n({\bar{x}})$ and $y':= {\bar{y}} - d_2 n({\bar{y}})$, so $x'$, $y'$ are points in D at distance $d_2$ from $\partial D$ and with the same projection on $\partial D$ as x, y, respectively.

By Lemma 4.8 we can choose $d_1$ sufficiently small so that $d_K(x', y')<\epsilon /3$. Invoking (4.7), since $h(x')=h(y')=d_2>\max \{h(x), h(y)\}$, yields

$$\begin{aligned} d_K(x,x')\le & {} \log (d_2/h(x)) + C( d_2 - h(x) )\qquad \text {and }\\ d_K(y,y')\le & {} \log (d_2/h(y)) + C( d_2 - h(y) ). \end{aligned}$$

Choose $d_1$ chosen sufficiently small so that $C d_2 \le \epsilon /3$.

Combining the previous bounds with the definition of g, we obtain the following estimates

$$\begin{aligned} d_K(x,y)-g(x,y)\le & {} d_K(x,x')+d_K(x',y')+d_K(y',y)-g(x,y) \\\le & {} \log (d_2/h(x)) + C( d_2 - h(x) ) + \epsilon /3 \\&+ \log (d_2/h(y)) + C( d_2 - h(y) ) - 2\ln \bigg ( \frac{d_2 + h(x) \wedge h(y))}{\sqrt{h(x) h(y)}}\bigg )\\\le & {} \epsilon -C h(x) - C h(y) - 2 \ln \bigg ( 1 + \frac{ h(x) \wedge h(y)}{d_2} \bigg ) < \epsilon , \end{aligned}$$

where we used that the terms $h(x),h(y) , \ln ( 1 + \frac{ h(x) \wedge h(y)}{d_2} )$ are positive. This conclude the proof of the upper bound in (4.2).

Proof of the lower bound in (4.2). Choose $\delta >0$ such that $\ln (1/(1+\delta ))<\epsilon$ and $r'>0$ small enough so that $\frac{\max (h(x),h(y))}{d_{CC}(p,q)} \le \delta$, for all $x \in D\cap B(p,r')$ and all $y \in D\cap B(q,r')$. Consider any arc $\gamma :[0,1]\rightarrow D$ joining x and y, and set $H:=\max _{z\in \gamma } h(z)$.

- If $H \ge d_{CC}({\bar{x}},{\bar{y}})$ then in view of Lemma 4.12 we have

$$\begin{aligned} d_K(x,y)- g(x,y)\ge & {} 2 \ln \bigg ( \frac{d_2}{\sqrt{h(x)h(y)} } \bigg )- C(2d_2- h(x)-h(y))) \nonumber \\&- 2\ln \bigg ( \frac{d_2 + h(x) \wedge h(y))}{\sqrt{h(x) h(y)}} \bigg )\nonumber \\\ge & {} 2 \ln \bigg ( \frac{1}{1+ \frac{\max (h(x),h(y))}{d_2}} \bigg ) - C(2d_2-h(x)-h(y)) \nonumber \\\ge & {} - (C+2) \epsilon . \end{aligned}$$

(4.14)

In this case the proof is concluded.

- If $H \le d_{CC}({\bar{x}},{\bar{y}})$ then it follows that H is smaller than the constants $\delta _0$ in Propositions 2.5 and 2.7. In particular we can assume without loss of generality that $CH<1/2$, where C is as in (2.8). Let $t_0\in [0,1]$ be such that $h(\gamma (t_0))=H$ and consider the two branches $\gamma _1,\gamma _2$ of $\gamma$ given by restrictions to $[0,t_0]$ and $[t_0,1]$. Given $\epsilon >0$ as in the statement, let $\theta \in (1,2]$ so that $\ln \theta <\epsilon$ and define $k\in {\mathbb {N}}$ such that

$$\begin{aligned} h(\gamma (0))\in \Bigg [\frac{H}{\theta ^k}, \frac{H}{\theta ^{k-1}}\Bigg ]. \end{aligned}$$

Following [3], we define $s_0,s_1,\ldots ,s_k\in [0,t_0]$ such that $s_0=0$ and

$$\begin{aligned} s_l = \min \Bigg \{s\in [0,t_0] \ \text { such that } h(\gamma (s))= \frac{H}{\theta ^{k-l}}\Bigg \}. \end{aligned}$$

Set $t_1=s_k\le t_0$ and for each $l=1,\ldots ,k$,

$$\begin{aligned} \nu _l^{-1}= \frac{d_{CC}({\bar{x}}, {\bar{y}}) \cdot (\theta -1) }{ 8\theta ^{k-l}}. \end{aligned}$$

For each of the two branches $\gamma _1,\gamma _2$, we distinguish two alternatives:

• Alternative $\# 1$ (All sub-arcs have large slope) In this alternative we assume that for every $l=1,\ldots ,k$ one has

  $$\begin{aligned} d_{CC}(\pi (\gamma (s_{l-1})), \pi (\gamma (s_l))) \le \nu _l^{-1} \end{aligned}$$

  (4.15)

  From the latter we draw two conclusions. The first is a simple application of the triangle inequality,

  $$\begin{aligned}&d_{CC} ({\bar{z}}, \pi (\gamma (t_1)) \\&\quad \le \sum _{l=1}^k d_{CC} (\pi (\gamma (s_{l-1})), \pi (\gamma (s_l))) \le (\theta -1) \frac{d_{CC}({\bar{x}},{\bar{y}})}{8 \theta ^k}\sum _{l=1}^k \theta ^l \le \frac{d_{CC}({\bar{x}},\bar{y})}{4}. \end{aligned}$$

  (A1 (i))

  On the other hand, in view of Lemma 4.3 one has

  $$\begin{aligned} l_K(\gamma |_{[0,t_1]}) \ge \ln \frac{h(\gamma (t_1))}{h(x)} -C(h(\gamma (t_1))-h(x)) = \ln \frac{H}{h(x)} -C(H-h(x)). \end{aligned}$$

  (A1 (ii))

• Alternative $\# 2$ (One sub-arc has small slope) In this alternative, we assume that there exists $l\in \{1,\ldots ,k\}$ such that

  $$\begin{aligned} d_{CC}(\pi (\gamma (s_{l-1})), \pi (\gamma (s_l))) > \nu _l^{-1} \end{aligned}$$

  (4.16)

  Note that if $s\in [s_{l-1},s_l]$ then from the definition of the points $s_l$, one has

  $$\begin{aligned} h(\gamma (s)) \le \theta ^{l-k}H\le \frac{8}{\theta -1} \nu _l^{-1}. \end{aligned}$$

  We then claim that there exists a constant ${\mathcal {C}}>0$ depending only on the defining function ${\varphi }$ such that

  $$\begin{aligned} l_K(\gamma |_{[s_{l-1},s_l]}) \ge {\mathcal {C}} (\theta -1)^2 \frac{d_{CC}({\bar{x}},{\bar{y}})}{H}. \end{aligned}$$

  (4.17)

  Indeed, arguing as in [3, page 521] we invoke (2.8), Lemma 2.10, and (4.16) and we bound as follows:

  $$\begin{aligned}&l_K(\gamma |_{[s_{l-1},s_l]}) \\&\quad \ge {\mathcal {C}} \frac{(1-CH)\theta ^{k-l}}{H} \int _{s_{l-1}}^{s_l} \bigg [ L_{{\varphi }}(\pi (\gamma (s)), [\pi (\gamma (s))]'_H) \\&\qquad +(\theta -1)^2 \nu _l^{2} |[\pi (\gamma (s))]'_N|^2 \bigg ]^{\frac{1}{2} }ds \\&\quad \ge \frac{{\mathcal {C}}}{2}(\theta -1) \frac{\theta ^{k-l}}{H} \int _{s_{l-1}}^{s_l} \bigg [ L_{{\varphi }}(\pi (\gamma (s)), [\pi (\gamma (s))]'_H) + \nu _l^{2} |[\pi (\gamma (s))]'_N|^2 \bigg ]^{\frac{1}{2} }ds \\&\quad \ge {\mathcal {C}} (\theta -1) \frac{\theta ^{k-l}}{H} d_{\nu _l} (\pi (\gamma (s_{l-1})), \pi (\gamma (s_l))) \\&\quad \ge {\mathcal {C}} (\theta -1) \frac{\theta ^{k-l}}{H} d_{CC} (\pi (\gamma (s_{l-1})), \pi (\gamma (s_l))) \\&\quad \ge {\mathcal {C}} (\theta -1)^2 \frac{d_{CC}({\bar{x}},\bar{y})}{H}, \end{aligned}$$

  where $d_{\nu _l}$ denotes the approximation of the Carnot–Carathéodory metric defined in (2.9).

  Next we claim that

  $$\begin{aligned} l_L(\gamma |_{[0,t_1]}) \ge \ln \bigg ( \frac{H}{h(y)} \bigg ) + \frac{{\mathcal {C}} (\theta -1)^2}{H} d_{CC}({\bar{x}},{\bar{y}})- C\big (H -h(y)\big ) -\epsilon . \end{aligned}$$

  (A2)

  Indeed, Lemma 4.3 and (4.17) yields

  $$\begin{aligned} l_L(\gamma |_{[0,t_1]})= & {} l_K(\gamma |_{[0,s_{l-1}]}) + l_K(\gamma |_{[s_{l-1},s_l]}) + l_K(\gamma |_{[s_l,t_1]} ) \\\ge & {} \ln \bigg ( \frac{H}{h(\gamma (s_l))} \frac{h(\gamma (s_{l-1}))}{h(x)} \bigg ) + \frac{{\mathcal {C}} (\theta -1)^2}{H} d_{CC}({\bar{x}},{\bar{y}}) \\&\quad -C\big (H-h(\gamma (s_{l}) + h (\gamma (s_{l-1})) -h(x)\big ) \\\ge & {} \ln \bigg ( \frac{H}{h(x)} \theta ^{-1} \bigg ) + \frac{{\mathcal {C}} (\theta -1)^2}{H} d_{CC}({\bar{x}},{\bar{y}})- C\big (H -h(x)\big ) \\\ge & {} \ln \bigg ( \frac{H}{h(x)} \bigg ) + \frac{{\mathcal {C}} (\theta -1)^2}{H} d_{CC}({\bar{x}},{\bar{y}})- C\big (H -h(x)\big )-\epsilon . \end{aligned}$$

Applying similar consideration to the branch $\gamma _2$ one obtains a $t_2\in [t_0,1]$ such that one of the following two alternatives hold: Either

$$\begin{aligned} d_{CC} ({\bar{y}}, \pi (\gamma (t_2)) \le \frac{d_{CC}({\bar{x}},{\bar{y}})}{4} \ \text { and } \ \ l_K(\gamma |_{[t_2,1]}) \ge \ln \frac{H}{h(y)} -C(H-h(y)). \end{aligned}$$

(B1)

or

$$\begin{aligned} l_L(\gamma |_{[t_2,1]}) \ge \ln \bigg ( \frac{H}{h(y)} \bigg ) + \frac{{\mathcal {C}} (\theta -1)^2}{H} d_{CC}({\bar{x}},{\bar{y}})- C\big (H -h(y)\big ) -\epsilon . \end{aligned}$$

(B2)

To conclude the proof we need to examine all possible combinations of these alternatives. We will show that in each case one obtains

$$\begin{aligned} l_K(\gamma )\ge 2 \ln \bigg (\frac{d_{CC}({\bar{x}},{\bar{y}})}{\sqrt{h(x)h(y)}} \bigg ) - C( 2d_{CC}({\bar{x}}, {\bar{y}}) -h(x)-h(y)) -\epsilon . \end{aligned}$$

(4.18)

• Suppose both (A1) and (B1) hold. Observe that

  $$\begin{aligned} d_{CC}(\pi (\gamma (t_1)),\pi (\gamma (t_2)))\ge & {} d_{CC}({\bar{x}},{\bar{y}}) - d_{CC}({\bar{x}},\pi (\gamma (t_1))) - d_{CC}({\bar{y}},\pi (\gamma (t_2))) \\\ge & {} \frac{ d_{CC}({\bar{x}},{\bar{y}})}{2}. \end{aligned}$$

  Repeating the argument in (4.17) for $l=k$ and invoking the Riemannian approximation lemma [3, Lemma 3.2] one has

  $$\begin{aligned} l_L(\gamma |_{[t_1,t_2]}) \ge {\mathcal {C}} (\theta -1)^2 \frac{d_{\nu _k}(\pi (\gamma (t_1)),\pi (\gamma (t_2)))}{H} \ge {\mathcal {C}} (\theta -1)^2 \frac{d_{CC}({\bar{x}},{\bar{y}})}{H}. \end{aligned}$$

  The latter, together with (A1 (ii)), and the second inequality in (B1) yields

  $$\begin{aligned} l_K(\gamma ) \ge 2 \ln \bigg (\frac{H}{\sqrt{h(x)h(y)}} \bigg ) + {\mathcal {C}} (\theta -1)^2 \frac{d_{CC}({\bar{x}},{\bar{y}})}{H} - C( 2H-h(x)-h(y)). \end{aligned}$$

  Since the right hand side is monotone decreasing in $H\le d_{CC}({\bar{x}},{\bar{y}})$ then one has

  $$\begin{aligned} l_K(\gamma )\ge & {} 2 \ln \bigg (\frac{d_{CC}({\bar{x}},{\bar{y}})}{\sqrt{h(x)h(y)}} \bigg ) + {\mathcal {C}} (\theta -1)^2 - C( 2d_{CC}({\bar{x}},{\bar{y}}) -h(x)-h(y)) \\\ge & {} 2 \ln \bigg (\frac{d_{CC}({\bar{x}},{\bar{y}})}{\sqrt{h(x)h(y)}} \bigg ) - C( 2d_{CC}({\bar{x}},{\bar{y}}) -h(x)-h(y)) \end{aligned}$$

  completing the proof of (4.18).

• Suppose both (A1) and (B2) hold. One immediately has

  $$\begin{aligned} l_K(\gamma )\ge & {} l_K(\gamma _{[0,t_1]}) + l_K(\gamma |_{[t_2,1]}) \\\ge & {} \ln \bigg ( \frac{H}{h(x)} \bigg ) + \frac{{\mathcal {C}} (\theta -1)}{H} d_{CC}({\bar{x}},{\bar{y}})- C[H -h(x)] \\&-\epsilon + \ln \frac{H}{h(y)} -C(H-h(y)). \end{aligned}$$

  Applying the same consideration as above we immediately deduce (4.18).

• Suppose both (A2) and (B1) hold. This combination is dealt with analogously to the previous case.

• Suppose both (A2) and (B2) hold. Estimate (4.18) follows immediately from (A2) and (B2).

To conclude the proof we need to consider the infimum of $l_K(\gamma )$ among all arcs $\gamma$ joining x and y and apply (4.18) to each. One has

$$\begin{aligned} d_K(x,y)-g(x,y)\ge & {} 2 \ln \bigg (\frac{d_{CC}({\bar{x}},{\bar{y}})}{\sqrt{h(x)h(y)} } \frac{1}{ \frac{d_{CC}({\bar{x}},{\bar{y}})}{\sqrt{h(x)h(y)}} +\frac{\max \{h(x),h(y)\} }{\sqrt{h(x)h(y)} }} \bigg ) \\&\quad - C( 2d_{CC}({\bar{x}},{\bar{y}}) -h(x)-h(y)) -\epsilon \\= & {} -2 \ln \bigg ( 1 +\frac{\max \{h(x),h(y)\} }{d_{CC}({\bar{x}},{\bar{y}}) } \bigg ) \\&\quad - C( 2d_{CC}({\bar{x}},{\bar{y}}) -h(x)-h(y)) -\epsilon . \end{aligned}$$

The proof is then concluded by applying the same argument as in (4.14). $\square$

5 Local biLipschitz equivalence of visual quasi-distances and proof of main result

In this section we prove Proposition 1.5 and the main result, Theorem 1.1.

Proof of Proposition 1.5

Let ${\bar{p}}$ as in the statement and choose $\epsilon >0$ such that $\exp (\frac{3}{2}\epsilon )\le 1+{\bar{\epsilon }}$. Invoke Theorem 4.1 in correspondence to the choice of ${\bar{p}}$ and $\epsilon$, to obtain the value $r>0$ and select any $\omega \in \partial D\cap B({\bar{p}},r){\setminus } \{{\bar{p}}\}$. In correspondence to this choice of $\omega$, Theorem 4.1 yields a smaller radius $0<r'<r$, so that if we choose $y\in D\cap B({\bar{p}},r')$ and $o\in D\cap B(\omega ,r')$ and then apply Theorem 4.1 to the quintuplet $({\bar{p}}, {\bar{p}}, \omega , y,o)$ we obtain

$$\begin{aligned} |g(y,o)-d_K(y,o)|<\epsilon , \qquad \text { for all } y\in D\cap B({\bar{p}},r') ,\ \text { and } o\in D\cap B(\omega ,r') \end{aligned}$$

Next, given $p,q\in \partial D\cap B({\bar{p}}, r')$ we similarly use Theorem  4.1 to infer the existence of a $r''>0$ for which, applying Theorem 4.1 to the quintuplet $({\bar{p}}, p, q x, y )$

$$\begin{aligned} |d_K(x,y) -g(x,y) |\le \epsilon , \quad \text { for all } x\in D\cap B( p,r'') ,\, \text { and for all } y\in D\cap B(q,r''). \end{aligned}$$

If $x_i$ (resp., $y_i$) is a sequence in D converging to p (resp., q), then for i large enough $x_i\in D\cap B( p,r'')$ and $y_i\in D\cap B(q,r'')$ and $x_i , y_i\in B({\bar{p}},r')$. From the above bounds one obtains

$$\begin{aligned} \left| \langle y_i,x_i\rangle _o^g- \langle y_i,x_i\rangle _o^K \right|= & {} \frac{1}{2} \left| g(y_i,o)-d_K(y_i,o) + g(x_i,o) \right. \\&\left. -d_K(x_i,o)+d_K(x_i,y_i) -g(x_i,y_i) \right| \\\le & {} \frac{3}{2}\epsilon . \end{aligned}$$

Consequently, if the sequences $x_i$, $y_i$ are taken so that $\langle p,q\rangle _o^g = \lim _{i\rightarrow \infty } \langle y_i,x_i\rangle _o^g$, we have

$$\begin{aligned} \dfrac{ \rho ^K_o(p,q)}{\rho ^g_o(p,q)}\le & {} \frac{\lim _{i\rightarrow \infty } \exp (-\langle y_i,x_i\rangle _o^K)}{ \lim _{i\rightarrow \infty }\exp (-\langle y_i,x_i\rangle _o^g)} \\= & {} \lim _{i\rightarrow \infty } \exp \left( \langle y_i,x_i\rangle _o^g- \langle y_i,x_i\rangle _o^K \right) \\\le & {} \exp ( \frac{3}{2}\epsilon ) \le 1+{\bar{\epsilon }}. \end{aligned}$$

And similarly, ${\rho ^g_o(p,q)}/{ \rho ^K_o(p,q)}$ is bounded by $1+ {\bar{\epsilon }}$. $\square$

Proof of Theorem 1.1

For any ${\bar{p}}\in \partial D_1$ and ${\bar{\epsilon }}>0$ we show that the boundary extension is $(1+{\bar{\epsilon }})-$quasi-conformal at ${\bar{p}}$, i.e. $H^*({\bar{p}},F, d_{CC}, d_{CC})\le 1+{\bar{\epsilon }}$, where $H^*$ is defined as in (2.1). Following the diagram (D) in the introduction, from (2.3) for every $o\in D_1$ we have

$$\begin{aligned}&H^*({\bar{p}},F, d_{CC}, d_{CC}) \nonumber \\&\quad \le H^*({\bar{p}}, \mathrm{Id}_{\partial D_1} , d_{\mathrm{CC}}, \rho ^{g}_o) H^*( {\bar{p}}, \mathrm{Id}_{\partial D_1} , \rho ^g_o, \rho ^K_o) H^*({\bar{p}}, F, \rho ^{K}_o, \rho ^{K}_{f(o)}) \nonumber \\&\quad \cdot H^*(F({\bar{p}}), \mathrm{Id}_{\partial D_2},\rho ^K_{f(o)}, \rho ^g_{f(o)} ) H^*(F({\bar{p}}),\mathrm{Id}_{\partial D_2}, \rho ^{g}_{f(o)}, d_{\mathrm{CC}}). \end{aligned}$$

(5.1)

Start by observing that for any $o\in D_1$ the pointed metric spaces $(D_1, d_{K}, o)$ and $(D_2,d_{K}, f(o))$ are isometric. Thus they give rise to visual boundaries that are isometric with respect to the induced distances $\rho ^{K}_o$ an $\rho ^{K}_{f(o)}$, as defined in (1.3). Consequently the induced extension map $F:(\partial D_1, \rho ^{K}_o) \rightarrow (\partial D_2, \rho ^{K}_{f(o)})$ is an isometry, and hence from (2.4)

$$\begin{aligned} H^*({\bar{p}}, F, \rho ^{K}_o, \rho ^{K}_{f(o)}) =1. \end{aligned}$$

(5.2)

Regarding the first and last term in the right-hand side of (5.1), in view of Proposition 1.4 we have that

$$\begin{aligned} H^*({\bar{p}}, \mathrm{Id}_{\partial D_1} , d_{\mathrm{CC}}, \rho ^{g}_o)=H^*(F({\bar{p}}),\mathrm{Id}_{\partial D_2}, \rho ^{g}_{f(o)}, d_{\mathrm{CC}})=1. \end{aligned}$$

(5.3)

We shall then prove that

$$\begin{aligned} H^*( {\bar{p}}, \mathrm{Id}_{\partial D_1} , \rho ^g_o, \rho ^K_o) \le 1+{\bar{\epsilon }} \, \text { and } \, H^*(F({\bar{p}}), \mathrm{Id}_{\partial D_2},\rho ^K_{f(o)}, \rho ^g_{f(o)} ) \le 1+{\bar{\epsilon }}, \end{aligned}$$

(5.4)

for some suitable choice of o. To prove this we will need to invoke Proposition 1.5 twice, in $D_1$ and in $D_2$, together with the observation (2.4). Namely, we shall prove that for a suitable choice of o The maps considered in (5.4) are $( 1+{\bar{\epsilon }} )$-biLipschitz in a neighborhood of the considered points.

First we apply Proposition 1.5 in a neighborhood of $F({\bar{p}}) \in \partial D_2$, thus yielding $r_2>0$ such that for all $\omega _2\in \partial D_2\cap B(F({\bar{p}}), r_2){\setminus } \{ F({\bar{p}})\}$ there exists $r_2'>0$ such that for all $o_2\in D_2 \cap B(\omega _2,r_2')$ one has that $\rho _{o_2}^g$ and $\rho _{o_2}^K$ are $(1+{\bar{\epsilon }})$-biLipschitz in $\partial D_2\cap B(F({\bar{p}}), r_2')$. For the moment we do not choose any specific $\omega _2$ and $o_2$, so $r_2'$ is still to be determined.

Next, we apply Proposition 1.5 to $D_1$ in a neighborhood of ${\bar{p}}$ and use it to choose $r_1>0$ such that for all $\omega _1\in \partial D_1\cap B({\bar{p}}, r_1){\setminus } \{ {\bar{p}}\}$ there exists $r_1'>0$ such that $o_1\in D_1 \cap B(\omega _1,r_1')$ one has that $\rho _{o_1}^g$ and $\rho _{o_1}^K$ are $(1+{\bar{\epsilon }})$-biLipschitz in $\partial D_1\cap B({\bar{p}}, r_1')$. By continuity of the map F we may have chosen $r_1$ small enough that $F(B({\bar{p}}, r_1)\cap D_1)\subset B(F({\bar{p}}), r_2)\cap D_2$.

We set $\omega _2:=F(\omega _1)$, which is then in $B(F({\bar{p}}), r_2)\cap D_2$ and is different than $F({\bar{p}})$ since F is a homeomorphism. Now we fix $r_2'$ accordingly, as we explained above. If needed we will select a smaller value for $r_1'$ so that we can assume $F(B(\omega _1,r_1')\cap D_1)\subset B(F(\omega _1), r_2')\cap D_2$.

To conclude, we can now select any base point $o\in B(\omega _1,r_1')\cap D_1$, so that $f(o)\in B(\omega _2, r_2')\cap D_2$ and and hence $\rho _{o_1}^g$ and $\rho _{o_1}^K$ are $(1+\bar{\epsilon })$-biLipschitz in $\partial D_1\cap B({\bar{p}}, r_1')$ and $\rho _{o_2}^g$ and $\rho _{o_2}^K$ are $(1+{\bar{\epsilon }})$-biLipschitz in $\partial D_2\cap B(F({\bar{p}}), r_2')$. Thus, (2.4) gives (5.4).

Using the estimates (5.2), (5.3), and (5.4) in (5.1) we get $H^*({\bar{p}},F, d_{CC}, d_{CC}) \le (1+{\bar{\epsilon }})^2$. By the arbitrariness of ${\bar{\epsilon }}$ we deduce $H^*({\bar{p}},F, d_{CC}, d_{CC}) = 1$. Finally, from Lemma 2.2 we conclude. $\square$