Conformal equivalence of visual metrics in pseudoconvex domains

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ 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 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.


Introduction
Let D ⊂ C n (n ≥ 2) be a strongly pseudo-convex domain with C ∞ -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 Carathéodory metric. In [BB00,BB99], 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 ∂D. They also show that the Carnot-Carathéodory metric d CC corresponding to the Levi form on ∂D, determines the canonical class of snowflake equivalent visual metrics on ∂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 [GdlH90,BH99] 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 [CCLDO16].
As in [BB00], 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 Section 2.2), with the invariant metric defined in the domain.
Theorem 1.1. Let D 1 , D 2 ⊂ C n be strongly pseudoconvex C ∞ -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 ) → (D 2 , d K ) is an isometry then the induced boundary map F : (∂D 1 , d CC ) → (∂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 Carathéodory metrics. Indeed, these distances satisfy (2.8) in view of the work in [BB00, BB99,Ma91].
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 ∈ D denote by h(x) := » d E (x, ∂D) and by π(x) ∈ ∂D a closest point in ∂D with respect to the Euclidean distance d E (·, ·), noting it is uniquely defined in a neighborhood of ∂D. Set (1.2) g(x, y) := 2 log Ç d CC (π(x), π(y)) + max(h(x), h(y)) This is an hyperbolic filling metric built from the metric space (∂D, d CC ) (see Bonk and Schramm [BS00]). Balogh and Bonk [BB00, Corollary 1.3], showed that g is a metric in a neighborhood of ∂D and that g and the invariant distance function d K are (1, C)-quasiisometric. 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 ρ K o associated to the invariant metric d K is in fact pointwise and asymptotically (1 + ǫ)-quasi-conformally equivalent to the Carnot-Carathéodory d CC metric. By pointwise and asymptotically we mean that for every point x ∈ ∂D in the boundary, and for every ǫ > 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 + ǫ at x. Following ideas in CAT(−1) spaces, given a pointed metric space (X, d, o) we consider the Bourdon function , where x, y o denotes the Gromov product in (X, d), see Section 2. Usually, ρ d o is called Bourdon distance since for CAT(−1) spaces it satisfies the triangle inequality. In our setting, ρ d o may not be a distance. Moreover, on a CAT(−1) space X Bourdon showed in [Bou95] that the visual boundaries (∂ ∞ X, ρ d o ) corresponding to diffent base points o, o ′ ∈ 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 [Kra13]) 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 equivalent 1 to the Bourdon function ρ g o associated to the hyperbolic filling metric g. In other words, the identity map (∂D, d CC ) → (∂D, ρ 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 ǫ > 0, one can find a base point o ∈ D such that the corresponding visual functions ρ K o and ρ g o are (1 + ǫ)-biLipschitz equivalent in a neighborhood of that point. In the following we denote Euclidean balls in C n with the notation B(x, r).
The proof of Proposition 1.5 and Theorem 1.1 are in Section 5. Theorem 1.1 follows rather directly from Propositions 1.4 and 1.5 and from the following diagram The result holds for any hyperbolic filling as in the work of Bonk and Schramm. See Section 3.2 At the center of this chain of compositions there is an isometry, the rest of the links are either (1 + ǫ) biLipschitz maps or conformal maps, so that the total distortion is at most ǫ away from being equal to 1 everywhere.
From the conformal equivalence theorem above and the results in [CCLDO16], one can immediately infer a result about boundary extensions for biholomorphisms between strictly pseudoconvex domains in C n , originally established by Fefferman [Fef74].
Corollary 1.6. Let D 1 , D 2 ∈ C n (n ≥ 2) be strongly pseudo-convex domains with C ∞smooth boundaries. If f : D 1 → D 2 is a biholomorphism then it extends to a smooth map F : ∂D 1 → ∂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 [Fef74] 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 [BL80, BC82, NWY80, Bar83,Kra15].
Rather than a simplification of Fefferman's original proof, our approach is a recasting of the result from the perspective of analysis in metric spaces and the circle of ideas at the core of Mostow rigidity [Mos73]. The differentiable structure is not used to show that the extension map is 1-quasi-conformal, and then it only enters in play coupled with the rigidity of 1-quasi-conformal mappings in higher dimension. Likewise, curvature enters into the arguments only in its synthetic (metric) form. In particular, our work can be seen as an instance of a dictionary, introduced by Bonk, Heinonen, and Koskela in [BHK01], 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 [Zim16].
Acknowledgements The recasting of Fefferman's result from the point of view of Mostow rigidity and metric hyperbolicity was the main motivation behind this work, and was outlined by Michael Cowling, back in 2007. The authors are very grateful to both Michael Cowling and to Loredana Lanzani for several key observations that have led to a better understanding of the problem.

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 → Y between general metric spaces, we consider the following quantities The quantity L F (x) is sometimes denoted by Lip F (x) and is called the pointwise Lipschitz constant. Within this paper, we define the distortion of f at a point x ∈ X as The homeomorphism f is said to be quasi-conformal if there exists K such that for all It is well-known that in the literature there are several other equivalent definitions of quasiconformality in 'geometrically nice' spaces, see [Wil12]. 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 Theorem 1.3 and Theorem 1.19 from [CCLDO16]. Namely, the additional subRiemannian structure allows to an unambiguous definition of 1-quasiconformality.
(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: The last equation follows from the fact that lim sup a n b n ≤ lim sup a n lim sup b n whenever a n , b n ≥ 0. Moreover, we trivially have that if f is an L-biLipschitz homeomorphism, then

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 [BB00].
Let D ⊂ C n , n ≥ 2 be a smooth, bounded open set. Let ϕ : C n → R denote the signed distance function from ∂D, negative in D and positive in its complement. Set Lemma 2.5 (Tubular Neighborhood Theorem). Let D ⊂ C n , n ≥ 2 be a bounded domain with smooth boundary. There exists δ 0 > 0 such that the projection π : N δo → ∂D is a smooth, well defined map and the distance function d E (·, ∂D) is smooth on N δ 0 .
We will denote by n(x) the outer unit normal at x ∈ ∂D, so that the fiber For p ∈ ∂D, one can define the tangent space T p ∂D = {Z ∈ C n |Re ∂ ϕ(p), Z = 0} and its maximal complex subspace is the hermitian product. By definition, the domain D is strictly pseudoconvex if for every p ∈ ∂D, the Levi form For each p ∈ ∂D one has a splitting C n = H p ∂D ⊕ N p ∂D, where N p ∂D is the complex one-dimensional subspace orthogonal to H p ∂D. This splitting at p induces a decomposition 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 Carathéodory metric (see [Kra13]). In all cases, for x ∈ D the length of a complex vector Z ∈ T x D = C n is given by a Finsler structure K(x, Z). We will rely on the following result, which can be found in [BB99] and also [BB00, Proposition 1.2].
Proposition 2.7 (Balogh-Bonk). Let D ⊂ C n , n ≥ 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 Carathéodory metric. For everyǭ > 0 there exists δ 0 , C > 0 such that for all x ∈ D with d E (x, ∂D) ≤ δ 0 and Z ∈ C n one has The subbundle H∂D is a contact distribution on ∂D and the triplet (∂D, H∂D, L ϕ ) yields a contact subRiemannian manifold. In this structure, the horizontal curves are those arcs in ∂D that are tangent to the contact distribution, and the Carnot-Carathéodory distance d CC (p, q) between p, q ∈ ∂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 [Gro96].
As in [BB00], we will need to use a family of Riemannian metrics on ∂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 ∂D as for every p ∈ ∂D and every Z = Z H + Z N ∈ T p ∂D. Here we just recall a basic comparison result (see for instance [BB00, 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 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 x, y o , is defined by For a Gromov hyperbolic space X one can define a boundary set ∂ ∞ X as follows, see Let g be as defined in (1.2) and let ρ g o be its Bourdon distance, as defined by (1.3). We begin by giving a computation of the distance ρ g o on two points p, q ∈ ∂D. We represent p and q by two sequences x i and y i ∈ D, respectively. Notice that since x i → p in C n then π(x i ) → p and h(x i ) → 0. In particular, we also have that max(h(x i ), h(o)) = h(o). Similar considerations apply to y i and q. We compute .
For every p ∈ ∂D one has so the limit exists, and the identity map (∂D, d CC ) → (∂D, ρ g o ) is 1-quasi-conformal.
3.2. Boundary distances of hyperbolic fillings. An important contribution of Bonk and Schramm [BS00], is that the functor X → ∂ ∞ X has an inverse functor, in the form of hyperbolic filling spaces Con(Z).
To be more precise, one defines Con(Z) = Z × (0, D), endowed with the metric given by The space (Con(Z), d 2 ) is Gromov hyperbolic, and its visual boundary is Z, with the canonical class of snowflake equivalent metrics given by d 1 . Here we note that a particular visual metric is actually conformal to d 1 . We will consider the particular visual metric generated by g given by the Bourdon distance. Choose a generic base point choose a base point o = (z, s), with z ∈ Z and s ∈ (0, D). For any two points x, y ∈ Z so that d 1 (x, y) < s. consider u, v ∈ (0, d 1 (x, y)). Following (1.3), the Bourdon distance d 2 (x, y) is defined as follows Notice that in general, Bourdon distances associated to the hyperbolic fillings are a quasidistance. 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 denotes the Bourdon distance 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→x d 1 (x, y)/d 2 (x, y) exists for every x ∈ Z. Fix any z ∈ Z and s ∈ (0, D). Let o = (z, s). Take two points x, y ∈ Z so that d 1 (x, y) < s. Take u, v ∈ (0, d 1 (x, y)).
The rest of the proof follows from We calculate lim y→x d 1 (x, y)/d 2 (x, y). Consider the quotient The latter implies that which gives the conclusion.

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 [BB00, Corollary 1.3]. We follow largely their arguments, but where in [BB00] 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 everyp ∈ ∂D and ǫ > 0 there exists r > 0 such that for all distinct p, q ∈ ∂D∩B(p, r) there exists r ′ > 0 such that for all x ∈ D∩B(p, r ′ ) and all y ∈ D∩B(q, r ′ ) In the rest of the paper we will refer to this result in connection with the quintuplet (p, p, q, x, y).
where C is the same constant as in (2.8). Moreover, if the curve is a segment γ(t) = x 1 + t(x 2 − p 1 ) ⊂ π −1 (p) for some p ∈ ∂D then one has .
On the other hand, if γ(t) = x 1 + t(x 2 − x 1 ), then we observe that γ ′ is parallel to the unit normal at π(x i ) and so has no tangent component, hence no horizontal component with respect to the splitting at π(x i ). Using the fact that and (2.8) one has which gives (4.5).
An immediate consequence of Lemma 4.3 is the following.
Corollary 4.6. Let δ 0 > 0 to be the constant in Proposition 2.5. If Moreover, if π(x 1 ) = π(p 2 ), then we also have 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 π(x 1 ), π(x 2 ).
Arguing as in the proof of [BB00, Lemma 2.2] yields the following relations between α ′ and γ ′ , In fact, from (4.10) one has γ ′ (t) H which, together with the bilinearity of the Levi form, yields (4.11). Consequently we have Setting h = d CC (x 1 , x 2 ) in the latter yields the conclusion.
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 γ joining two points x 1 , x 2 ∈ D will travel at a distance further than the Carnot-Carathéodory distance between their projections.
Lemma 4.12. Let δ 0 > 0 be smaller than the similarly named constants in Propositions 2.5 and 2.7. Consider two points x 1 , x 2 ∈ D with d E (x i , ∂D) < δ 0 . Set p i = π(x i ) ∈ ∂D, and let γ : [0, 1] → D denote an arc joining x 1 and x 2 . If max z∈γ h(z) ≥ d CC (p 1 , p 2 ) then where C is the same constant as in (2.8).
Next we invoke Lemma 4.3 to deduce which is the desired bound (4.13).

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 allp ∈ ∂D and ǫ > 0 one can choose r > 0 small enough so that for all distinct p, q ∈ ∂D ∩ B(p, r) one can find r ′ ∈ (0, r) such that (4.2) holds for all x ∈ D ∩ B(p, r ′ ) and all y ∈ D ∩ 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 δ 0 in Propositions 2.5 and 2.7 and so that d 1 is small enough to be determined later. Denote byx, andȳ the projections on the boundary of x and y, respectively. Note that since the projections are the closest points in ∂D, thenx ∈ B(p, 2r ′ ) andȳ ∈ B(q, 2r ′ ). Set d 2 := d CC (x,ȳ). Notice that as r ′ → 0 we have d 2 → d 1 . We shall choose r ′ sufficiently small so that r ′ < d 2 and d 2 ∈ (d 1 /2, 2d 1 ). In particular, if r was chosen small enough, then d 2 is positive and smaller than the constants δ 0 in Propositions 2.5 and 2.7. Proof of the upper bound in (4.2). Set x ′ :=x − d 2 n(x) and y ′ :=ȳ − d 2 n(ȳ), so x ′ , y ′ are points in D at distance d 2 from ∂D and with the same projection on ∂D as x, y, respectively.
Choose d 1 chosen sufficiently small so that Cd 2 ≤ ǫ/3. Combining the previous bounds with the definition of g, we obtain the following estimates where we used that the terms h(x), h(y), ln(1 + h(x)∧h(y) d 2 ) are positive. This conclude the proof of the upper bound in (4.2).
In this case the proof is concluded.
-If H ≤ d CC (x,ȳ) then it follows that H is smaller than the constants δ 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 ∈ [0, 1] be such that h(γ(t 0 )) = H and consider the two branches γ 1 , γ 2 of γ given by restrictions to [0, t 0 ] and [t 0 , 1]. Given ǫ > 0 as in the statement, let θ ∈ (1, 2] so that ln θ < ǫ and define k ∈ N such that Set t 1 = s k ≤ t 0 and for each l = 1, ..., k, For each of the two branches γ 1 , γ 2 , we distinguish two alternatives: • Alternative #1 (All sub-arcs have large slope) In this alternative we assume that for every l = 1, ..., k one has (4.15) d CC (π(γ(s l−1 )), π(γ(s l ))) ≤ ν −1 l From the latter we draw two conclusions. The first is a simple application of the triangle inequality, On the other hand, in view of Lemma 4.3 one has • 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 (γ) among all arcs γ joining x and y and apply (4.18) to each. One has The proof is then concluded by applying the same argument as in (4.14).

Local biLipschitz equivalence of Bourdon functions 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. Letp as in the statement and choose ǫ > 0 such that exp( 3 2 ǫ) ≤ 1 +ǭ. Invoke Theorem 4.1 in correspondence to the choice ofp and ǫ, to obtain the value r > 0 and select any ω ∈ ∂D ∩ B(p, r) \ {p}. In correspondence to this choice of ω, Theorem 4.1 yields a smaller radius 0 < r ′ < r, so that if we choose y ∈ D ∩ B(p, r ′ ) and o ∈ D ∩ B(ω, r ′ ) and then apply Theorem 4.1 to the quintuplet (p,p, ω, y, o) we obtain for all y ∈ D ∩ B(p, r ′ ), and o ∈ D ∩ B(ω, r ′ ) Next, given p, q ∈ ∂D ∩ B(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 (p, p, qx, y) for all x ∈ D ∩ B(p, r ′′ ), and for all y ∈ D ∩ B(q, r ′′ ).
If x i (resp., y i ) is a sequence in D converging to p (resp., q), then for i large enough x i ∈ D ∩ B(p, r ′′ ) and y i ∈ D ∩ B(q, r ′′ ) and x i , y i ∈ B(p, r ′ ). From the above bounds one obtains Consequently, if the sequences x i , y i are taken so that p, And similarly, ρ g o (p, q)/ρ K o (p, q) is bounded by 1 +ǭ. Regarding the first and last term in the right-hand side of (5.1), in view of Proposition 1.4 we have that We shall then prove that (5.4) H * (p, Id ∂D 1 , ρ g o , ρ K o ) ≤ 1 +ǭ and H * (F (p), Id ∂D 2 , ρ K f (o) , ρ g f (o) ) ≤ 1 +ǭ, 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 +ǭ)-biLipschitz in a neighborhood of the considered points.