Conformal equivalence of visual metrics in pseudoconvex domains

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Conformal equivalence of visual metrics in pseudoconvex domains © Authors, 2020 Published version Capogna, Luca; Le Donne, Enrico Capogna, L., & Le Donne, E. (2020). Conformal equivalence of visual metrics in pseudoconvex domains. Mathematische Annalen, 377(3-4), 1643-1672. https://doi.org/10.1007/s00208-02001968-9 2020 Mathematische Annalen https://doi.org/10.1007/s00208-020-01962-1 Mathematische Annalen Conformal equivalence of visual metrics in pseudoconvex domains Luca Capogna1·Enrico Le Donne2,3 Received: 19 December 2018 / Revised: 20 January 2020 © The Author(s) 2020 Abstract We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between bounded, smooth strongly pseudoconvex domains in Cn 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. Mathematics Subject Classification 32T15 ·32Q45 ·32H40 ·53C23 ·53C17 Communicated by Ngaiming Mok. Luca Capogna was partially funded by NSF award DMS 1503683 and Simons collaboration grant for mathematicians 585688. Enrico Le Donne was partially supported by the Academy of Finland (Grant 288501 ‘Geometry of subRiemannian groups’ and by grant 322898 ‘Sub-Riemannian Geometry via Metric-geometry and Lie-group Theory’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’). BEnrico Le Donne [email protected] Luca Capogna [email protected] 1Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA 2Dipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy 3Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box (MaD), 40014 Jyväskylä, Finland 123 L. Capogna, E. Le Donne 1 Introduction Let D⊂Cn(n≥2)be a bounded, strongly pseudo-convex domain with C∞-smooth boundary. Denote by dKthe 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,dK)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 dCC 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 [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 gdefined 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 D1,D2⊂Cnbe bounded strongly pseudoconvex C∞-smooth domains and denote by dKthe distance function corresponding to a Finsler structure K satisfying (2.8), and by dCC the Carnot–Carathéodory distance on the boundaries induced by the Levi form. If f :(D1,dK)→(D2,dK)is an isometry then the induced boundary map F :(∂ D1,dCC)→(∂ D2,dCC)is a diffeomorphism, conformal with respect to the metric dCC. We emphasize that the result holds when dKis the Bergman, the Kobayashi, or the inner Carathéodory metrics. Indeed, these distances satisfy (2.8)inviewofthework 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 dKand the visual distance of an ad-hoc hyperbolic filing metric, built through the Carnot–Carathéodory distance: For x∈D denote by h(x):= √dE(x,∂D)and by π(x)∈∂Da closest point in ∂Dwith respect to the Euclidean distance dE(·,·), noting it is uniquely defined in a neighborhood of ∂D. Set g(x,y):= 2logdCC(π(x), π(y)) +max(h(x), h(y)) √h(x)h(y).(1.2) This is an hyperbolic filling metric built from the metric space (∂ D,dCC)(see Bonk and Schramm [10]). Balogh and Bonk [3, Corollary 1.3], showed that gis a metric in 123 Conformal equivalence of visual metrics... a neighborhood of ∂Dand that gand the invariant distance function dKare (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 ρK oassociated to the invariant metric dKis in fact pointwise and asymptotically (1+)-quasi-conformally equivalent to the Carnot–Carathéodory dCC metric. By pointwise and asymptotically we mean that for every point x∈∂Din the boundary, and for every >0, one can choose a base point ofor the definition of the visual distances so that the identity map has distortion less than 1+at x. Following ideas in CAT(−1)spaces, given a pointed metric space (X,d,o)we consider the visual function ρd o(x,y)=exp(−x,yo), (1.3) where x,yodenotes the Gromov product in (X,d), see Sect. 2. Usually, ρd ois called Bourdon distance since for CAT(−1) spaces it satisfies the triangle inequality. In our setting, ρd omay not be a distance. Moreover, Bourdon showed in [9] that on a CAT(−1) space Xthe visual boundaries (∂∞X,ρd o)corresponding to different base points o,o∈Xare conformally equivalent, thus implying immediately that any isometry of Xextends 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 1.1is achieved in two steps: First one shows that the Carnot–Carathéodory distance is conformally equivalent1to the function ρg oassociated to the hyperbolic filling metric g. Proposition 1.4 For any o ∈D, the functions dCC and ρg oare conformally equivalent. In other words, the identity map (∂ D,dCC)→(∂ D,ρg 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 >0, one can find a base point o∈Dsuch that the corresponding visual functions ρK oand ρg oare (1+)- biLipschitz equivalent in a neighborhood of that point. In the following we denote Euclidean balls in Cnwith the notation B(x,r). Proposition 1.5 For any ¯p∈∂D and ¯>0there exists r >0such that for all ω∈∂D∩B(¯p,r)\{ ¯p}there exists r>0such that for all o ∈D∩B(ω, r)the two functions ρg oand ρK oare (1+¯)-biLipschitz on ∂D∩B(¯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 1The result holds for any hyperbolic filling as in the work of Bonk and Schramm. See Sect. 3.1 123 L. Capogna, E. Le Donne (D1,dK)f iso (D2,dK) (∂ D1,ρg o)id (1+)BL (∂ D1,ρK o)F iso (∂ D2,ρK f(o))id (1+)BL(∂ D2,ρg f(o)) idconf (∂ D1,dCC) id conf F(∂ D2,dCC) (D) At the center of this chain of compositions there is an isometry, the rest of the links are either (1+)-biLipschitz maps or conformal maps, so that the total distortion is at most 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 Cn, originally established by Fefferman [15]. Corollary 1.6 Let D1,D2∈Cn(n≥2)be bounded strongly pseudo-convex domains with C∞-smooth boundaries. If f :D1→D2is a biholomorphism then it extends to a smooth map F :∂D1→∂D2that 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 technicallevel, but ratherintwonewinsights:namely,that one candeduce 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, introducedby Bonk, Heinonen,and Koskelain[7],translating backand forthproblems 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. 123 Conformal equivalence of visual metrics... 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→Ybetween general metric spaces, we consider the following quantities LF(x):= lim sup x→x d(F(x), F(x)) d(x,x)and F(x):= lim inf x→x d(F(x), F(x)) d(x,x). The quantity LF(x)is sometimes denoted by LipF(x)and is called the pointwise Lipschitz constant. Within this paper, we define the distortion of fat a point x∈X as H∗(x,F,dX,dY):= LF(x) F(x).(2.1) The homeomorphism fis said to be quasi-conformal if there exists Ksuch that for all x∈Xone has lim sup r→0 sup{dY(F(p), F(q)) :dX(p,q)≤r} inf{dY(F(p), F(q)) :dX(p,q)≥r}≤K. 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 Kabove) 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→Y be a quasiconformal homeomorphism between two equiregular subRiemannian manifolds. (i) The requirement H∗(·,F,dX,dY)≡1is equivalent to other notions 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: H∗(x,F1◦F2)≤H∗(x,F2)H∗(F2(x), F1). (2.3) The last equation follows from the fact that lim supanbn≤lim supanlim sup bn whenever an,bn≥0. Moreover, we trivially have that if Fis an L-biLipschitz homeomorphism, then H∗(x,F)≤L2.(2.4) 123 L. Capogna, E. Le Donne 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⊂Cn,n≥2 be a smooth, bounded open set. Let ϕ:Cn→Rdenote the signed distance function from ∂D,negativeinDand positive in its complement. Set Nδ={x∈D|dE(x,∂D)<δ}. Lemma 2.5 (Tubular Neighborhood Theorem) Let D ⊂Cn,n ≥2be a bounded domain with smooth boundary. There exists δ0>0such that the projection π: Nδo→∂D is a smooth, well defined map and the distance function dE(·,∂D)is smooth on Nδ0. We will denote by n(x)the outer unit normal at x∈∂D, so that the fiber π−1(x)∩ Nδ0={x+sn(x)|s∈(0,δ 0)}. For p∈∂D, one can define the tangent space Tp∂D={Z∈Cn|Re¯ ∂ϕ(p), Z= 0}and its maximal complex subspace Hp∂D={Z∈Cn|¯ ∂ϕ(p), Z=0}, where Z,Z=n i=1Zi¯ Z iis the hermitian product. By definition, the domain Dis strictly pseudoconvex if for every p∈∂D,theLevi form Lϕ(p,Z):= n  α,β=1 ∂2 zα¯zβϕ(p)Zα¯ Zβ(2.6) is positive definite on Hp∂D. For each p∈∂Done has a splitting Cn=Hp∂D⊕Np∂D, where Np∂Dis the complex one-dimensional subspace orthogonal to Hp∂D. This splitting at pinduces a decomposition Z=ZH+ZNfor all Z∈Cn. 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∈Dthe length of a complex vector Z∈TxD=Cnis 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 ⊂Cn,n ≥2be 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 ¯>0there exists δ0,C>0such that for all x ∈D with dE(x,∂D)≤δ0and Z∈Cnone has (1−CdE(x,∂D))|ZN|2 4d2 E(x,∂D)+(1−¯)Lϕ(π(x), ZH) dE(x,∂D)1 2 ≤K(x,Z) ≤(1+CdE(x,∂D))|ZN|2 4d2 E(x,∂D)+(1+¯)Lϕ(π(x), ZH) dE(x,∂D)1 2 ,(2.8) where Z =ZH+ZNis the splitting at π(x). 123 Conformal equivalence of visual metrics... The subbundle H∂Dis a contact distribution on ∂Dand the triplet (∂ D,H∂D,Lϕ) yields a contact subRiemannian manifold. In this structure, the horizontal curves are those arcs in ∂Dthat are tangent to the contact distribution, and the Carnot– Carathéodory distance dCC(p,q)between p,q∈∂Dis 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 ∂Dthat 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 gkon T∂Das g2 k(p,Z):= Lϕ(p,ZH)+k2|ZN|2,(2.9) for every p∈∂Dand every Z=ZH+ZN∈Tp∂D. Here we just recall a basic comparison result (see for instance [3, Lemma 3.2]) relating the distance function dk associated to gkto the Carnot–Carathéodory distance dCC. Lemma 2.10 There exists a constant C >0such that for all k >0, and for all points p,q∈∂D, with dCC(p,q)≥k−1one has C−1dk(p,q)≤dCC(p,q)≤Cdk(p,q). (2.11) 2.3 Gromov hyperbolicity Let x,y,obe three points in a metric space (X,d). Then the Gromov product of x and yat o, denoted x,yo, is defined by x,yo=1 2d(x,o)+d(y,o)−d(x,y). Then Xis called Gromov hyperbolic if there exists δ≥0 such that x,yo≥min{x,zo,z,yo}−δ, for all x,y,z,o∈X. For a Gromov hyperbolic space Xone can define a boundary set ∂∞Xas follows, see [6, p.431-2]. Fix a base point o∈X. A sequence (xi)in Xis said to converge at infinity if limi,j→∞xi,xjo=∞. Two sequences (xi)and (yi)converging at infinity are called equivalent if limxi,yio=∞.These notions do not depend on the choice of the base point o.Theset∂∞Xis now defined as the set of equivalence classes of sequences converging at infinity. For p,q∈∂∞Xand o∈Xwe define p,qo=sup lim inf i→∞ xi,yio, 123 L. Capogna, E. Le Donne where the supremum is taken over all sequences (xi)and (yi)representing the boundary points pand q, respectively. Actually, there exists such sequences (xi)and (yi) for which p,qo=limi→∞xi,yio,see[6, Remark 3.17]. Balogh and Bonk have proved that if D⊂Cn,n≥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,dK)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→∂∞X has an inverse functor, in the form of hyperbolic filling spaces Con(Z).Tobemore precise, one defines Con(Z)=Z×(0,D), endowed with the metric given by g((x,u), (y,v))=2logd1(x,y)+max(u,v) √uv.(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 d1. In this section we prove that a particular visual metric, i.e. the visual metric generated by gthrough the formula (1.3), is actually conformal to d1. Choose a generic base point o=(z,s), with z∈Zand s∈(0,D). For any two points x,y∈Zso that d1(x,y)<s. consider u,v ∈(0,d1(x,y)). Following (1.3), we let d2(x,y)be defined as follows d2(x,y)=lim u,v→0e−(x,u),(y,v)o. 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 d1a distance on a bounded space Z. If d2is defined as in (1.3), associated to the hyperbolic filling for d1, then d1and d2are conformally equivalent. Proof In order to show that d1,d2are conformally equivalent it suffices to prove that the limit limy→xd1(x,y)/d2(x,y)exists for every x∈Z.Fixanyz∈Zand s∈(0,D).Leto=(z,s). Take two points x,y∈Zso that d1(x,y)<s.Take u,v ∈(0,d1(x,y)). The rest of the proof follows from (x,u), (y,v)o=1 2(d2((x,u), o)+d2((y,v),o)−d2((x,u), (y, v))) =log d1(x,z)+max(u,s) √us +log d1(y,z)+max(v, s) √vs 123 Conformal equivalence of visual metrics... h(γ (0)) ∈H θk,H θk−1. Following [3], we define s0,s1,...,sk∈[0,t0]such that s0=0 and sl=min s∈[0,t0]such that h(γ (s)) =H θk−l. Set t1=sk≤t0and for each l=1,...,k, ν−1 l=dCC(¯x,¯y)·(θ −1) 8θk−l. For each of the two branches γ1,γ 2, we distinguish two alternatives: •Alternative #1 (All sub-arcs have large slope) In this alternative we assume that for every l=1,...,kone has dCC(π(γ (sl−1)), π(γ (sl))) ≤ν−1 l(4.15) From the latter we draw two conclusions. The first is a simple application of the triangle inequality, dCC(¯z,π(γ(t1)) (A1 (i)) ≤ k  l=1 dCC(π(γ (sl−1)), π(γ (sl))) ≤(θ −1)dCC(¯x,¯y) 8θk k  l=1 θl≤dCC(¯x,¯y) 4. On the other hand, in view of Lemma 4.3 one has lK(γ |[0,t1])≥ln h(γ (t1)) h(x)−C(h(γ (t1)) −h(x)) =ln H h(x)−C(H−h(x)). (A1 (ii)) •Alternative #2 (One sub-arc has small slope) In this alternative, we assume that there exists l∈{1,...,k}such that dCC(π(γ (sl−1)), π(γ (sl))) > ν−1 l(4.16) Note that if s∈[sl−1,sl]then from the definition of the points sl, one has h(γ (s)) ≤θl−kH≤8 θ−1ν−1 l. We then claim that there exists a constant C>0 depending only on the defining function ϕsuch that lK(γ |[sl−1,sl])≥C(θ −1)2dCC(¯x,¯y) H.(4.17) 123 L. Capogna, E. Le Donne Indeed, arguing as in [3, page 521] we invoke (2.8), Lemma 2.10, and (4.16) and we bound as follows: lK(γ |[sl−1,sl]) ≥C(1−CH)θk−l Hsl sl−1Lϕ(π(γ (s)), [π(γ(s))]H) +(θ −1)2ν2 l|[π(γ(s))]N|21 2 ds ≥C 2(θ −1)θk−l Hsl sl−1Lϕ(π(γ (s)), [π(γ(s))]H)+ν2 l|[π(γ(s))]N|21 2 ds ≥C(θ −1)θk−l Hdνl(π(γ (sl−1)), π(γ (sl))) ≥C(θ −1)θk−l HdCC(π(γ (sl−1)), π(γ (sl))) ≥C(θ −1)2dCC(¯x,¯y) H, where dνldenotes the approximation of the Carnot–Carathéodory metric defined in (2.9). Next we claim that lL(γ |[0,t1])≥ln H h(y)+C(θ −1)2 HdCC(¯x,¯y)−CH−h(y)−. (A2) Indeed, Lemma 4.3 and (4.17) yields lL(γ |[0,t1])=lK(γ |[0,sl−1])+lK(γ |[sl−1,sl])+lK(γ |[sl,t1]) ≥ln H h(γ (sl)) h(γ (sl−1)) h(x)+C(θ −1)2 HdCC(¯x,¯y) −CH−h(γ (sl)+h(γ (sl−1)) −h(x) ≥ln H h(x)θ−1+C(θ −1)2 HdCC(¯x,¯y)−CH−h(x) ≥ln H h(x)+C(θ −1)2 HdCC(¯x,¯y)−CH−h(x)−. Applying similar consideration to the branch γ2one obtains a t2∈[t0,1]such that one of the following two alternatives hold: Either dCC(¯y,π(γ(t2)) ≤dCC(¯x,¯y) 4and lK(γ |[t2,1])≥ln H h(y)−C(H−h(y)). (B1) 123 Conformal equivalence of visual metrics... or lL(γ |[t2,1])≥ln H h(y)+C(θ −1)2 HdCC(¯x,¯y)−CH−h(y)−. (B2) To conclude the proof we need to examine all possible combinations of these alternatives. We will show that in each case one obtains lK(γ ) ≥2lndCC(¯x,¯y) √h(x)h(y)−C(2dCC(¯x,¯y)−h(x)−h(y)) −. (4.18) •Suppose both (A1) and (B1) hold. Observe that dCC(π(γ (t1)), π(γ (t2))) ≥dCC(¯x,¯y)−dCC(¯x,π(γ(t1))) −dCC(¯y,π(γ(t2))) ≥dCC(¯x,¯y) 2. Repeating the argument in (4.17)forl=kand invoking the Riemannian approximation lemma [3, Lemma 3.2] one has lL(γ |[t1,t2])≥C(θ −1)2dνk(π(γ (t1)), π(γ (t2))) H≥C(θ −1)2dCC(¯x,¯y) H. The latter, together with (A1 (ii)), and the second inequality in (B1) yields lK(γ ) ≥2lnH √h(x)h(y)+C(θ −1)2dCC(¯x,¯y) H−C(2H−h(x)−h(y)). Since the right hand side is monotone decreasing in H≤dCC(¯x,¯y)then one has lK(γ ) ≥2lndCC(¯x,¯y) √h(x)h(y)+C(θ −1)2−C(2dCC(¯x,¯y)−h(x)−h(y)) ≥2lndCC(¯x,¯y) √h(x)h(y)−C(2dCC(¯x,¯y)−h(x)−h(y)) completing the proof of (4.18). •Suppose both (A1) and (B2) hold. One immediately has lK(γ ) ≥lK(γ[0,t1])+lK(γ |[t2,1]) ≥ln H h(x)+C(θ −1) HdCC(¯x,¯y)−C[H−h(x)] −+ln H h(y)−C(H−h(y)). Applying the same consideration as above we immediately deduce (4.18). 123 L. Capogna, E. Le Donne •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 lK(γ ) among all arcs γ joining xand yand apply (4.18) to each. One has dK(x,y)−g(x,y)≥2lndCC(¯x,¯y) √h(x)h(y) 1 dCC(¯x,¯y) √h(x)h(y)+max{h(x),h(y)} √h(x)h(y) −C(2dCC(¯x,¯y)−h(x)−h(y)) − =−2ln1+max{h(x), h(y)} dCC(¯x,¯y) −C(2dCC(¯x,¯y)−h(x)−h(y)) −. The proof is then concluded by applying the same argument as in (4.14).  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 ¯pas in the statement and choose >0 such that exp(3 2) ≤1+¯. Invoke Theorem 4.1 in correspondence to the choice of ¯pand , 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 |g(y,o)−dK(y,o)|<, 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) |dK(x,y)−g(x,y)|≤, for all x∈D∩B(p,r), and for all y∈D∩B(q,r). If xi(resp., yi) is a sequence in Dconverging to p(resp., q), then for ilarge enough xi∈D∩B(p,r)and yi∈D∩B(q,r)and xi,yi∈B(¯p,r). From the above bounds one obtains yi,xig o−yi,xiK o=1 2|g(yi,o)−dK(yi,o)+g(xi,o) −dK(xi,o)+dK(xi,yi)−g(xi,yi)| ≤3 2. 123 Conformal equivalence of visual metrics... Consequently, if the sequences xi,yiare taken so that p,qg o=limi→∞yi,xig o,we have ρK o(p,q) ρg o(p,q)≤limi→∞ exp(−yi,xiK o) limi→∞ exp(−yi,xig o) =lim i→∞exp yi,xig o−yi,xiK o ≤exp(3 2) ≤1+¯. And similarly, ρg o(p,q)/ρK o(p,q)is bounded by 1 +¯. Proof of Theorem 1.1 For any ¯p∈∂D1and ¯>0 weshowthattheboundaryextension is(1+¯)−quasi-conformalat ¯p,i.e. H∗(¯p,F,dCC,dCC)≤1+¯,where H∗isdefined as in (2.1). Following the diagram (D) in the introduction, from (2.3) for every o∈D1 we have H∗(¯p,F,dCC,dCC) ≤H∗(¯p,Id∂D1,dCC,ρg o)H∗(¯p,Id∂D1,ρg o,ρK o)H∗(¯p,F,ρK o,ρK f(o)) ·H∗(F(¯p), Id∂D2,ρK f(o),ρg f(o))H∗(F(¯p), Id∂D2,ρg f(o),dCC). (5.1) Start by observing that for any o∈D1the pointed metric spaces (D1,dK,o)and (D2,dK,f(o)) areisometric.Thustheygiverisetovisualboundariesthatareisometric with respect to the induced distances ρK oan ρK f(o), as defined in (1.3). Consequently the induced extension map F:(∂ D1,ρK o)→(∂ D2,ρK f(o))is an isometry, and hence from (2.4) H∗(¯p,F,ρK o,ρK f(o))=1.(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 H∗(¯p,Id∂D1,dCC,ρg o)=H∗(F(¯p), Id∂D2,ρg f(o),dCC)=1.(5.3) We shall then prove that H∗(¯p,Id∂D1,ρg o,ρK o)≤1+¯and H∗(F(¯p), Id∂D2,ρK f(o),ρg f(o))≤1+¯, (5.4) for some suitable choice of o. To prove this we will need to invoke Proposition 1.5 twice, in D1and in D2, together with the observation (2.4). Namely, we shall prove that for a suitable choice of oThe maps considered in (5.4)are(1+¯)-biLipschitz in a neighborhood of the considered points. First we apply Proposition 1.5 in a neighborhood of F(¯p)∈∂D2, thus yielding r2>0 such that for all ω2∈∂D2∩B(F(¯p), r2)\{F(¯p)}there exists r 2>0 such that for all o2∈D2∩B(ω2,r 2)one has that ρg o2and ρK o2are (1+¯)-biLipschitz in ∂D2∩B(F(¯p), r 2). For the moment we do not choose any specific ω2and o2,sor 2 is still to be determined. 123 L. Capogna, E. Le Donne Next, we apply Proposition 1.5 to D1in a neighborhood of ¯pand use it to choose r1>0 such that for all ω1∈∂D1∩B(¯p,r1)\{ ¯p}there exists r 1>0 such that o1∈D1∩B(ω1,r 1)one has that ρg o1and ρK o1are (1+¯)-biLipschitz in ∂D1∩ B(¯p,r 1). By continuity of the map Fwe may have chosen r1small enough that F(B(¯p,r1)∩D1)⊂B(F(¯p), r2)∩D2. We set ω2:= F(ω1), which is then in B(F(¯p), r2)∩D2and is different than F(¯p) since Fis a homeomorphism. Now we fix r 2accordingly, as we explained above. If neededwewillselecta smallervalueforr 1sothat wecanassume F(B(ω1,r 1)∩D1)⊂ B(F(ω1), r 2)∩D2. To conclude, we can now select any base point o∈B(ω1,r 1)∩D1, so that f(o)∈ B(ω2,r 2)∩D2and and hence ρg o1and ρK o1are (1+¯)-biLipschitz in ∂D1∩B(¯p,r 1) and ρg o2and ρK o2are (1+¯)-biLipschitz in ∂D2∩B(F(¯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∗(¯p,F,dCC,dCC)≤ (1+¯)2. By the arbitrariness of ¯we deduce H∗(¯p,F,dCC,dCC)=1. Finally, from Lemma 2.2 we conclude.  Acknowledgements Open access funding provided by University of Jyväskylä (JYU). 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. We also want to acknowledge the thoughtfulness and care of the anonymous referees, whose suggestions we have incorporated in the paper, improving the exposition. In particular, thanks to a referee’s remarks we have streamlined Sect. 3, and simplified the proof of Proposition 1.4. 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