# A Gehring-Hayman inequality for strongly pseudoconvex domains

Łukasz Kosiński  Ł. Kosiński
Institute of Mathematics, Faculty of Mathematics and Computer Science,
Jagiellonian University, Łojasiewicza 6, 30-348 Kraków, Poland
<EMAIL_ADDRESS>,  Nikolai Nikolov  N. Nikolov
Institute of Mathematics and Informatics
Bulgarian Academy of Sciences
Acad. G. Bonchev 8, 1113 Sofia, Bulgaria
Faculty of Information Sciences
State University of Library Studies and Information Technologies
Shipchenski prohod 69A, 1574 Sofia, Bulgaria<EMAIL_ADDRESS>and  Pascal J.
Thomas  P.J. Thomas
Institut de Mathématiques de Toulouse; UMR5219
Université de Toulouse; CNRS
UPS, F-31062 Toulouse Cedex 9, France<EMAIL_ADDRESS>
###### Abstract.

We prove that if $D$ is a strongly pseudoconvex domain with
$\mathcal{C}^{2,\alpha}$-smooth boundary, then the length of a geodesic for
the Kobayashi-Royden infinitesimal metric between two points is bounded by a
constant multiple of the Euclidean distance between the points.

###### Key words and phrases:

Kobayashi hyperbolic spaces; visibility; strongly pseudoconvex domains;
geodesics

###### 2010 Mathematics Subject Classification:

32F45

The first named author was supported by the NCN grant SONATA BIS no.
2017/26/E/ST1/00723. The second named author was partially supported by the
Bulgarian National Science Fund, Ministry of Education and Science of Bulgaria
under contract KP-06-N52/3. The second and third author enjoy the support of
the PHC Rila program 48135TJ, which made possible the stay at the Institute
for Mathematics and Informatics of the Bulgarian Academy of Sciences, Sofia,
during which this work was started.

##  1\. Introduction and motivation

In 1962, F.W. Gehring and W.K. Hayman [9] proved a conjecture of G. Piranian:
there is an absolute constant $C$ such that for any simply connected domain
$U$ strictly included in the complex plane, for any $z_{1},z_{2}\in U$, if
$\gamma_{H}$ is the hyperbolic geodesic from $z_{1}$ to $z_{2}$, and $\gamma$
any rectifiable curve from $z_{1}$ to $z_{2}$ contained in $U$, then

$l\left(\gamma_{H}\right)\leq Cl\left(\gamma\right),$

where $l$ denotes Euclidean length. When $U$ is convex, this just means that
$l\left(\gamma_{H}\right)\leq C|z-w|$. There are also versions of this result
for the diameters of the curves and for other metrics, see for instance the
survey [23].

In the several complex variable settings, it is natural to ask similar
questions about geodesics for the Kobayashi-Royden metric

$\kappa_{D}(z;X)=\inf\\{|\alpha|:\exists\varphi\in\mathcal{O}(\Delta,D),\varphi(0)=z,\alpha\varphi^{\prime}(0)=X\\},$

and it seems realistic to restrict oneself to domains with some regularity.
The best bounds known so far were of the form $C|z-w|^{1/m}$ for $m$-convex
domains and strongly pseudoconvex domains, see [20], [19] and [21]. However,
in the case of the unit ball, it is easy to see that geodesics, since they lie
inside affine complex lines, must have length bounded by $\frac{\pi}{2}|z-w|$,
which is much smaller in the critical case when $z$ and $w$ are close to each
other. The goal of this note is to show that this type of bound still holds in
regular enough strongly pseudoconvex domains.

######  Theorem 1.

Let $D$ be a strongly pseudoconvex domain with $\mathcal{C}^{2,\alpha}$-smooth
boundary. Then there is $C>0$ such that

$|z-w|\leq l(\gamma_{z,w})\leq C|z-w|$

for any $z,w\in D$ and any Kobayashi geodesic $\gamma_{z,w}$ joining $z$ and
$w$.

Observe that $C$ here depends on the domain. For this result, it does not
matter whether we consider the lengths of all curves or simply the Euclidean
distance: since the domain is bounded, we may always assume that $|z-w|\leq
c_{0}$ for come $c_{0}>0$, the inequality being trivial otherwise, and that
$z,w$ are close to $\partial D$, since when $z,w$ are contained in some
compact set $K\subset\subset D$, the Kobayashi-Royden and Euclidean metrics
are equivalent, and so $l(\gamma_{z,w})$ is comparable to $|z-w|$. And for two
points close to each other and to a $\mathcal{C}^{2}$ boundary, the lengths of
curves connecting the points are comparable to their Euclidean distance.

The optimal class of domains to which this result can apply remains an open
question: can we lower the regularity requirement? Can we allow finite type or
even mere pseudoconvexity? Is that property even required? For really
irregular domains, do we have a bound involving the lengths of connecting
curves rather than the Euclidean distance? Can we have a universal constant if
the bound involves the lengths of connecting curves? Of course, geodesics for
the Kobayashi-Royden metric do not always exist, and the question would have
to be rephrased using the existing notions of quasi-geodesics.

##  2\. A construction of Fridman-Ma and Lempert’s scaling

Within this section, we shall outline the _global scaling_ process which
provides much more precise estimates than standard methods used so far (e.g.
the ones involving squeezing functions).

In his seminal paper [15], Lempert showed that in a strongly convex domain $D$
with $\mathcal{C}^{6}$-boundaries any two points lie on a complex geodesic
disc. To show the existence of a geodesic he used the scaling method: for a
point $p$ in the topological boundary $\partial D$ of a strongly pseudoconvex
domain $D$ he found a neighborhood $U$ and a biholomorphic map $F$ on $U$ such
that $F(\partial D\cap U)$ was close in $\mathcal{C}^{6}$-topology to a big
part of the Euclidean sphere. The main weakness of the above construction is
that it is not defined globally on $D$.

A similar construction was independently applied by Fridman and Ma [8] to the
problem of exhaustion of smooth pseudoconvex domains at points of global
strict convexity.

We will construct a global scaling on strongly pseudoconvex domains. _Scaling_
here means that we use a precise automorphism of the Euclidean ball to scale
while _global_ means that scaling maps are defined on the whole
$\overline{D}$, so that images of geodesics under scaling maps are still
geodesics, while an explicit form for automorphisms is essential in carrying
out precise estimates.

We shall try to keep the notation from both papers: $\mathbb{B}$ denotes the
unit Euclidean ball and $B$ denotes the ball of radius $1$ centered at
$(-1,0)$, that is $B=\mathbb{B}-(1,0)$.

Fridman-Ma construction. Let $D$ be a strongly pseudoconvex domain with
$\mathcal{C}^{2,\alpha}$-smooth boundary, $\alpha\in[0,1]$. Note that for
$\alpha=0$ this covers the $\mathcal{C}^{2}$ case. Thanks to [7], that any
boundary point $p\in\partial D$ can be exposed, that is to say, there is a
biholomorphism $\Phi$ on a neighborhood of $\overline{D}$ such that for
$D^{\prime}=\Phi(D)$, $p^{\prime}=\Phi(p)$, $p^{\prime}$ is a point of _global
strong convexity_ , that is, there is a ball $U$ such that $D\subset U$ and
$p^{\prime}\in\partial D\cap\partial U$. This will be the starting point for
the Fridman-Ma construction [8], which we shall recall, but in addition we
will keep track of the dependence of the transformations on the starting point
$p$.

Let us start with the following definition.

######  Definition 2 (See [8, Definition 3.1]).

Let $N,k$ be natural numbers, $N\geq 4$, $k\leq 2(N-4)$. A $\mathcal{C}^{2}$
smooth domain $D\subset\mathbb{C}^{n}$, $0\in\partial D$ is of type $C(N,k)$
if there is a defining function $\rho$ of $D$ near $0$ of the form

$\rho(z)=2\operatorname{Re}z_{1}+\operatorname{Re}\sum_{i,j=2}^{n}a_{ij}z_{i}z_{j}+d|z_{1}|^{2}+N|z^{\prime}|^{2}+o(|z|^{2+\alpha}),$

where $z^{\prime}=(z_{2},\dots,z_{n})$, the numbers $N,d$ and $a_{ij}$ satisfy
$d>1$ and

(1)
$\operatorname{Re}\sum_{i,j=2}^{n}a_{ij}z_{i}z_{j}+(N-4-k/2)|z^{\prime}|^{2}\geq
0\quad\text{for }z^{\prime}\in\mathbb{C}^{n-1}.$

######  Remark 3.

Note that (1) is equivalent to
$|\sum_{i,j=2}^{n}a_{ij}z_{i}z_{j}|\leq(N-4-k/2)|z^{\prime}|^{2}$. In
particular, if $k=2(N-4)$, then $a_{ij}=0$.

Since $p$ is a point of global strong convexity, the same is true of nearby
points. The construction is outlined in five steps below.

Step 1. The translation

  1. $i)$

$z\mapsto T_{p}(z):=z-p$,

the scaling (locally independent of $p$) and the unitary transformation

  1. $ii)$

$z\mapsto A(p)z$,

which can be chosen so that if we write $\Phi_{1}:=A(p)\circ T_{p},$
$\Phi_{1}(D)\subset B$ and it admits a defining function $\rho$ with expansion
at $0$ of the form

$\rho(z)=2\operatorname{Re}z_{1}+\operatorname{Re}\sum_{i,j=1}^{n}a_{ij}(p)z_{i}z_{j}+\\\
\operatorname{Re}\sum_{j=1}^{n}c_{j}(p)z_{1}\bar{z}_{j}+\sum_{j=2}^{n}N_{j}(p)|z_{j}|^{2}++o(|z|^{2+(1-\epsilon)\alpha}),\forall\epsilon>0,$

where we also have used $A(p)$ to diagonalize the possible hermitian terms
$\sum_{i,j=2}^{n}c_{i,j}z_{i}\bar{z}_{j}$. The unitary matrix depends on the
first and second-order derivatives of the defining function of $D$ at $p$, so
$a_{ij}$, $c_{j}$ and $N_{j}$ are functions of class $\mathcal{C}^{\alpha}$ of
$p$.

Step 2. This is precisely [8, Lemma 3.2]. Namely, there is an integer $N\geq
4$ such that $D$ is biholomorphic to a domain of the type $C(N,0)$.

This is achieved by Fridman and Ma using an affine map

  1. $iii)$

$\Phi_{2}(z_{1},z_{2},\dots,z_{n})=\left(z_{1},z_{2}+\frac{c_{2}(p)}{2N_{2}(p)}z_{1},\dots,z_{n}+\frac{c_{n}(p)}{2N_{n}(p)}z_{1}\right)$

the coefficients of which are clearly of class $\mathcal{C}^{\alpha}$, as are
their inverses; rescaling maps $\Phi_{3}$

  1. $iv)$

$z_{j}\mapsto t_{j}(p)z_{j}$,

chosen so that $\Phi_{3}\circ\Phi_{2}\circ\Phi_{1}(D)\subset B$; the $t_{j}$
are of class $\mathcal{C}^{\alpha}$ with respect to $p$. Then, as in [8,
(3.7), p. 391] $\Phi_{3}\circ\Phi_{2}\circ\Phi_{1}(D)$ admits a defining
function of the form

(2)
$\rho(z)=2\operatorname{Re}z_{1}+2\operatorname{Re}\sum_{j=1}^{n}b_{j}(p)z_{1}z_{j}+\\\
\operatorname{Re}\sum_{i,j=1}^{n}a_{ij}(p)z_{i}z_{j}+d|z_{1}|^{2}+M|z^{\prime}|^{2}++o(|z|^{2+(1-\epsilon)\alpha}),\forall\epsilon>0.$

We then apply an automorphism $\Phi_{4}=\Phi_{4}^{\epsilon}$ of $B$ of the
form

  1. $v)$

$\Phi_{4}^{\epsilon}:z\mapsto\left(\frac{\epsilon
z_{1}}{2-\epsilon+(1-\epsilon)z_{1}},\frac{\sqrt{\epsilon(2-\epsilon)}}{2-\epsilon+(1-\epsilon)z_{1}}z^{\prime}\right).$

where $\epsilon$ is such that $\epsilon\sum_{j}|b_{j}(p)|$ is smaller than a
given positive number $\lambda$ chosen as in [8, Lemma 3.1, p. 390]
($\lambda=1/8$ will do). In particular, since the $b_{j}$ are continuous w.r.t
to $p$, $\epsilon$ can be chosen locally independently of $p$. The resulting
domain $\Phi_{4}\circ\cdots\Phi_{1}(D)$ admits a defining function of the form
(2) with new coefficients $b^{\prime}_{j}$ satisfying the hypotheses of [8,
Lemma 3.1] and so applying a map
$\Phi_{5}(z)=(z_{1}+\sum_{j=1}^{n}b^{\prime}_{j}z_{1}z_{j},z^{\prime})$, then
a contraction $\Phi_{6}(z)=\frac{M}{N}z$ with $N$ an integer, $N>8M$ we obtain
that $\Phi_{6}\circ\cdots\Phi_{1}(D)$ is of type $C(N,0)$ [8, end of proof of
Lemma 3.2, p. 392]. Again, it is straightforward to see that $N$ can be chosen
locally constant.

Step 3. The goal is to show that any domain of type $C(N,k)$ is biholomorphic
to $C(N,k+1)$. This is relevant only when $N\geq 5$ and $k\leq 2(N-4)-1$. Then
the statement is precisely [8, Lemma 3.4]. The transformations used to achieve
this statement are:

  1. $vi)$

$z\mapsto(z_{1}+\sum_{i.j=2}^{n}b_{ij}(p)z_{i}z_{j},z^{\prime})$,

where $b_{ij}(p)=\frac{1}{4(N-4-k/2)}a_{ij}(p)$,

  1. $vii)$

an automorphism $\Phi_{4}^{\epsilon}$ of the ball $B$,

where $\epsilon$ is chosen independently of $p$, and

  1. $viii)$

$z\mapsto\left(\frac{14}{15}z_{1},\sqrt{\frac{14}{15}}z^{\prime}\right)$. Call
$\Phi_{7}$ the composition of those maps.

Step 4. Repeating Step 3 exactly $2(N-4)$ times, with maps
$\Phi_{7},\dots,\Phi_{2N-2}$, we get a domain whose defining function near $0$
is of the form

(3)
$\rho(z)=2\operatorname{Re}z_{1}+d(p)|z_{1}|^{2}+N|z^{\prime}|^{2}++o(|z|^{2+(1-\epsilon)\alpha}),\forall\epsilon>0.$

Step 5. Applying a map

  1. $ix)$

$\Phi_{2N-1}(z)=(d(p)z_{1},(d(p)N)^{1/2}z^{\prime}),$

we can assume that $d(p)=N=1$. Also, the coefficients are of class
$\mathcal{C}^{\alpha}$. Thus $\Phi_{2N-1}\circ\cdots\Phi_{1}(D)$ which admits
a defining function of the form

(4)
$\rho(z)=2\operatorname{Re}z_{1}+|z|^{2}++o(|z|^{2+(1-\epsilon)\alpha}),\forall\epsilon>0,\quad\text{as
}z\to 0,$

and that is contained in the ball $rB$ of radius $r$ centered at $(-r,0)$ for
some $r\geq 1$.

######  Definition 4.

If $D$ is a $\mathcal{C}^{2}$ domain and $p\in\partial D$ a point of strong
pseudoconvexity, we denote $F_{p}:=\Phi_{2N-1}\circ\cdots\Phi_{1}$ the map
which results from the Fridman-Ma construction.

######  Remark 5.

It follows from the above construction that we can locally write
$F_{q}(z)=\tilde{F}(z,\alpha_{1}(q),\ldots,\alpha_{M}(q))$, where $\tilde{F}$
is a holomorphic mapping and $\alpha_{j}$ are of class $\mathcal{C}^{\alpha}$
(here $M$ denotes the number of $\mathcal{C}^{\alpha}$ smooth functions that
appear in the construction of $F_{q}$). The same remains true for
$F_{q}^{-1}(z)$.

In particular, the Jacobian of $F_{q}$ with respect to $z$ as well as that of
its inverse $F_{q}^{-1}$ vary locally $\mathcal{C}^{\alpha}$-continuously. In
particular, $F_{q}$ locally does not change the Euclidean distance between
points.

For later purposes, we need to know that we can choose an appropriate $p$ to
make the Fridman-Ma construction.

######  Lemma 6.

For any $z\in D$ that is sufficiently close to $p_{0}\in\partial D$, there
exists $q\in\partial D$, close to $p_{0}$, so that
$F_{q}(z)\in(-1,0)\times\\{0\\}^{n-1}.$

###### Proof.

Applying one Fridman-Ma transformation we can assume that $p_{0}=0$ and a
defining $\rho$ function of $D$ near $p_{0}$ is of the form
$\rho(z)=2\operatorname{Re}z_{1}+|z|^{2}++o(|z|^{2+(1-\epsilon)\alpha}),$ as
as $z\to 0$.

As in Remark 5 write
$F_{q}(z)=\tilde{F}(z,\alpha_{1}(q),\ldots,\alpha_{M}(q))$, where $\alpha_{j}$
are $\mathcal{C}^{\alpha}$-continuous, $\alpha_{j}(0)=0$, $j=1,\ldots,M$.

Let us consider the map

$\varphi:(-1,\delta)\times\partial D\ni(t,q)\mapsto
F_{q}^{-1}(t,0)\in\mathbb{C}^{n},$

which is well defined for $\delta>0$ sufficiently small. In particular,

$\varphi(t,q)=G(t,\alpha_{1}(q),\ldots,\alpha_{M}(q))$

for $q\in\partial D$ close $0$, where $G$ is holomorphic near $0$ in
$\mathbb{C}^{1+M}$. Moreover,

$\varphi(t,0)=(rt+o(t^{2}),0)\ \text{for some $r>0$, }\ \text{and
}\varphi(0,q)=q$

(the first equality above follows from formulas $(iii)-(ix)$). Consequently,
we can write

(5) $\varphi(t,q)=(rt,0)+q+t\Gamma(t,\alpha_{1}(q),\ldots,\alpha_{M}(q)),$

for $\Gamma$ holomorphic near $0$, $\Gamma(0)=0$.

We need to show that there is $\epsilon>0$ and a neighborhood $U$ of $p$ in
$\partial D$ such the range $\varphi((-\epsilon,\epsilon)\times U)$ contains a
neighborhood of $0$. This is already claimed in the proof of [6, Theorem 4.1].
The authors deduced this fact by claiming that $\varphi$ is a local
diffeomorphism near $t=0$ and $q=0$. This, however, is not the case, if
$\alpha<1$, as the mapping $\varphi$ is only $\mathcal{C}^{\alpha}$-smooth
with respect to the variable $q$.

Nevertheless, a topological argument proves the above assertion also in the
case when $\alpha=0$, i.e. when $\partial D$ is $\mathcal{C}^{2}$ smooth (in
particular [6, Theorem 4.1] is salvaged in its full generality). Let us
present how it can be done.

Let us write $\partial D$ near $0$ as
$\operatorname{Re}z_{1}=\tilde{\rho}(\operatorname{Re}z_{2},z^{\prime})$,
where $\tilde{\rho}$ is $\mathcal{C}^{2,\alpha}$ smooth. For simplicity of
notation, we shall identify $\mathbb{C}^{n}$ with $\mathbb{R}^{2n}$ writing
$(z_{1},\ldots,z_{n})=(s,x)=(s,x_{2},\ldots,x_{2n})$. Let $V$ be a
neighborhood of $0$ in $\mathbb{R}^{2n-1}$ and put
$\psi(s,x)=(s+\tilde{\rho}(x),x)$ if $x\in V$ so that $\psi$ is a
$\mathcal{C}^{2}$-smooth map that sends $\\{0\\}\times V$ diffeomorphically to
$U$.

Consider $\Phi:(-\epsilon,\epsilon)\times V\to\mathbb{R}^{2n}$ given by
$\Phi(t,x)=\psi^{-1}(\varphi(t,\psi(0,x))).$ Our aim is to show that
$\Phi((-\epsilon,\epsilon)\times V)$ is a neighborhood of $0$. In other words,
we want to show $(s,y_{1},\ldots,y_{2n})$ is in the image of $\Phi$, i.e. that
there is $(t,x)$ such that

(6) $\Phi(t,x)=(s,y)$

providing that $s$ and $y$ are small. Write
$\Phi=(\Phi_{1},\ldots,\Phi_{2n})$.

It follows from (5) that

$\Phi(t,x)=(rt,x)+t\gamma(t,x),$

where $\gamma$ is continuous and $\gamma(0,0)=0$.

For a fixed $x$ we are looking at the equation $\Phi_{1}(t,x)=s$. It follows
from the implicit function theorem ($\Phi_{1}$ depends smoothly on $t$ as well
as on continuous functions $\alpha_{j}$ that we treat as coefficients) that
this equation has a solution $t=t(s,x)$ which is of the form

(7) $t=s/r+s\delta(s,x)$

where $\delta$ is continuous, $\delta(0)=0$.

Plugging $t=s/r+s\delta(s,x)$ to $(\Phi_{2},\ldots,\Phi_{n})(t,x)$ we get the
mapping of the form

$x\mapsto x+s\beta(s,x),$

where again $\beta$ is continuous and $\beta(0)=0$. Fix a closed ball in $V$
with $0$ being its center and denote it by $K$. Take $s$ sufficiently small so
that $s\beta(s,x)$ is in $\frac{1}{2}K$ for any $x\in K$. Then, by the Brouwer
fixed point theorem, $\frac{1}{2}K$ is contained in the image of $x\mapsto
x+s\beta(s,x)$ (this is because for any $y\in\frac{1}{2}K$ the mapping
$x\mapsto y-s\beta(s,x)$ sends $K$ to $K$, so must have a fixed point). Thus,
if $s$ is small and $y\in\frac{1}{2}K$, equation (6) has a solution $x$; then
$t$ is given by (7).

∎

Lempert’s scaling method.

We will use automorphisms of the ball $\mathbb{B}$

(8)
$A_{t}(z)=\left(m_{t}(z_{1}),\sqrt{1-t^{2}}\frac{z^{\prime}}{1+tz_{1}}\right),\text{
where }m_{t}(z_{1})=\frac{z_{1}+t}{1+tz_{1}}.$

Given a domain of the form $F_{p}(D)$, with $\partial D$ of class
$\mathcal{C}^{2,\alpha}$ and $F_{p}$ as in Definition 4, we consider
$D_{0}=T_{e_{1}}(D)$ where $e_{1}=(1,0)\in\mathbb{C}\times\mathbb{C}^{n-1}$
and $T_{e_{1}}$ is the translation $z\mapsto z+e_{1}$. Then $D_{0}$ admits a
defining function of the form
$\rho_{0}(z)=-1+|z|^{2}++o(|z|^{2+(1-\epsilon)\alpha}),\forall\epsilon>0,$ as
$z\to e_{1},$ and that is contained in the ball $B^{\prime}$ of radius
$1+s^{\prime}$ centered at $(-s^{\prime},0)$ for some $s^{\prime}\geq 0$. Note
that $A_{t}^{-1}$ is well defined on $B^{\prime}$ and $A_{t}^{-1}(z)\to(-1,0)$
uniformly on compact subsets of $B^{\prime}\setminus\\{e_{1}\\}$ as $t\to 1$.
Define

(9)
$D_{t}=A_{t}^{-1}(D_{0})\quad\text{and}\quad\rho_{t}(z)=\frac{|1+tz_{1}|^{2}}{1-t^{2}}\rho_{0}(A_{t}(z)),\
t\in(0,1).$

Clearly $D_{t}=\\{\rho_{t}<0\\}$ and $\rho_{t}(z)$ converges to $-1+|z|^{2}$
locally uniformly in $\\{\operatorname{Re}z_{1}>-\alpha\\}$ for any $\alpha<1$
as $t\to 1$.

Summing up, for any $\beta>-1$:

  1. A1)

$D_{t}\cap\\{\operatorname{Re}z_{1}>\beta\\}$ converges to
$\mathbb{B}\cap\\{\operatorname{Re}z_{1}>\beta\\}$ in
$\mathcal{C}^{2,\alpha}$-topology (here and in the sequel this means
convergence of defining functions),

  2. A2)

$D_{t}$ converges to the ball in the Hausdorff topology.

We now must restrict ourselves to the case when $\partial D$ is of class
$\mathcal{C}^{2,\alpha}$ with $\alpha>0$. The reason is that we must use
Lempert’s work on complex geodesics [15, 17, 18] (see also a discussion in
[10, Remark B]) and the arguments about the dependence and regularity of
complex geodesics fail if we merely assume $\mathcal{C}^{2}$-smoothness [17,
remark after the Main Theorem, p. 561].

In [17] and [18] Lempert considered the class $S^{k}$, where $k>1$ is non-
integer; we refer the reader to [17, p. 561] for a precise definition. If
$D\in S^{m+\alpha}$, $m\in\mathbb{N}$, $0<\alpha<1$, then $\partial D$ is of
class $\mathcal{C}^{m,\alpha}$; if $\partial D$ is of class
$\mathcal{C}^{m,\alpha}$, then $D\in S^{m-1+\alpha}$. Thus it is proven in
[17, Corollary 3.3] that any complex geodesic in a
$\mathcal{C}^{2,\alpha}$-smoothly bounded strictly convex domain $D$ is in
$\mathcal{C}^{1,\alpha}(\overline{\mathbb{D}})$. Also, the
$\mathcal{C}^{1/2}(\overline{\mathbb{D}})$ norm of a complex geodesic $f$ is
uniformly bounded when $f(0)$ is within a compact subset of $K$ of $D$, with
bounds depending only on the diameter of $D$ and normal curvatures of
$\partial D$ and on $K$ [15, Proposition 13], so there is local uniformity
with respect to the domain as well, in the sense of $\mathcal{C}^{2,\alpha}$
convergence of defining functions.

Using those a priori estimates in the $\mathcal{C}^{6}$ case and approximation
(see [17, p. 562–563]), Lempert showed in [18, Proposition 3] that the bound
on $\mathcal{C}^{1,\alpha}(\overline{\mathbb{D}})$ norm is in fact locally
uniform. This implies the following.

######  Proposition 7.

Let $D_{n}$ be a sequence of strictly convex domains such that their defining
functions converge in the $\mathcal{C}^{2,\alpha}$ sense to the defining
function of a strictly convex domain $D$. Let $K$ be a compact subset of $D$
and $f_{n}$ a sequence of complex geodesics in $D_{n}$ such that $f_{n}(0)\in
K$. Then $\mathcal{C}^{1,\alpha}(\overline{\mathbb{D}})$-norms of $f_{n}$ are
uniformly bounded, for $n$ big enough, and a subsequence of $(f_{n})$
converges in the $\mathcal{C}^{1}(\overline{\mathbb{D}})$ topology to a
complex geodesic we denote by $f$.

###### Proof.

It follows from the hypothesis that the
$\mathcal{C}^{\alpha}(\overline{\mathbb{D}})$ norms of $f_{n}^{\prime}$ are
uniformly bounded as well. Applying the Arzelà-Ascoli theorem to
$f_{n}^{\prime}$ we can assume that a subsequence converges uniformly on
$\overline{\mathbb{D}}$ to an analytic disc, say $g$. Also, a subsequence of
$(f_{n})$ converges on $\overline{\mathbb{D}}$ to an analytic disc that we
shall denote by $f$. Clearly $f^{\prime}=g$ and a subsequence $f_{n}$
converges to $f$ in $\mathcal{C}^{1}(\overline{\mathbb{D}})$ topology. Note
that $f$ is a (unique) complex geodesic in $D$, because it verifies the
extremality conditions from [15]. ∎

Actually, something can still be said about complex geodesics in $D$ in the
case when $\alpha=0$, that is when a strongly convex domain $D$ has
$\mathcal{C}^{2}$-smooth boundary. Chirka, Coupet and Sukhov established in
[5, Corollary 1.5] the $\mathcal{C}^{1-\epsilon}$ continuity up to the
boundary for a holomorphic map $f$ from $\mathbb{D}$ with boundary values in a
totally real $\mathcal{C}^{1}$ manifold $M$. The manifold to be considered
here is, in the terminology of [17], $\widehat{\partial
D}:=\\{(z,T_{z}^{\mathbb{C}}\partial D),z\in\partial
D\\}\subset\mathbb{C}^{n}\times\mathbb{P}^{n-1}$, while if we have a geodesic
$\phi$, the map has to be $(\phi,\tilde{\phi})$, see [17, proof of Lemma 3.1].
Also, their result implies that one can control $\mathcal{C}^{1-\epsilon}$
norms of geodesic also under $\mathcal{C}^{2}$-perturbations of $\partial D$
(see the discussion following [5, Corollary 1.5]).

##  3\. Proof of Theorem 1

We shall divide the proof into two cases, according to whether the direction
generated by $z$ and $w$ is “tangential” or “normal”. In order to give this a
precise meaning, first recall that it is enough to prove our theorem for $z,w$
in a neighborhood $U$ of $\partial D$, which we will take small enough so that
for any $z\in U\cap D$, there exists a unique point $\pi(z)\in\partial D$ such
that $|z-\pi(z)|=\min\\{|z-\zeta|,\zeta\in\mathbb{C}^{n}\setminus D\\}$. We
denote by $n_{z}$ the outer unit normal to $\partial D$ at $\pi(z)$, and
write, for any vector $v$, $v_{z}:=\langle v,n_{z}\rangle n_{z}$ where
$\langle\cdot,\cdot\rangle$ stands for the usual Hermitian inner product.

We first deal with the (almost) “tangential” case.

######  Lemma 8.

Let $D$ be a strongly pseudoconvex domain with $\mathcal{C}^{2,\alpha}$
boundary. There exists $\epsilon_{0}>0$ such that if
$0<\epsilon<\epsilon_{0}$, and if $z,w\in D$ satisfy $\delta(z)<\epsilon$,
$|z-w|<\epsilon$ and $|(z-w)_{z}|$ $<\epsilon|z-w|$, then $z$ and $w$ lie on a
stationary map (thus on a complex geodesic) whose diameter is $o(1)$ as
$\epsilon\to 0$. Moreover, $|(x-y)_{x}|/|x-y|$ $=o(1)$ for any points $x,y$ of
this complex geodesic.

Part of this result was proven in [3] under a $\mathcal{C}^{3}$ smoothness
assumption (see also [11]). Below we shall show how to deduce it in the
$\mathcal{C}^{2,\alpha}$ setting directly from Lempert’s work.

###### Proof.

First make the additional assumption that $D$ is convex. Since it is strongly
pseudoconvex (bounded), it enjoys the visibility property ([2], or use [1] and
the fact that the Gromov boundary of a Gromov hyperbolic space is always
visible): given two distinct points $p,q\in\partial D$, and sequences
$p_{n}\to p$, $q_{n}\to q$, there exists a compact set $K\subset D^{\prime}$
such that for $n$ large enough, the real geodesic from $p_{n}$ to $q_{n}$
intersects $K$. It is easy to see that this can be made uniform: as soon as
$|p-q|\geq\delta>0$, there is a $K_{\delta}\subset\subset D^{\prime}$ such
that geodesics from $p_{n}$ to $q_{n}$ eventually intersect $K_{\delta}$. In
particular, if the diameter of the complex geodesic through $z$ and $w$ is big
enough, and $z$ and $w$ are close to the boundary, the complex geodesic needs
to intersect a fixed compactum $K$, and thus meets the boundary transversely
(uniformly with respect to the point in $K$, by [15, Proposition 14]).
Therefore, for points on such a complex geodesic $z,w$ which are close enough
to each other and to the boundary, $|(z-w)_{z}|\geq C|z-w|,$ where $C>0$ is
uniform. This contradicts the assumptions with $\epsilon_{0}$ small enough, we
would have a contradiction, so the assertion about the diameter is proved.

For general strongly pseudoconvex domains one can derive the assertion by
exposing points as in [7]: using a biholomorphism on a neighborhood of
$\overline{D}$, we can assume that $D\subset\mathbb{B}$, $p\in\partial
D\cap\partial\mathbb{B}$. Furthermore by the regularity assumptions $\partial
D$ remains strongly convex in a neighborhood of $p$, so taking $\epsilon_{0}$
small enough we may assume $z_{n},w_{n},q\in U$, a neighborhood of $p$ such
that $D\cap U$ is smooth strongly convex and $U\cap\partial D$ is close to
$U\cap\partial\mathbb{B}$. Consider the complex geodesic
$f:\mathbb{D}\longrightarrow D\cap U$ obtained for the convex domain $D\cap U$
by the argument above. This is a stationary map in the sense of Lempert [17,
Section 3, p. 562, points 1–3], meaning in particular that there exists a
$\mathcal{C}^{1/2}$ positive function $p(\zeta)$,
$\zeta\in\partial{\mathbb{D}}$, such that the function
$\tilde{f}(\zeta):=\zeta p(\zeta)\overline{n_{f(\zeta)}}$ extends to a
holomorphic function on ${\mathbb{D}}$. Therefore it is also stationary as a
map $f:\mathbb{D}\longrightarrow D$.

Making $z,w$ even closer to $p$ as needed, we may assume that
$\overline{f({\mathbb{D}})}\cap\partial(D\cap U)$ is contained in $\partial D$
and close enough to $p$ so that for any $z\in D$, $\operatorname{Re}\langle
z-f(\zeta),n_{f(\zeta)}\rangle<0$. A winding number argument then shows that
the equation $\langle z-f(\zeta),\overline{\tilde{f}(\zeta)}\rangle=0$ admits
a unique solution $F(z)$ in ${\mathbb{D}}$, which has to be holomorphic, and
this uniqueness shows that $F(f(\zeta))=\zeta$ for any $\zeta\in{\mathbb{D}}$
(in other words we have a left-inverse, see [14]) and therefore the map $f$ is
indeed a complex geodesic for $z,w$ with respect to $D$.

The last statement in the Lemma follows from [12, Theorem 3]: if $x,y$ are in
the range of a complex geodesic $\varphi$, then
$|(x-y)_{x}|\lesssim|x-y|\operatorname{diam}(\varphi)$111As usual, $f\lesssim
g$ means that $f\leq Cg$ for some uniform constant $C>0$ (depending only on
$D$); $f\simeq g$ is understood analogously. ∎

######  Corollary 9.

Let $D$ be a domain in $\mathbb{C}^{n}$ and $p\in\partial D$ a strongly
pseudoconvex point. If $z,w\in D$ are close to $p$ and $|(z-w)_{z}|/|z-w|$ is
sufficiently small, then there is a unique real geodesic for $z,w$ and it is
induced by a complex one.

###### Proof.

Recall there is a neighborhood $U$ of $p$ so that there exists a
biholomorphism on a neighborhood of $\overline{D\cap U}$ making $D$ convex.

Let $\varphi$ be any complex geodesic passing through $z,w$. By Lemma 8, the
range of $\varphi$ is close to the boundary, its diameter is also small, so we
may assume $\varphi(\mathbb{D})\subset U$. So after biholomorphism we can
reduce ourselves to stationary discs in a strictly convex domain $D^{\prime}$.

Inside the complex geodesic $\varphi$ there can only be one real geodesic,
that we shall denote by $\gamma$, $\gamma(0)=z$, $\gamma(t_{1})=w$. Suppose
that there is another real geodesic between $z$ and $w$, say $\eta$. Then for
some $t_{0}$ the point $\eta(t_{0})$ must lie outside the range of $\varphi$
and we can choose it so that $|(\eta(0)-\eta(t))_{\eta(0)}|/|\eta(0)-\eta(t)|$
is small for all $t\in[0,1]$. As in Lemma 8 take a complex geodesic $\psi$
passing through $\eta(0)$ and $\eta(t_{0})$; it has to be close to $\eta$ when
$t_{0}$ is chosen close to $\sup\\{t:\eta(t)\notin\varphi(\mathbb{D})$.
Consider the one-parameter family of maps from the unit disc to $D^{\prime}$
given by $(1-\theta)\varphi+\theta\psi$, then they are extremal for $z,w$ and
for $\theta$ close to $0$, [14, Lemma 3.2] shows that they must be proper into
$D^{\prime}$, a contradiction. ∎

###  3.1. Theorem 1 – tangential case

Let $z_{n},w_{n}\in D$ be such that $z_{n},w_{n}\to p\in\partial D$, and
$|(z_{n}-w_{n})_{z_{n}}|/|z_{n}-w_{n}|\to 0$. Let $\varphi_{z_{n},w_{n}}$ be
the unique complex geodesic that passes through $z_{n}$ and $w_{n}$. Let
$\gamma_{z_{n},w_{n}}$ be the unique real geodesic for $z_{n},w_{n}$, induced
by $\varphi_{z_{n},w_{n}}$.

We shall follow the notation from Section 2. For each $n$, take a point
$p_{n}\in\overline{\varphi_{z_{n},w_{n}}({\mathbb{D}})}$ so that $p_{n}\to p$
and then apply the transformation $F_{p_{n}}$. Let $D_{n}:=F_{p_{n}}(D)$. So
we are reduced to the case where $p=(1,0)\in\partial D_{n}$ lies in the range
of the closed analytic disc $\overline{\varphi_{z_{n},w_{n}}({\mathbb{D}})}$
and $D_{n}$ near the point $p$ is given by the inequality

(10) $\\{|z|^{2}+o(|z-p|^{2+\alpha})<1\\}.$

We now drop the subscript from the notation and write $D$ for $D_{n}$ and
$z,w$ for $z_{n},w_{n}$. For $t\in(0,1)$, $D_{t}:=A_{t}^{-1}(D)$ as in (9).

Then for $t$ well chosen, close to $1$,
$\psi_{z,w}:=A_{t}^{-1}\circ\varphi_{z,w}$ is a complex geodesic in $D_{t}$
that lies entirely in $\\{\operatorname{Re}z_{1}\geq 0\\}$, with its boundary
intersecting $\\{\operatorname{Re}z_{1}=0\\}$; it is close in
$\mathcal{C}^{1}$ topology on $\overline{\mathbb{D}}$ to a geodesic in
${\mathbb{B}}_{n}$, after a possible reparametrization. Its range intersects a
fixed compact subset $K$ of $\mathbb{B}$, independent of $n$, and contains
$(1,0)$ in its closure.

Complex geodesics in $\mathbb{B}_{n}$ are intersections of $\mathbb{B}_{n}$
with complex affine lines. Therefore, a simple argument shows that any complex
geodesic $\psi=(\psi_{1},\tilde{\psi})$ in $\mathbb{B}_{n}$ contained in
$\\{\operatorname{Re}z_{1}\geq 0\\}$ that passes through $(1,0)$, and such
that $\psi(0)\in K$, satisfies the following uniform estimates:

$\left|\frac{\tilde{\psi}(\mu)}{1+t\psi_{1}(\mu)}-\frac{\tilde{\psi}(\lambda)}{1+t\psi_{1}(\lambda)}\right|\simeq|\mu-\lambda|\simeq|\tilde{\psi}(\mu)-\tilde{\psi}(\lambda)|.$

Trivially,

$|\psi_{1}(\mu)-\psi_{1}(\lambda)|\lesssim|\lambda-\mu|,$

again uniformly. By $\mathcal{C}^{1}$ convergence argument, all these uniform
estimates remain true for the geodesics $\psi_{z,w}$ (after proper
reparametrizations). In particular, if
$x=(x_{1},x^{\prime}),y=(y_{1},y^{\prime})$ are two points lying in the range
of any $\psi_{z,w}$, then

(11)
$\left|\frac{y^{\prime}}{1+ty_{1}}-\frac{x^{\prime}}{1+tx_{1}}\right|\simeq|y^{\prime}-x^{\prime}|\simeq|y-x|.$

Let $x,y$ be in the range of $\eta_{z,w}:=A_{t}^{-1}(\gamma_{z,w})$. Clearly,

$A_{t}(x)-A_{t}(y)=\left(\frac{(1-t^{2})(y_{1}-x_{1})}{(1+tx_{1})(1+ty_{1})},(1-t^{2})^{1/2}(\frac{y^{\prime}}{1+ty_{1}}-\frac{x^{\prime}}{1+tx_{1}})\right).$

In particular, by the estimates from (11):

(12) $|A_{t}(y)-A_{t}(x)|\simeq(1-t^{2})^{1/2}|y-x|,$

where $\simeq$ is uniform.

As already mentioned, $\eta_{z,w}$ are real geodesics, induced by
$\psi_{z,w}$. The geodesics $\psi_{z,w}$ intersect a fixed compact set $K$. By
Proposition 7 and its proof, they are uniformly
$\mathcal{C}^{1+\alpha}$-smooth on the closed disc in the sense that their
$\mathcal{C}^{1+\alpha}$-norms (so $\mathcal{C}^{1}$ as well) are uniformly
bounded, which implies the assertion of Theorem 1 for $\eta_{z,w}$. From this
and (12) one can deduce the assertion for $\gamma_{z,w}=A_{t}(\eta_{z,w})$.

###  3.2. Theorem 1 – non-tangential case: preparatory results

A visibility lemma for the class of “Goldilocks” domains (which includes
finite type domains) was given by Bharali-Zimmer [2]. The following visibility
lemma is essentially due to Bracci-Fornaess-Wold [3], but we need a slightly
more general form:

######  Lemma 10.

Let $\alpha>-1$ and $D_{n}$ be a family of domains $D_{n}\subset\mathbb{B}$
such that $D_{n}\cap\\{\operatorname{Re}z_{1}\geq\alpha\\}$ converges to
$\mathbb{B}\cap\\{\operatorname{Re}z_{1}\geq\alpha\\}$ in $\mathcal{C}^{2}$
topology. Let $z_{n},w_{n}\in D_{n}$ converge to $p$ and $q$ respectively,
where $p,q\in\partial\mathbb{B}\cap\\{\operatorname{Re}z_{1}>\alpha\\}$,
$p\neq q$. Then there is $K\subset\subset\mathbb{B}$ such that any real
geodesic $\sigma_{n}$ in $D_{n}$ for $z_{n},$ $w_{n}$ intersects $K$.

If instead of a sequence of domains $D_{n}$ a fixed one is considered, the
result was proven in [3] (see also [2]). A glimpse at their reasoning ensures
us that estimates achieved there depend just on the normal curvature of the
domain.

###### Proof.

We can reparametrize $\sigma_{n}:[a_{n},b_{n}]\to D_{n}$ by arc-length so that
$\delta_{n}(\sigma_{n}(0))\leq\delta_{n}(\sigma_{n}(t))$, where
$\delta_{n}(x)=\operatorname{dist}(x,\partial D_{n})$. Since $D_{n}$ are
uniformly bounded, $|\sigma_{n}^{\prime}(t)|\leq L$ for some uniform constant
$L$.

On the other hand, for $n$ big enough the normal curvature of $\partial D_{n}$
restricted to $\\{\operatorname{Re}z_{1}\geq\alpha+\epsilon\\}$, where
$\epsilon$ is small enough, is close to the one of $\mathbb{B}$. In
particular, all $D_{n}$ satisfy an exterior ball condition with a uniform
constant. From this we deduce the existence of $C>0$ such that

(13) $k_{D_{n}}(x,v)\geq C\frac{||v||}{\sqrt{\delta_{n}(x)}}$

for $x\in D_{n}\cap\\{\operatorname{Re}z_{1}\geq\alpha+\epsilon\\}$.

The same curvature-type argument shows that $\partial D_{n}$ restricted to
$\\{\operatorname{Re}z_{1}\geq\alpha+\epsilon\\}$ satisfies a uniform interior
ball condition. Therefore, there is a uniform $k\in\mathbb{N}$ so that any two
points in $D_{n}$ can be connected with a chain of $k$ balls of a fixed
radius. From this, we gain an estimate

$K_{D_{n}}(z,w)\leq
C+\frac{1}{2}\log\frac{1}{\delta_{n}(z)}+\frac{1}{2}\log\frac{1}{\delta_{n}(w)},$

with a uniform $C>0$.

Now it is enough to notice that a curve that remains too close to the boundary
at all times has length bigger than the above estimate for the Kobayashi
distance because of (13), or to repeat an argument from [3, Proof of
Proposition 2.2] or [2, Section 5]. ∎

######  Lemma 11.

Suppose that $D_{t_{n}}$ converges to $\mathbb{B}$ as in A1) and A2) when
$t_{n}\to 1$. Let $z_{n},w_{n}\in D_{t_{n}}$ be such that $z_{n}\to(1,0)$ and
$w_{n}\to(-1,0)$. Let $\varphi_{n}$ be a real geodesic for $k_{D_{n}}$ joining
$z_{n}$ and $w_{n}$, parametrized so that for some $\alpha_{n},\beta_{n}>0$,
$\varphi_{n}(-\alpha_{n})=w_{n}$, $\varphi_{n}(\beta_{n})=z_{n}$, and
$\operatorname{Re}\varphi_{n}(0)_{1}=0$. Then inside any compact subset
$K\subset\mathbb{B}^{N}$, the geodesics $\varphi_{n}$ converge in the
$\mathcal{C}^{1}$ topology to a geodesic line in $\mathbb{B}$ between $(-1,0)$
and $(1,0)$, and for any $\epsilon,\delta_{0}>0$, there exists
$n(\epsilon,K,\delta_{0})$ such that for any $z^{\prime},w^{\prime}\in
K\cap\varphi_{n}$ with $|z^{\prime}-w^{\prime}|\geq\delta_{0}$, then for
$n\geq n(\epsilon,K,\delta_{0})$,
$l(\varphi_{n}|_{[z^{\prime},w^{\prime}]})\leq(1+\epsilon)|z^{\prime}-w^{\prime}|$.

The same holds if we change the assumption on $z_{n}$ to the following:
$z_{n}\in(0,1)\times\\{0\\}^{N-1}$ (possibly replacing the geodesic line by a
ray).

###### Proof.

Take $\tilde{\beta}<\beta<1$ close to $1$. Take
$s_{n}:=\max\\{t:\operatorname{Re}\varphi_{n}(t)_{1}=-\beta\\}$,
$\tilde{s}_{n}:=\min\\{t:\operatorname{Re}\varphi_{n}(t)_{1}=-\tilde{\beta}\\}$,
$\tilde{t}_{n}:=\max\\{t:\operatorname{Re}\varphi_{n}(t)_{1}=\tilde{\beta}\\}$,
$t_{n}:=\min\\{t:\operatorname{Re}\varphi_{n}(t)_{1}=\beta\\}$, and
corresponding points $x_{n}=\varphi_{n}(s_{n})$,
$\tilde{x}_{n}=\varphi_{n}(\tilde{s}_{n})$,
$\tilde{y}_{n}=\varphi_{n}(\tilde{t}_{n})$ and $y_{n}=\varphi_{n}(t_{n})$. It
follows from the visibility lemmas applied to $x_{n},\tilde{x}_{n}$ and to
$y_{n},\tilde{y}_{n}$ that there are compact sets
$K_{1}\subset\\{-\beta<\operatorname{Re}z_{1}<-\tilde{\beta}\\}$ and
$K_{2}\subset\\{\tilde{\beta}<\operatorname{Re}z_{1}<\beta\\}$ that intersect
the ranges of the geodesics $\varphi_{n}$.

Since the $\varphi_{n}$ form an equicontinuous family, we get that a
subsequence of $\varphi_{n}$ converges to a real geodesic $\varphi$ in the
ball that intersects $K_{1}$ and $K_{2}$ and passes through $0$. Letting
$\tilde{\beta}$ tend to $1$, we get the convergence on all compact sets. The
statement about lengths comes from the fact that the real geodesic for the
ball between $z^{\prime}$ and $w^{\prime}$ is close to a line.

The second assertion can be deduced in a similar way. ∎

###  3.3. Proof by contradiction: Claims

Suppose that the estimate in Theorem 1 fails. Then there are sequences of
points $z_{n},w_{n}\in D$, joined by a geodesic
$\gamma_{n}=\gamma_{z_{n},w_{n}}$ such that

(14) $l(\gamma_{n})\geq a_{n}|z_{n}-w_{n}|,\mbox{ where }a_{n}\to\infty.$

Our goal is to obtain Claim 1, and then to prove Claim 2 which contradicts the
previous one.

Since the ratio of the Euclidean and Kobayashi-Royden infinitesimal metrics
are bounded above and below on any compactum, it is enough to consider the
case at least one of the points $z_{n},w_{n}$ escapes from any compact set, so
passing to a subsequence and exchanging $z_{n}$ and $w_{n}$ if needed, we may
assume $z_{n}\to p\in\partial D$.

Then we may reduce ourselves to the case where $w_{n}$ converges to the same
boundary point. Indeed, if the distance between $z_{n}$ and $w_{n}$ did not
converge to 0, then by (14) the lengths of the curves $\gamma_{n}$ would go to
infinity and switching $z_{n}$ and $w_{n}$, if necessary, we could pick points
$w^{\prime}_{n}\in\gamma_{n}$, such that $|w^{\prime}_{n}-z_{n}|\to 0$ while
$l(\gamma_{n}|_{[w_{n}^{\prime},z_{n}]})$ remains big.

Claim 0. It is enough to focus on the case when
$|\left(\gamma_{n}(t)-\gamma_{n}(s)\right)_{\gamma_{n}(t)}|\geq
C|\gamma_{n}(t)-\gamma_{n}(s)|$, for all $s,t$, for some uniform constant $C$,
where normal length is taken with respect to $p\in\partial D$.

###### Proof of the claim.

Actually, if the opposite inequality were true for some $t,s$, where $C$ is
small enough, we would deduce that $\gamma_{n}$ is contained in a complex
geodesic, according to Corollary 9. From Lemma 8 we would get that this is
actually covered by the tangential case. ∎

It follows from Claim 0 that it suffices to measure the length of $\gamma_{n}$
projected to the normal direction. Denote this normal length by $l_{N}$.

Claim 1. There exist a sequence of domains $D_{n}$, points $\zeta_{n}$,
$\nu_{n}\in D_{n}$ and real geodesics $\eta_{n}$ in $D_{n}$ joining them such
that $\zeta_{n}=\eta_{n}(t_{n})$, $\nu_{n}=\eta_{n}(-s_{n})$, and

  1. (1)

For any $\beta>-1$, the domains $D_{n}$ converge to $\mathbb{B}$ in
$\mathcal{C}^{2,\alpha}$ topology when restricted to
$\\{\operatorname{Re}z_{1}\geq\beta\\}$;

  2. (2)

$(1,0)\in\partial D_{n}$ and the defining functions of $D_{n}$ near $(1,0)$
are of the form (10);

  3. (3)

$\zeta_{n}\in(0,1)\times\\{0\\}^{N-1}$, $\zeta_{n}\to(1,0)$, and
$\nu_{n}\to(-1,0)$, and setting $\xi_{n}=\eta_{n}(0)$,
$\operatorname{Re}(\xi_{n})_{1}=0$, $\xi_{n}\to(0,x)$ as $n\to\infty$;

  4. (4)

$l_{N}(\eta_{n}|_{[0,t_{n}]})\to\infty$, as $n\to\infty$.

We shall prove Claim 1 in several steps.

The following simple observation will be used several times, so let us state
it as a separate result. Recall that $m_{t}$ is defined in (8).

######  Lemma 12.

1) Let $\delta:I\to{\mathbb{D}}$ be a curve. Let $t\in(0,1)$ be such that
$\tilde{\delta}:=m_{t}^{-1}\circ\delta$ is contained in
$\\{\operatorname{Re}\lambda\geq\beta\\}$, $\beta>-1$. Then

$|\delta(s_{1})-\delta(s_{2})|\simeq(1-t^{2})|\tilde{\delta}(s_{1})-\tilde{\delta}(s_{2})|,\quad
s_{1},s_{2}\in I.$

Consequently,

$l(\delta)\simeq(1-t^{2})l(\tilde{\delta}).$

The estimates above depend only on $\beta$.

2) Suppose additionally

(15) $\left|\tilde{\delta}(s_{2})-\tilde{\delta}(s_{1})\right|\leq
l(\tilde{\delta}|_{[s_{1},s_{2}]})\leq(1+\epsilon)\operatorname{Re}(\tilde{\delta}(s_{2})-\tilde{\delta}(s_{1})),$

where $s_{1},s_{2}\in I$. Assume also that
$|\operatorname{Im}\tilde{\delta}(s)|<\epsilon$, $s\in I$, where $\epsilon>0$
is small enough. Then

$l(\delta|_{[s_{1},s_{2}]})\leq
C_{\beta}(1+\epsilon)\operatorname{Re}(\delta(s_{2})-\delta(s_{1})).$

###### Proof.

If $\operatorname{Re}x_{1},\operatorname{Re}y_{1}\geq-\beta$, then
$|1+tx_{1}|$ and $|1+ty_{1}|$ are between $1-\beta$ and 2, so by direct
computations $|m_{t}(x_{1})-m_{t}(y_{1})|\simeq(1-t^{2})|y_{1}-x_{1}|$.

The second part is technical, as well. The first part and assumption (15)
imply that $l(\delta|_{[s_{1},s_{2}]})\leq
c_{\beta}(1+\epsilon)(1-t^{2})\operatorname{Re}(\tilde{\delta}(s_{2})-\tilde{\delta}(s_{1})).$

To get the assertion, compute

$\operatorname{Re}(\delta(s_{2})-\delta(s_{1}))=(1-t^{2})\operatorname{Re}\left(\frac{\tilde{\delta}(s_{2})-\tilde{\delta}(s_{1})}{(1+t\tilde{\delta}(s_{1}))(1+t\tilde{\delta}(s_{2}))}\right)$

Since $|\operatorname{Im}\tilde{\delta}(s)|<\epsilon$, the argument of the
denominator is bounded by $C\epsilon$, and by (15), so is the argument of the
numerator. So we get

$\operatorname{Re}(\delta(s_{2})-\delta(s_{1}))\geq(1-t^{2})(1-C\epsilon)\operatorname{Re}\left(\tilde{\delta}(s_{2})-\tilde{\delta}(s_{1})\right),$

which finishes the proof. ∎

Let us come back to the proof of Claim 1. Let $p_{n}\in\partial D$ minimize
the distance from $z_{n}$ to $\partial D$. Clearly $p_{n}\to p$. Using a
Fridman-Ma transformation, we can assume that $p_{n}=(1,0)$ and that $\partial
D$ is near $(1,0)$ of the form (10). Let $A_{t}$ be given by (8).

Choose $t_{n}$ such that
$A^{-1}_{t_{n}}(\gamma_{n})\subset\\{\operatorname{Re}z_{1}\geq 0\\}$ and
$A^{-1}_{t_{n}}(\gamma_{n})\cap\\{\operatorname{Re}z_{1}=0\\}\neq\emptyset$.

Step 1. $t_{n}\to 1$ as $n\to\infty$.

###### Proof.

This comes from the fact that a Gromov hyperbolic metric space $X$ (with
boundary) does not have geodesic loops, i.e. isometric maps $\gamma$ from
$\mathbb{R}$ such that $\lim_{t\to\pm\infty}\gamma(t)=p\in\partial_{G}X$,
where $\partial_{G}X$ is the Gromov boundary of $X$ and the convergence is in
the sense defined for that boundary. Recall that a bounded strongly
pseudoconvex domain in $\mathbb{C}^{n}$ endowed with the Kobayashi distance is
Gromov hyperbolic and its Gromov boundary coincides with the Euclidean
boundary [1].

Suppose that some subsequence, again denoted by $t_{n}$, remains bounded away
from $1$. We may restrict attention to a neighborhood of $p$ and thus assume
it is a point of global strong convexity. The geodesics $(\gamma_{n})$ pass
through $z_{n}$ and points on the surface
$A_{t_{n}}\left(\\{\operatorname{Re}z_{1}=0\\}\right)$, which are far away
from each other. Thus the visibility applies and allows us to choose
$K\subset\subset\mathbb{B}$ such that every $\gamma_{n}$ intersects it.

Choose a sequence $s_{n}\to 1$ slowly enough so that $A_{s_{n}}^{-1}(z_{n})$
and $A_{s_{n}}^{-1}(w_{n})$ still tend to $p$, so the starting and ending
points of $A_{s_{n}}^{-1}(\gamma_{n})$ are both close to $(1,0)$. But
$A_{s_{n}}^{-1}(K)$ converges to the point $(-1,0)$. Whence
$A_{s_{n}}^{-1}(\gamma_{n})$ is convergent to a nontrivial geodesic line in
$\mathbb{B}$ with both extremities tending to the same boundary point, a
contradiction with the non existence of geodesic loops. ∎

Let $\tilde{z}_{n}=A_{t_{n}}^{-1}(z_{n})$,
$\tilde{w}_{n}=A_{t_{n}}^{-1}(w_{n})$, and
$\tilde{\gamma}_{n}:=A^{-1}_{t_{n}}(\gamma_{n})$.

Step 2. The following uniform estimates hold:

$l_{N}(\gamma_{n})\simeq(1-t_{n}^{2})l_{N}(\tilde{\gamma}_{n}),$

and

$|z_{n}-w_{n}|_{z_{n}}\simeq(1-t_{n}^{2})|\tilde{z}_{n}-\tilde{w}_{n}|$

as $n\to\infty$. Consequently,

(16) $l(\tilde{\gamma}_{n})/|\tilde{z}_{n}-\tilde{w}_{n}|\simeq
l(\gamma_{n})/|z_{n}-w_{n}|\geq a_{n}.$

###### Proof.

This is a direct consequence of Lemma 12. ∎

Step 3. There is no $K\subset\subset\mathbb{B}$ such that both $\tilde{z}_{n}$
and $\tilde{w}_{n}$ lie in $K$ for any $n\in\mathbb{N}$ large enough.

###### Proof.

If there was, there would be a constant $C_{K}>1$ such that
$C_{K}^{-1}k_{D_{n}}(z;v)\leq\|v\|\leq C_{K}k_{D_{n}}(z;v)$ for $z\in K$, and
therefore $C^{\prime}_{K}>1$ such that
$(C^{\prime}_{K})^{-1}|\tilde{z}_{n}-\tilde{w}_{n}|\leq
l(\tilde{\gamma}_{n})\leq C^{\prime}_{K}|\tilde{z}_{n}-\tilde{w}_{n}|$, which
contradicts (16). ∎

Step 4. $\liminf_{n\to\infty}|\tilde{z}_{n}-\tilde{w}_{n}|>0$.

###### Proof.

If this statement were false, then (possibly passing to a subsequence)
$\tilde{z}_{n},\tilde{w}_{n}$ would converge to $(1,0)$, according to Step 3
and the fact that $\tilde{z}_{n}\in(0,1)\times\\{0\\}^{N-1}$. Since the
geodesics $\tilde{\gamma}_{n}$ touch $\\{\operatorname{Re}z_{1}=0\\}$ (which
is far from $(1,0)$), the argument from Step 1 provides us with a
contradiction. ∎

Step 5. $l_{N}(\tilde{\gamma}_{n})\to\infty$ as $n\to\infty$.

###### Proof.

A direct consequence of Steps 2 and 4. ∎

###### End of proof of Claim 1.

From Step 4 and Lemma 10 we deduce that $\tilde{\gamma}_{n}$ intersects a
fixed compact subset $K_{0}$ of $\mathbb{B}$. For each $n$, choose $x_{n}\in
K_{0}\cap\tilde{\gamma}_{n}$. Step 2 ensures that the lengths of curves from
$x_{n}$ to $\tilde{z}_{n}$ or from $x_{n}$ to $\tilde{w}_{n}$ converge to
infinity. If the first possibility holds, then by the arguments from Step 3
$\tilde{z}_{n}$ goes to the boundary, and we put
$\zeta^{\prime}_{n}=\tilde{z}_{n}$ and $\nu^{\prime}_{n}=x_{n}$.

If the second one holds, we apply Fridman-Ma construction with respect to
points $q_{n}\in\partial D_{n}$ from Lemma 6. It follows from the Fridman-Ma
construction that it preserves the fact that lengths go to infinity or not.
Then define $\zeta^{\prime}_{n}$ to be (the new) $\tilde{w}_{n}$, and
$\nu^{\prime}_{n}$ to be (the new) $x_{n}$.

Now we need to redefine $\zeta^{\prime}_{n}$ and $\nu^{\prime}_{n}$ a bit so
that $\nu_{n}$ escapes to $(-1,0)$. It will be straightforward: the
$\nu^{\prime}_{n}$ remain within a fixed compact set that will be pushed
towards $(-1,0)$ under appropriate automorphisms of the ball.

We set $\eta_{n}:=A_{s_{n}}^{-1}(\tilde{\gamma}_{n})$,
$\zeta_{n}:=A_{s_{n}}^{-1}(\zeta^{\prime}_{n})$, and
$\nu_{n}:=A_{s_{n}}^{-1}(\nu^{\prime}_{n})$ where $(s_{n})_{n}\subset(0,1)$ is
to be chosen increasing to $1$, so that $\nu_{n}$ tends to $(-1,0)$. Now let
$\xi_{n}$ be a point on $\eta_{n}\cap\\{\operatorname{Re}z_{1}=0\\}$. We can
pick $(s_{n})_{n}$ converging to $1$ slowly enough so that the lengths of the
curves $\eta_{n}$ from $\xi_{n}$ to $\zeta_{n}$ still tend to infinity.

The assertion on normal lengths is clear (compare Claim 0 and Step 2). ∎

Claim 2. Let $D_{n}$, $\eta_{n}$ be as in Claim 1 and satisfy properties (1),
(2), and (3). Then

(17)
$\frac{l_{N}(\eta_{n}|_{[0,t_{n}]})}{|\eta_{n}(0)-\eta_{n}(t_{n})|_{\eta_{n}(0)}}=O(1),$

as $n\to\infty$.

###### Proof of Claim 2.

Let $A^{-k}:=A_{1/2}^{-1}\circ\cdots\circ A_{1/2}^{-1}$ and
$m^{-k}:=m_{1/2}^{-1}\circ\cdots\circ m_{1/2}^{-1},$ where the automorphism
$A_{1/2}$ and Möbius map are defined in (8).

First note that composing with $A^{-k}$ makes domains $D_{n}$ closer to
$\mathbb{B}$ when $k\to\infty$. For the simplicity of notation write
$\delta=\eta_{n}^{1}$ (the first complex coordinate of $\eta_{n}$).

Fix $\epsilon>0$. Since $\zeta_{n}\in(0,1)\times\\{0\\}^{N-1}$, we have
$m^{-k}((\zeta_{n})_{1})>0$ for $k\leq k(n)$, and since $m^{-k}(\nu_{n})$ is
closer to $(-1,0)$ than $\nu_{n}$ for $k\geq 0$, it follows from the second
assertion of Lemma 11 applied to $m^{-k}(\eta_{n})$ that if $n$ is big enough,
then $|\operatorname{Im}(m^{-k}(\delta(s)))|<\epsilon$ for $s$ and $k\geq 0$
such that $\operatorname{Re}(m^{-k}(\delta(s)))=0$. Fix
$-1<\beta_{1}<\beta_{2}<0$ so that
$m^{-1}(i[-\epsilon,\epsilon])\subset\\{\beta_{1}<\operatorname{Re}z_{1}<\beta_{2}\\}.$

By the Denjoy-Wolff theorem $m^{-k}((\zeta_{n})_{1})\to-1$ as $k\to\infty$.
Therefore we can pick a finite number of points
$0=t_{0}<s_{1}<\ldots<s_{k}<t_{n}\leq s_{k+1}$ that satisfy the following
equalities $\operatorname{Re}m^{-1}(\delta(s_{1}))=0$,
$\operatorname{Re}m^{-2}(\delta(s_{2}))=0,\ldots,\operatorname{Re}m^{-(k+1)}(\delta(s_{k+1}))=0$.

It follows from our construction that
$\beta_{1}<\operatorname{Re}(m^{-(j+1)}(\delta(s_{j})))<\beta_{2}$ while
$\operatorname{Re}(m^{-(j+1)}(s_{j+1}))=0$. by Lemma 11, we have a bound on
the imaginary part of $m^{-(j+1)}\circ\delta$ on $[s_{j},s_{j+1}]$. So (15) is
satisfied, and Lemma 12 applied to
$\tilde{\delta}|_{[s_{1},s_{2}]}:=m^{-(j+1)}\circ\delta|_{[s_{j},s_{j+1}]}$
and $m^{-1}_{t}:=m^{-(j+1)}$ provides us with the estimate

$l(\delta_{[s_{j},s_{j+1}]})\leq
C(1+\epsilon)\operatorname{Re}(\delta(s_{j+1})-\delta(s_{j})),$

where the constant $C$ is uniform. From this, we trivially get the assertion.
∎

###### Proof of Theorem 1.

Claims 1 and 2 contradict each other. ∎

Acknowledgments. The authors wish to thank the anonymous referees for their
multiple observations which greatly improved the exposition of this paper.

## References

  * [1] Z.M. Balogh, M. Bonk, Gromov hyperbolicity and the Kobayashi metric on strictly pseudoconvex domains, Comment. Math. Helv. 75 (2000), 504–533.
  * [2] G. Bharali, A. Zimmer, Goldilocks domains, a weak notion of visibility, and applications, Adv. Math. 310 (2017), 377–425.
  * [3] F. Bracci, J.E. Fornaess, E.F. Wold, Comparison of invariant metrics and distances on strongly pseudoconvex domains and worm domains, Math. Z. 292 (2019), no. 3-4, 879–893.
  * [4] F. Bracci, A. Saracco, S. Trapani, The pluricomplex Poisson kernel for strongly pseudoconvex domains, Adv. Math. 380 (2021), paper No. 107577, 39 pp.
  * [5] E.M. Chirka, B. Coupet, A.B. Sukhov, On boundary regularity of analytic discs, Michigan Math. J. 46(2), 271–279.
  * [6] F. Deng, Q. Guan, L. Zhang, Properties of squeezing functions and global transformations of bounded domains, Trans. Amer. Math. Soc. 368 (2016), no. 4, 2679–-2696.
  * [7] K. Diederich, J.E. Fornaess, E.F. Wold, Exposing points on the boundary of a strictly pseudoconvex or a locally convexifiable domain of finite 1-type, J. Geom. Anal. 24 (2014), no. 4, 2124–2134.
  * [8] B. L. Fridman, D. Ma, On exhaustion of domains, Indiana Univ. Math. J. 44 (1995), no. 2, 385–395.
  * [9] F.W. Gehring, W.K. Hayman, An inequality in the theory of conformal mapping, J. Math. Pures Appl. 41 (1962), 353–361.
  * [10] X. Huang, Revisiting a non-degeneracy property for extremal mappings, Acta Math. Sci. 41B (2021), 1829–1838.
  * [11] Ł. Kosiński, Comparison of invariant functions and metrics, Arch. Math. (Basel) 102 (2014), 271–281.
  * [12] Ł. Kosiński, N. Nikolov, Lower estimates of the Kobayashi distance and limits of complex geodesics, 2023, Mathematische Annalen, to appear, arXiv:2208.13239.
  * [13] Ł. Kosiński, T. Warszawski, Lempert theorem for strongly linearly convex domains, Annales Polonici Mathematici 107 (2013), no. 2, 167–216.
  * [14] Ł. Kosiński, W. Zwonek, Nevanlinna-Pick problem and uniqueness of left inverses in convex domains, symmetrized bidisc and tetrablock, J. Geom. Anal. 26 (2016), no. 3, 1863–1890.
  * [15] L. Lempert, La métrique de Kobayashi et la représentation des domains sur la boule, Bull. Soc. Math. France 109 (1981), 427–474.
  * [16] L. Lempert, Intrinsic distances and holomorphic retracts, Complex analysis and applications ’81 (Varna, 1981), 341–364, Publ. House Bulgar. Acad. Sci., Sofia, 1984.
  * [17] L. Lempert, A precise result on the boundary regularity of biholomorphic mappings, Math. Z. 193 (1986), 559–579.
  * [18] L. Lempert, Erratum: ”A precise result on the boundary regularity of biholomorphic mappings” [Math. Z. 193 (1986), no. 4, 559–579; MR0867348]. Math. Z. 206 (1991), no. 3, 501–504.
  * [19] J. Liu, X. Pu, H. Wang, Bi-Hölder extensions of quasi-isometries on pseudoconvex domains of finite type in $\mathbb{C}^{2}$, arXiv:2301.06411, 2023.
  * [20] J. Liu, X. Pu, H. Wang, Q. Zhou, Bi-Hölder Extensions of Quasi-isometries on Complex Domains, J. Geom. Anal. 32, article no. 38 (2022).
  * [21] N. Nikolov, A.Y. Ökten, Strongly Goldilocks domains, quantitative visibility, and applications, arXiv:2206.08344, 2022.
  * [22] N. Nikolov, P.J. Thomas, Quantitative localization and comparison of invariant distances of domains in $\mathbb{C}^{n}$, J. Geom. Anal. 33, article no. 35 (2023). https://doi.org/10.1007/s12220-022-01086-9
  * [23] C. Pommerenke, S. Rohde, The Gehring-Hayman inequality in conformal mapping, Quasiconformal mappings and analysis (Ann Arbor, MI, 1995), 309–319, Springer, New York, 1998.