On isometries of the carathéodory and Kobayashi metrics on strongly pseudoconvex domains

On isometries of the Carath´eodory and

Kobayashi metrics on strongly pseudoconvex domains

HARISHSESHADRI ANDKAUSHALVERMA

Abstract. Let1and2be strongly pseudoconvex domains inCⁿ and f : 1→2an isometry for the Kobayashi or Carath´eodory metrics. Suppose that f extends as aC¹map to¯1. We then prove that f|_∂1 :∂1→∂2is a CR or anti-CR diffeomorphism. It follows that1and2must be biholomorphic or anti-biholomorphic.

Mathematics Subject Classification (2000): 32F45 (primary); 32Q45 (sec- ondary).

1.

Introduction

Complex Finsler metrics such as the Carath´eodory and Kobayashi [14] metrics and K¨ahler metrics such as the Bergman and Cheng-Yau K¨ahler-Einstein metrics [5]

have proved to be very useful in the study functions of several complex variables.

Since biholomorphic mappings are isometries for these metrics, they are referred to as ‘intrinsic’.

This work is motivated by the question of whether (anti)-biholomorphic map- pings are theonlyisometries for these metrics,i.e., is any isometry f : 1 →2

between two domains1and2in

C

ⁿ(on which the appropriate intrinsic metrics are non-degenerate) holomorphic or anti-holomorphic?

To be more precise by what we mean by an isometry, let Fandddenote an intrinsic Finsler metric and the induced distance on a domain. In this paper by aC⁰-isometry we mean a distance-preserving bijection between the metric spaces (1,d₁) and(2,d₂). For k ≥ 1, a C^k-isometry is a C^k-diffeomorphism f from1to2with f^∗(F₂)= F₁. AC^k-isometry,k ≥1, is aC⁰-isometry and if the Finsler metric comes from a smooth Riemannian metric (as is the case with the Bergman and the Cheng-Yau metrics), the converse is also true by a classical theorem of Myers and Steenrod.

Both authors were supported by DST (India) Grant No. SR/S4/MS-283/05.

Received December 7, 2005; accepted in revised form July 28, 2006.

The question above makes sense for a large class of domains (for example bounded domains). However, we confine ourselves to bounded strongly pseudoconvex do- mains in this paper.

We note that the question has been answered in the affirmative (for strongly pseudoconvex domains) for the Bergman and the K¨ahler-Einstein metrics in [12].

The proof is essentially based on the fact that the metric under consideration is a K¨ahler metric whose holomorphic sectional curvatures tend to −1 as one ap- proaches the boundary of the domain. Note that the Bergman metric and the K¨ahler- Einstein metric both have this property.

The case of the Carath´eodory and the Kobayashi metrics is more delicate. A technical reason is that these metrics are Finsler, not Riemannian, and moreover they are just continuous and not smooth for general strongly pseudoconvex do- mains. Despite these issues, the results in this paper indicate that the answer to the main question might be in the affirmative.

Before stating our results we remark that all domains we consider have at least C²-boundaries. Our main theorem asserts that, an isometry is indeed a holomorphic mapping at ‘infinity’.

Theorem 1.1. Let f :1→ 2be a C¹-isometry of two bounded strongly pseu- doconvex domains in

C

ⁿ equipped with the Kobayashi metrics. Suppose that f extends as a C¹map to¯1. Then f|_∂1 :∂1 →∂2is a CR or anti-CR diffeo- morphism. Hence1and2must be holomorphic or anti-biholomorphic.

A similar statement holds for the inner-Carath´eodory metric if we assume that

∂1and∂2are C³-smooth.

A few comments about the C¹-extension assumption: any C¹-isometry be- tween strongly pseudoconvex domains1and2equipped with the Kobayashi or inner-Carath´eodory metrics extends to a C^1/2 (H¨older continuous with exponent 1/2) map of1 by the results of [1]. See also the remark after Lemma 3.2. A key ingredient in the proof of this result is that strongly pseudoconvex domains equipped with the Kobayashi or inner-Carath´eodory metrics are Gromov hyper- bolic.

It would be interesting and desirable to prove theC¹- extension property for Kobayashi/inner-Carath´eodory isometries and hence render the assumption in The- orem 1.1 unnecessary. More precisely, what needs to be investigated is whether C^1/2-boundary regularity impliesC¹-boundary regularity of the givenC¹-isometry.

This is known to be valid for holomorphic mappings and the reader is referred to [19, 24] and [8] among others for further details. Thus the study of isometries seems to fit in naturally with the study of the boundary behaviour of holomorphic mappings.

We now summarize the ideas behind the proof of Theorem 1.1. The main idea is to use the rescaling technique of Pinchuk [25] to study the derivative of the isometry at a boundary point. We construct a sequence of rescalings of the isometry near a boundary point p and show that this sequence converges (see Proposition 3.2 and 3.3) to an (anti)-holomorphic automorphism of the unbounded realization

of the ball in

C

ⁿ. On the other hand, we observe that the horizontal components of these rescalings converge to the horizontal component of the derivative of the isometry at p. Here by a ‘horizontal’ vector we mean a vector in the maximal complex subspace of the tangent space at a boundary point of the domain. In fact, we show that therestriction of the derivative to the horizontal subspace at p can be related to the values of the holomorphic automorphism acting on certain points in the ball(see Lemma 4.2). This fact is crucial; indeed, in conjunction with the (anti)-holomorphic limit of the scaled isometries, it allows us to show thecomplex- linearity of the derivative on the horizontal subspace of the tangent space of p.

Much of the technical work in the proof is in showing the convergence of metrics under Pinchuk rescalings which is needed to show that the scaled isometries are a

‘normal’ family. This problem is not faced when dealing with scaled holomorphic mappings.

To the best of our knowledge, this interpretation (cf. Lemma 4.2) of Pinchuk rescaling that relates the derivative on the boundary in the horizontal direction to values of the (anti)-holomorphic limit at certain points in the ball is new. It may be possible to use this interpretation to re-formulate the question about boundary regularity of holomorphic mappings in terms of the scaling method.

A related theme that has been studied in detail is the following: What can be said about a holomorphic mapping between two domains in

C

ⁿ which is known to be an isometry in some fixed Finsler metric in the domain and range, at a single point? The reader is referred to [4, 11, 20] and [29] which study this question, the conclusion in all being that the holomorphic mapping is biholomorphic. The main theorem dispenses with the hypothesis of having a global holomorphic mapping and replaces it with a global isometry at the expense of assuming some apriori boundary regularity.

The first important technical lemma that we need is about the behaviour of the distance to the boundary under isometries (Lemma 2.2). Here we use the two-sided estimates for the Kobayashi distance obtained in [1].

2.

Preliminaries

2.1. The Kobayashi and Carath´eodory metrics

Letdenote the open unit disc in

C

^{and letρ(a,b)}denote the distance between two pointsa,b∈with respect to the Poinc´are metric (of constant curvature−4).

Let be a domain in

C

ⁿ. The Kobayashi, the Carath´eodory and the inner- Carath´eodory distances on, denoted byd^K, d^C andd^C˜ respectively, are defined as follows:

Letz∈andv∈T_za tangent vector atz. Define the associated infinitesi- mal Kobayashi and Carath´eodory metrics as

F^K(z, v)= inf 1

α : α >0, φ∈

O

(, )withφ(0)=z, φ(0)=αv

and

F^C˜(z, v)= sup{|d f(z)v|: f ∈

O

(, )},

respectively. The inner-Carath´eodory length and the Kobayashi length of a piece- wiseC¹curveσ : [0,1]→are given by

l^C˜(σ )= ₁

0

F^C˜(σ, σ)dt and l^K(σ)= ₁

0

F^K(σ, σ)dt,

respectively. Finally the Kobayashi and inner-Carath´eodory distances betweenp,q are defined by

d^K(p,q)= inf{l^K(p,q)} and d^C˜(p,q)= inf{l^C˜(p,q)},

where the infimums are taken over all piece-wiseC¹curves injoining pandq.

The Carath´eodory distance is defined to be

d^C(p,q)= sup{ρ(f(p), f(q)): f ∈

O

(, )}.

We note the following well-known and easy facts:

• If is a bounded domain, thend^K, d^C andd^C˜ are non-degenerate and the topology induced by these distances is the Euclidean topology.

• These distance functions are invariant under biholomorphisms. More gener- ally, holomorphic mappings are distance non-increasing. The same holds for F^K(z, v)andF^C˜(z, v).

• We always have

d^C(p,q)≤ d^C˜(p,q)≤d^K(p,q).

• If =

B

ⁿ, all the distance functions above coincide and are equal to the dis- tance function of the Bergman metricg₀on

B

ⁿ. Here the Bergman metric is a complete K¨ahler metric normalized to have constant holomorphic sectional cur- vature−4. Also, for

B

ⁿ, the infinitesimal Kobayashi and Carath´eodory metrics are both equal to the quadratic form associated tog₀.

2.2. Convexity and Pseudoconvexity

Suppose is a bounded domain in

C

^{n, n ≥ 2, with C2}-smooth boundary. Let ρ:

C

^{n →}

R

be a smooth defining function for, i.e,ρ=0 on∂,dρ=0 at any point of∂andρ⁻¹(−∞,0)=.

A domain withC²smooth boundaryis said to bestrongly convexif there is a defining functionρfor∂such that the real Hessian ofρis positive definite as a bilinear form onT_p∂, for every p∈∂.

isstrictly convexif the interior of the straight line segment joining any two points inis contained in. Note that we do not demand the boundary ofbe smooth. Strong convexity implies strict convexity.

Let be a bounded domain. A holomorphic mapφ : → is said to be an extremal discor acomplex geodesicfor the Kobayashi metric (or distance) if it is distance preserving,i.e. d^K(φ(p), φ(q))=ρ(p,q)for all p,q∈.

The following fundamental theorem about complex geodesics in strictly con- vex domains will be repeatedly used in Section 3 of this paper:

Theorem 2.1 (L. Lempert [16]). Letbe a bounded strictly convex domain in

C

^n. (1) Given p ∈ and v ∈

C

ⁿ, there exists a complex geodesicφ withφ(0) = p andφ(0) = v (or dφ (T₀) = P_v, where P_v ⊂ T_pis the real -two plane generated by the complex vectorv).

φ also preserves the infinitesimal metric,i.e., F^K(φ(q);dφ(w)) = F(q;w) for allw∈T_q.

(2) Given p and q in, there exists a complex geodesicφwhose image contains p and q.

(3) The Kobayashi, Carath´eodory and inner-Carath´eodory distances coincide on . Also, the Kobayashi and Carth´eodory infinitesimal metrics coincide on.

TheLevi formof the defining functionρat p∈

C

ⁿis defined by L_p(v)=

n i,j=1

∂²ρ

∂z_i∂z¯_j(p)vivj for v=(v1, .., vn)∈

C

^n.

For p∈∂, the maximal complex subspace of the tangent spaceT_p∂is denoted by H_p(∂)and called thehorizontal subspace at p. By definition,is strongly pseudoconvexifL_pis positive definite onH_p(∂)for allp∈∂. It can be checked that strong convexity implies strong pseudoconvexity.

For a strongly pseudoconvex domain, theCarnot-Carath´eodory metricon∂ is defined as follows. A piecewiseC¹curveα: [0,1]→∂is calledhorizontalif

˙

α(t)∈ H_α(t)(∂)whereverα(˙ t)exists. The strong pseudoconvexity ofimplies that∂is connected and, in fact, any two points can be connected by a horizontal curve. TheLevi-lengthof a curveαis defined byl(α)=₁

0 L_α(t)(α(˙ t))¹²dt. Finally theCarnot-Carath´eodory metricis defined, for any p,q∈∂, by

d_H(p,q)= inf

α l(α),

where the infimum is taken over horizontal curvesα: [0,1]→∂withα(0)= p andα(1)=q.

2.3. Notation

◦ := {z ∈

C

^{:|z|<1},r := {z∈}

C

^:|z|<r}.

◦ ρ=distance function onof the Poinc´are metric of curvature−4.

◦ Forn≥2,

B

^{n := {z∈}

C

^{n :|z|<1}andBa(r)= {z∈}

C

^{n :|z−a|<r}.}

◦ z = {z = (z₁, . . . ,z_n) ∈

C

^{n : 2 Re zn + |z1|2+. . .+ |zn}−1|² < 0}, the unbounded realization of the ball

B

ⁿ, which is a Siegel domain.

◦ H_p(∂)⊂T_p∂denotes the horizontal subspace ofT_p∂, p∈∂.

◦ Given p∈∂, for anyv∈

C

^{n, v=vH+vN} corresponds to

C

^{n = Hp(∂)⊕} H_p(∂)^⊥.

◦ z=(˜z,z_n)corresponds to

C

ⁿ⁼

C

^n−1×

C

^.

◦ δ(x)=d(x, ∂)denotes Euclidean distance ofx∈to∂.

◦ d^K, d^C andd^C˜ denote the Kobayashi, Carath´eodory and inner-Carath´eodory distances on.

◦ F^K andF^C denote the Kobayashi and Carath´eodory infinitesimal metrics on.

◦ If f : 1 → 2 is a smooth map between domains1 and2 in

C

^{n, then} d f_p :

R

^{2n →}

R

²ⁿdenotes the derivative at p∈1.

Finally, the letters C or cwill be used to denote an arbitrary constant throughout this article and which is subject to change, even within the limits of a given line, unless otherwise stated.

2.4. An estimate for the distance to the boundary

We prove thatC⁰-isometries approximately preserve the distance to the boundary.

This is needed for the convergence of Pinchuk rescalings in Section 3. For a domain and a pointx ∈ ,δ(x)denotes the Euclidean distanceδ(x)= d(x, ∂). Our proof uses the results and notations of [1] in a crucial way and we refer the reader to it for further details.

We note that in the lemma below we do not need to assume that the isometry has aC¹-extension to the closure of the domain.

Lemma 2.2. Let 1 and 2 be strongly pseudoconvex domains in

C

^{n and f :} 1 → 2 a C⁰ isometry of the Kobayashi on1 and2. There exist positive constants A and B such that

Bδ(x)≤ δ(f(x)) ≤ Aδ(x)

for all x ∈1. A similar statement holds for an isometry of the inner-Carath´eodory distance if we assume that∂1and∂2are C³smooth.

Proof. Sincehas aC²boundary, givenx ∈sufficiently close to the boundary, there exists a unique pointπ(x) ∈ ∂such that |x −π(x)| = δ(x). Extend the domain of π to be all of . Such an extension is not uniquely defined but any extension will work for our purposes.

Following [1], define for any strongly pseudoconvex, the functiong:×→

R

^by

g(x,y)=2 log

d_H(π(x), π(y)) + max{h(x),h(y)}

√h(x)h(y)

, whereh(x)=√

δ(x)andd_H is the Carnot-Carath´eodory metric on∂.

The Box-Ball estimate (Proposition 3.1 of [1]) implies that the topology in- duced byd_Hon∂agrees with the Euclidean topology. Hence,(∂,d_H)is com- pact and, in particular, has finite diameter, say D. This implies that

2 log

h(y)

h(x) ≤g(x,y)≤2 log D+S

√h(x)h(y)

where we have used max{h(x),h(y)} ≥ h(y) in the first inequality and where S=sup_x∈h(x). Hence

h(y)

h(x) ≤e^g(x,y)≤ E

h(x)h(y). (2.1)

Now we consider the functions g₁ andg₂ associated to1 and2. By Corollary 1.3 of [1], there exists a constantC₁such that

g₁(x,y)−C₁≤d^K

1(x,y)≤g₁(x,y)+C₁ (2.2) for allx,yin1.

According to [2], such an estimate holds for the inner-Carath´eodory distance as well, if one assumesC³-regularity of the boundaries.

Combining (2.2) and (2.1) gives A₁h(y)

h(x) ≤e^dK1(x,y) ≤ B₁ h(x)h(y).

A similar inequality holds on2(with A₂, etc). Fixingy∈1, usingd^K

1(x,y)= d^K

2(f(x), f(y)), and comparing the inequalities on1and2, we get the required estimates. The proof for the inner-Carath´eodory distance is the same.

An immediate corollary of Lemma 2.2 is that for C¹-isometries which have C¹- extensions, the derivative of the boundary map preserves the horizontal distribution ofT. Note that necessarily f(∂1)⊂∂2, by Lemma 2.2.

Lemma 2.3. Let f : 1 → 2 be a C¹-isometry of strongly pseudoconvex do- mains equipped with the Kobayashi metric. If f extends to a C¹-map of1, then

d f_p(H_p(∂1))⊂ H_f(p)(∂2),

for any p ∈ ∂. This holds for an isometry of the inner-Carath´eodory metric as well if we assume that∂1and∂2are C³-smooth.

Proof. By [17], there existsδ0 > 0 such that for anyx ∈1withδ(x) ≤δ0and for allv=vH+vN ∈

C

ⁿ(where this decomposition is taken atπ(x)), we have

|vN|²

8δ(x)² + L_π(x)(vH)

4δ(x) ≤ F^K

1(x, v)2

≤ |vN|²

2δ(x)² + 4L_π(x)(vH)

δ(x) . (2.3) One has similar estimates ford f_x(v)=d f_x(v)H+d f_x(v)N. Since f is an isometry we have F^K

1(x, v) = F^K

2(f(x),d f_x(v)). Now assume that v ∈ H_p(∂1), i.e., v=vH. Comparing the estimates (corresponding to (2.3)) forvandd f_x(v), we get

|d f_x(v)N|²

8δ(f(x))² + L_π(f(x))(d f_x(v)H)

4δ(f(x)) ≤4L_π(x)(vH)

δ(x) . (2.4)

We can assume thatL_π(x)(w)≤c|w|²for allw∈H_q(∂1), q ∈∂1. Combining this with Lemma 2.2 and (2.4), we get

|d f_x(v)N| ≤C δ(x)|v|

for some uniform positive constantC. Lettingx → pand using the continuity of d f we obtaind f_p(v)N =0.

Remark: Once it known thatδ(x)≈ δ(f(x))uniformly for allx ∈1, it follows by well known arguments that f ∈C^1/2(1). Indeed, from (2.3) and the fact that

f is an isometry we get C |d f_x(v)|

√δ(f(x)) ≤F^K₂(f(x),d f_x(v))= F^K₁(x, v)≤ |v|

δ(x) for all tangent vectorsvatx. Hence

|d f_x(v)| ≤C |v|

√δ(x).

Integrating this along a polygonal path as in [22] yields

|f(p)− f(q)| ≤C |p−q|^1/2 uniformly for all p,qin1.

3.

A metric version of Pinchuk rescaling

Throughout this section, we will assume that the boundary of the domain under con- sideration isC³-smooth when dealing with the inner-Carth´eodory distance. Other- wise we assume that∂isC².

Let p ∈ ∂, and fix a sequence{p_n}inconverging to p. It has been shown in Lemma 2.2 that

d(pn, ∂1)≈d(f(pn), ∂2). (3.1) In particular{f(p_n)}will cluster only on∂2. By passing to a subsequence we can assume thatq_n = f(p_n) → q ∈ ∂2. Fix a defining functionρ for∂1that is strongly plurisubharmonic and of classC²in some neighbourhood of1. Similarly letρbe such a function for2. The following lemma in [23] will be vital for what follows.

Lemma 3.1. Letbe a strongly pseudoconvex domain,ρa defining function for

∂, and p ∈ ∂. Then there exists a neighbourhood U of p and a family of biholomorphic mappings h_ζ :

C

^{n →}

C

ⁿ depending continuously onζ ∈ ∂∩U that satisfy the following:

(i) h_ζ(ζ )=0.

(ii) The defining functionρ_ζ =ρ◦h⁻_ζ¹of the domain^ζ :=h_ζ()has the form ρ_ζ(z) = 2 Re(z_n+K_ζ(z))+H_ζ(z)+α_ζ(z)

where K_ζ(z) = _n

i,j=1a_{i j}(ζ )z_iz_j, H_ζ(z)= _n

i,j=1a_i¯j(ζ )z_iz¯_j andα_ζ(z)= o(|z|²)with K_ζ(˜z,0)≡0and H_ζ(˜z,0)≡ |˜z|².

(iii) The mapping h_ζ takes the real normal to∂atζto the real normal{˜z= y_n= 0}to∂^ζ at the origin.

To apply this lemma select ζk ∈ ∂1, closest to p_k and wk ∈ ∂2 closest to q_k = f(p_k). Forklarge, the choice ofζk andwk is unique since∂1and∂2are sufficiently smooth. Moreoverζk → pandwk →q. Leth_k :=h_ζk andg_k := g_wk be the biholomorphic mappings provided by the lemma above. Let

^k₁ :=h_k(1), ^k₂ :=g_k(2) and f_k :=g_k◦ f ◦h⁻_k¹:^k₁→^k₂. Note that f_kis also an isometry for the Kobayashi distance on^k₁and^k₂.

LetT_k :

C

^{n →}

C

ⁿbe the anisotropic dilatation map given by T_k(˜z,z_n)=

1

√δk

˜ z, 1

δk

z_n

and let

˜^k₁:=T_k(^k₁), ˜^k₂:=T_k(^k₂) and k :=T_k◦ f_k◦T_k⁻¹: ˜^k₁→ ˜^k₂. Againk is an isometry. Let us note that the explicit expression fork is

k(z)= 1

√δk

f˜_k(√

δkz˜, δkz_n), 1 δk

fˆ_k(√

δkz˜, δkz_n)

.

For notational convenience, let us denote the compositions of the rotations and the scalings by

H_k :=T_k◦h_k and I_k :=T_k◦g_k ⇒ k = I_k◦ f ◦H_k. (3.2) Note that the defining functions for˜^k₁and˜^k₂are given by

˜

ρk(z)= 1

√δk

ρk(√

δkz˜, δkz_n), ρ˜k(w)= 1

√δk

ρk(√

δkw, δ˜ kwn) respectively.

The family of functions{h_k}converges uniformly on compact subsets of

C

^nto the identity mapping, as do their inversesh⁻_k¹. Thus it follows that fork1

1

B ≤ d(h_k(z), ∂^k₁)

d(z, ∂1) ≤ B and 1

B ≤ d(g_k(w), ∂^k₂)

d(w, ∂2) ≤ B (3.3) for some constantBindependent ofzandk.

Combining (3.1) and (3.3) shows that fork1 1

c d(z, ∂^k₁) ≤ d(f_k(z), ∂^k₂) ≤ c d(z, ∂^k₁) (3.4) wherecis independent ofk, fork1 andz∈^k₁. Moreover, sinceρandρ(and henceρk :=ρ_ζk =ρ◦h⁻_k¹andρ_k :=ρ_wk =ρ◦g_k⁻¹) are smooth, it follows that there exists a uniform constantc>0 such that

1

c ≤ d(z, ∂^k₁)

|ρk(z)| ≤ c and 1

c ≤ d(w, ∂^k₂)

|ρ_k(w)| ≤ c (3.5) fork 1 andz∈^k₁,w∈^k₂. Letδk=d(h_k(p_k), ∂^k₁)andγk =d(g_k(q_k), ∂^k₂). Two observations can be made at this stage: first, for k 1, h_k(p_k) = (0,−δk), g_k(q_k) = (0,−γk)and f_k(0,−δk)= (0,−γk)as follows from Lemma 3.1, and secondly (3.4) shows that

1

c δk ≤ γk ≤ cδk (3.6)

for somec>0.

It has been shown in [23] that the sequence of domains{ ˜^k₁}converges to the unbounded realization of the unit ball, namely to

z= {z∈

C

^{n : 2 Rezn+ |˜z|2 <0}.}

The convergence is in the sense of Hausdorff convergence of sets. Similarly{ ˜^k₂} will converge to_w, the unbounded realization of the ball inwcoordinates.

Proposition 3.2. Let1, 2be smoothly bounded strongly pseudoconvex domains in

C

ⁿ. Suppose that f :1 →2is a C⁰-isometry with respect to the Kobayashi (respectively inner-Carath´eodory)distances on1and2. Define the sequence of domains˜^k₁, ˜^k₂and mappingsk :˜^k₁→ ˜^k₂. Then there exists a subsequence of {k}that converges uniformly on compact subsets ofzto a continuous mapping :z→

C

^n.

Proof. The case whenis an isometry with respect to the Kobayashi distance will be dealt with first. By construction k(0,−1) = (0,−γk/δk) and (3.6) shows that{k(0,−1)}is bounded. The domainz can be exhausted by an increasing union{S_i}of relatively compact convex domains each containing (0,−1). Fix a pairS_i0

^Si0+1and writeS₁= S_i0andS₂= S_i0+1for brevity. Since˜^k₁converges tozit follows thatS₁

^S2

^˜k1for allk1. It will suffice to show that{k} restricted to S₁ is uniformly bounded and equicontinuous. Fix s₁,s₂ in S₁. The following inequalities hold for largek:

d^K_˜

^k₂(k(s₁), k(s₂))=d^K_˜

^k₁(s₁,s₂)≤d_S^K

2(s₁,s₂)≤c|s₁−s₂| (3.7) for c > 0 independent of k. Indeed the equality holds for allk since k is an isometry and the inequalities are a result of the following observations: first, the inclusionS₂ → ˜^k₁is distance decreasing for the Kobayashi distance and second, sinceS₂is convex, the infinitesimal Kobayashi metricF_S^K

2(z, v)satisfies F_S^K

2(z, v)≤ |v|

δ_v(z) (3.8)

wherez ∈ S₂,vis a tangent vector atzandδ_v(z)is the distance ofzto∂S₂in the direction alongv. Now joinings₁ands₂by a straight line pathγ (t)and integrating (3.8) alongγ (t)yields the last inequality in (3.7).

To estimated^K_˜

^k₂(k(s₁), k(s₂))note from (3.2) that d^K_˜

^k₂(k(s₁), k(s₂))=d^K

2(f ◦H_k⁻¹(s₁), f ◦H_k⁻¹(s₂)) (3.9) since I_k is an isometry. Since f is continuous at p ∈∂1, choose neighborhoods U₁, U₂of p, q = f(p)respectively so that f(U₁∩1)⊂U₂∩2. Note that p_n, andζnas chosen earlier lie inU₁∩1eventually. Forklarge,H_k⁻¹(S₁)⊂U₁∩1

and hence both f ◦ H_k⁻¹(s₁)and f ◦ H_k⁻¹(s₂)lie in U₂∩2. It is well known that the Kobayashi distance can be localized near strongly pseudoconvex points in the sense that for every choice ofU₂, there is a smaller neighborhood p ∈U₃,U₃ relatively compact inU₂, andc>0 such that

c d_U^K

2∩2(x,y)≤d^K

2(x,y) (3.10)

for allx,y∈U₃∩2. We apply this tox = f ◦H_k⁻¹(s₁)andy = f ◦H_k⁻¹(s₂), both of which belong toU₃∩2for largek, by shrinkingU₁if necessary.

Moreover, thanks to the strong pseudoconvexity of ∂2 nearq, it is possible to chooseU₂small enough so that fork1,

g_k(U₂∩2)⊂ {w∈

C

^{n : |wn+R|2+ | ˜w|2< R2} ⊂ ˜} where:= {w∈

C

^{n : 2R}(Rewn) <−| ˜w|²}for some fixedR>1.

Note that˜ is invariant under the dilatation T_k for all k and moreover˜ is biholomorphic to

B

^{n. Thus,Tk◦gk(U2∩2)⊂ ˜and hencek(s1), k(s2)both} lie in˜ forklarge. From (3.9) and (3.10) it follows that

c d^K_˜

(k(s₁), k(s₂))≤d^K_˜

^k₂(k(s₁), k(s₂)) (3.11) forklarge. Combining (3.7) and (3.11) gives

d^K_˜

(k(s₁), k(s₂))≤c|s₁−s₂| (3.12) fors₁,s₂∈S₁andk1.

Let ψ : ˜ →

B

ⁿ be a biholomorphic mapping. To show that {k(S₁)} is uniformly bounded, choose s₁ ∈ S₁ arbitrarily and s₂ = (0,−1). Then (3.11) shows that

d^K_˜

(k(s₁), k(0,−1))≤c|s₁−s₂|<∞.

Since{k(0,−1)}is bounded and

B

^{n (and hence˜}) is complete in the Kobayashi distance, it follows that{k(s₁)}is bounded.

To show that{k}restricted toS₁is equicontinuous observe that the Kobayashi distance in

B

^{nbetweenψ◦k(s1)andψ◦k(s2)equals}

d^K_˜

(k(s₁), k(s₂))≤c|s₁−s₂|.

Using the explicit formula for the Kobayashi distance between two points in

B

^n, this gives

ψ◦k(s₁)−ψ◦k(s₂) 1−ψ◦k(s₁) ψ◦k(s₂)

≤ exp(2c|s₁−s₂|)−1 exp(2c|s₁−s₂|)+1.

Since {k(S₁)}is relatively compact in ˜ fork 1, it follows that so is {ψ ◦ k(S₁)}in

B

^{n. Let}{ψ◦k(S₁)} ⊂G

B

^{n. Hence}

|1−ψ◦k(s₁)ψ◦k(s₂)| ≥c>0 forklarge and this shows that

|ψ◦k(s₁)−ψ◦k(s₂)| ≤exp(c|s₁−s₂|)−1

exp(c|s₁−s₂|)+1 ≤c|s₁−s₂|

This shows that{k}is equicontinuous onS₁and hence there is a subsequence of {k}that converges uniformly on compact subsets ofzto a continuous mapping :z→

C

^n.

It may be observed that the same proof works when f : 1 → 2is an isometry in the inner-Carath´eodory distance on the domains. Indeed, the process of defining the scaling does not depend on the intrinsic metric that is used as a distance func- tion. Moreover the Carath´eodory distance enjoys the same functorial properties as the Kobayashi distance and even the quantitative bounds used in (3.8) and (3.10) remain the same. Hence the same proof works verbatim for the inner-Carath´eodory distance.

Proposition 3.3. Let f : 1 → 2 be an isometry in the Kobayashi distance on smoothly bounded strongly pseudoconvex domains 1, 2 in

C

ⁿ. Then the limit map map:z→

C

ⁿconstructed above satisfies:

(i) (z)⊂_w

(ii) :z→_wis a C⁰-isometry for the Kobayashi distance.

The same conclusions hold when f : 1 → 2 is an isometry in the inner- Carath´eodory distance on the domains, provided the boundaries are at least C³ smooth . In particular : z → _w is an isometry in the inner-Carath´eodory distance (which equals the Kobayashi distance) onzand_w.

Proof. Letk :˜^k₁→ ˜^k₂be the sequence of scaled mappings as before. Without loss of generality we assume thatk → uniformly on compact subsets ofz. The defining equations for˜^k₂and˜^k₁are respectively given by

˜

ρ_k(w)= 1

√δk

ρk(√

δkw, δ˜ kwn) , ρ˜k(z)= 1

√δk

ρk(√

δkz˜, δkz_n).

It is shown in [23] that these equations simplify as

˜

ρk(w)=2 Rewn+ | ˜w|²+ ˜B_k(w) , ρ˜k(w)=2 Rez_n+ |˜z|²+ ˜A_k(w) in neighborhoods of the origin where

| ˜B_k(w)| ≤ |w|²(c√

δk+η(δk|w|²)) , | ˜A_k(z)| ≤ |z|²(c√

δk+η(δk|z|²)) withη(t)=o(1)ast →0 andc>0 is uniform for allklarge.

Fix a compact subset ofz, sayS. Then fork1 andz∈S

2 Re(k)n(z)+ | ˜k(z)|²+ ˜B_k(k(z)) <0 (3.13) where

| ˜B_k(k(z))| ≤ |k(z)|²(c√

δk+η(δk|k(z)|²)).

By the previous proposition{k}is uniformly bounded onSand henceδk|k(z)|² converges to 0, with the result thatη(δk|k(z)|²))→ 0 ask → ∞. Passing to the limit ask→ ∞in (3.13) shows that

2 Ren(z)+ | ˜(z)|²≤0

which means exactly that(C) ⊂ _w. Since S ⊂ z is arbitrary it follows that :z →_w. Ifwere known to be holomorphic it would follow at once by the maximum principle that:z→_w. Howeveris known to be just continuous.

Let D ⊂ z be the set of all points z such that (z) ∈ _w. D is non- empty since(0,−1)∈ D as can be seen from (3.6) and the fact thatk(0,−1)= (0,−γk/δk). Sinceis continuous, Dis open inz.

Claim.It suffices to show that

d^K_z(z₁,z₂)=d^K_w((z₁), (z₂)) (3.14) forz₁,z₂∈ D. Indeed, ifz₀∈∂D∩z, choose a sequencez_j ∈Dthat converges toz₀. If the claim were true, then

d^K_z(z_j, (0,−1))=d^K_w((z_j), (0,−1)) (3.15) for all j. Sincez₀ ∈∂D, (z_j)→∂_wand as_wis complete in the Kobayashi distance, the right side in (3.15) becomes unbounded. However the left side remains bounded, again because of the completeness ofz. This contradiction would show thatD=z, knowing which the claim would prove assertion(ii)as well.

It is already known that d^K_˜

^k₁(z₁,z₂)=d^K_˜

^k₂((z₁), (z₂))

fork 1. To prove the claim it suffices to take limits on both sides in the equal- ity above. This is an issue of the stability of the Kobayashi distance, to under- stand which we need to study the behaviour of the infinitesimal Kobayashi metric _˜^k

1(z, v)ask → ∞. To do this, we will use ideas from [13]. Once this is done, an integration argument will yield information about the global metricK_˜k

1. Step 1. It will be shown that

klim→∞F^K_˜

^k₁(a, v)=F^K_z(a, v) (3.16) for(a, v)∈z×

C

ⁿ. Moreover, the convergence is uniform on compact subsets of z×

C

^n.

LetS⊂zandG ⊂

C

ⁿbe compact and suppose that the desired convergence does not occur. Then there are points a_k ∈ S converging to a ∈ S and vectors vk ∈Gconverging tov∈Gsuch that

0< ε0<|F^K_˜

^k₁(a_j, vj)−F^K

z(a_j, vj)|

for j large. This inequality holds for a subsequence only, which will again be de- noted by the same symbols. Further, since the infinitesimal metric is homogeneous of degree one in the vector variable , we can assume that|vj| =1 for all j. It was