# Topological invariants and Holomorphic Mappings

R. E. Greene, K.-T. Kim and N. V. Shcherbina  (Greene) Department of
Mathematics, University of California, Los Angeles, CA 90095 U. S. A.
<EMAIL_ADDRESS>(Kim) Department of Mathematics, Pohang University of
Science and Technology, Pohang City 37673 South Korea<EMAIL_ADDRESS>(Shcherbina) Department of Mathematics, University of Wuppertal, Wuppertal
42119 Germany<EMAIL_ADDRESS>
> Abstract. Invariants for Riemann surfaces covered by the disc and for
> hyperbolic manifolds in general involving minimizing the measure of the
> image over the homotopy and homology classes of closed curves and maps of
> the $k$-sphere into the manifold are investigated. The invariants are
> monotonic under holomorphic mappings and strictly monotonic under certain
> circumstances. Applications to holomorphic maps of annular regions in
> $\mathbb{C}$ and tubular neighborhoods of compact totally real submanifolds
> in general in $\mathbb{C}^{n}$, $n\geq 2$, are given. The contractibility of
> a hyperbolic domain with contracting holomorphic mapping is explained.

###### Contents

  1. 1 Introduction
  2. 2 The $\ell_{1}$-invariant
  3. 3 Holomorphic maps of Riemann surfaces of general type
  4. 4 Non-Riemannian Kobayashi hyperbolic case
  5. 5 Holomorphic mappings of tubular domains
  6. 6 Higher homotopy invariants
  7. 7 Illustration of the $k$-homotopy invariant
  8. 8 Monotonicity of the Kobayashi metric
  9. 9 The example from Section 7 concluded
  10. 10 Tubular neighborhoods in general
  11. 11 Contraction Mapping and Homotopy

##  1\. Introduction

It is an elegant and effective technique in Riemannian geometry to consider
the minimum-length curve in each nontrivial free homotopy class of closed
curves in a compact Riemannian manifold. Such a minimum-length curve always
exists and is a smooth closed geodesic. This idea is, for example, the basic
step in proving that a compact, orientable manifold of even dimension with
everywhere positive sectional curvature is simply connected (cf. [21] p. 172,
Theorem 26). If the Riemannian manifold is not compact, then a minimal length
curve may not exist, but it remains of interest to consider the infimum of the
lengths of closed curves in the free homotopy class and also the infimum over
all nontrivial free homotopy classes as well.

The purpose of this paper is to examine this general idea in the context of
complex manifolds and also to consider the corresponding possibilities for the
situation of maps of the $k$-sphere which are homotopically nontrivial, with
length replaced by a suitable idea of $k$-dimensional measure.

For Riemann surfaces which are covered by the unit disc $\Delta$—that is, all
Riemann surfaces except
$\mathbb{C},\mathbb{C}\setminus\\{0\\},\mathbb{C}\cup\\{\infty\\}$, and
compact surfaces of genus 1—the situation becomes one of Riemannian metrics:
If $\pi\colon\Delta\to M$ is a holomorphic covering map of the Riemann
surface, then the Poincaré metric on the unit disc $\Delta$ “pushes down” to a
smooth Riemannian (Hermitian) metric on $M$, i.e., there is a unique metric on
$M$ such that $\pi$ is a local isometry. This canonical push-down metric on
such $M$’s has the property that, if $F\colon M_{1}\to M_{2}$ is a holomorphic
map, then $F$ is metric nonincreasing. This property will be exploited early
in the paper to recover various classical results on maps of annular regions
in $\mathbb{C}$ from the viewpoint of minimization of lengths of curves in
free homotopy classes.

Extension of these ideas to higher dimensional complex manifolds and to
$k$-homotopy classes, $k>1$, involves new features: The natural metric to
consider is of course the “hyperbolic metric” introduced by S. Kobayashi,
which defines a natural extension to all dimensions of the canonical metric
construction for Riemann surfaces just discussed. But a new aspect arises: the
Kobayashi metric on a hyperbolic manifold (in the sense of Kobayashi) need not
be a Riemannian metric in dimension $\geq 2$. It is, however, a Finsler metric
with an infinitesimal form known as the Kobayashi-Royden metric [23]. But for
such an infinitesimal metric, which is only upper semi-continuous and does not
satisfy the triangle inequality in general, it is necessary to exercise care
in defining $k$-dimensional volume measure to the image of a $k$-sphere in the
manifold. We shall consider primarily the details of this matter only for some
subsets of $\mathbb{C}^{n}$, $n\geq 2$ (cf. Sections 9 and 10), although in
principle greater generality would be possible with other hypotheses. This
relevant general idea is $k$-dimensional Hausdorff measures. These ideas are
applied in the final sections to obtain results for holomorphic mappings of
tubular neighborhoods of totally real $k$-spheres in $\mathbb{C}^{n}$,
analogous to the earlier results on maps of annular regions (cf. Section 3).
It is also natural to consider the tubular neighborhoods of more general
compact totally real submanifolds. For such a general case, the degree of
maps, a homology invariant is more appropriate and has been investigated.

Some Notation

  * •

$F^{\textrm{Kob}}_{U}$ : the Kobayashi-Royden metric of an open set $U$ in
$\mathbb{C}^{n}$

  * •

$\mu^{k}_{X,d}(A)$ : the $k$-dimensional Hausdorff measure of the subset $A$
in the metric space $(X,d)$

  * •

$\mu^{k}_{\textrm{Euc}}(B)$ : the $k$-dimensional Hausdorff measure of the
subset $B$ in $\mathbb{C}^{n}$ with respect to the Euclidean norm

  * •

$\mu^{k}_{\textrm{Kob},U}(C)$ : the $k$-dimensional Hausdorff measure of the
subset $C$ of an open set $U$ in $\mathbb{C}^{n}$ with respect to the
Kobayashi distance

##  2\. The $\ell_{1}$-invariant

Let $(X,\rho)$ be a metric space with the metric $\rho$. For a continuous
curve $\alpha\colon[a,b]\to X$, and a partition of the interval $[a,b]$

$P:=\\{t_{k}\colon k=0,1,\cdots,N,\textrm{ with
}a=t_{0}<t_{1}<\cdots<t_{N}=b\\}$

for some positive integer $N$, let

$s(\alpha,P):=\sum_{k=1}^{N}\rho(\alpha(t_{k-1}),\alpha(t_{k})).$

If the set $\\{s(\alpha,P)\colon P\textrm{ a partition of }[a,b]\\}$ is
bounded above, we say that $\alpha$ is rectifiable and define the length
$L(\alpha)$ of $\alpha$ by

$L(\alpha)=\sup\\{s(\alpha,P)\colon P\textrm{ a partition of }[a,b]\\}.$

As usual, reparametrizations of rectifiable curves are rectifiable and the
length is independent of parametrization.

######  Definition 2.1.

A map $F\colon X_{1}\to X_{2}$ from a metric space $(X_{1},\rho_{1})$ to
$(X_{2},\rho_{2})$ is called distance nonincreasing if
$\rho_{2}(F(x),F(y))\leq\rho_{1}(x,y)$ for all $x,y\in X_{1}$.

Notice that, if a map $F$ is distance nonincreasing, then the $F$-images of
rectifiable curves are rectifiable and the $F$-images have length $\leq$ the
length of the original curve, i.e., length $F(\gamma(t))\leq$ length
$\gamma(t)$, for all rectifiable curves $\gamma(t)$ in $X_{1}$.

######  Definition 2.2 (The $\ell_{1}$-invariant).

Consider a metric space $(X,\rho)$ which is not simply connected. Then the
$\ell_{1}$-invariant of the metric space $X$ is defined to be

$\ell_{1}(X):=\inf\\{L(\alpha)\colon\alpha~{}\textrm{a non-contractible
rectifiable loop in }X\\},~{}\inf\varnothing=+\infty.$

######  Proposition 2.1.

If $f\colon(X_{1},\rho_{1})\to(X_{2},\rho_{2})$ is a distance nonincreasing
map, and if the induced homomorphism
$f_{*}\colon\pi_{1}(X_{1})\to\pi_{1}(X_{2})$ is injective, then
$\ell_{1}(X_{2})\leq\ell_{1}(X_{1})$.

###### Proof.

For a loop $\alpha$ in $X_{1}$ denote by $[\alpha]$ the set of all loops that
are free homotopic to $\alpha$ in $X_{1}$. Set

$\ell_{1}([\alpha])=\inf\\{L(\beta)\colon\beta\in[\alpha]\\}.$

Then

$\ell_{1}(X_{1})=\inf\\{\ell_{1}([\alpha])\colon\alpha\textrm{ non-
contractible in }X_{1}\\}.$

Let $\epsilon>0$. Take a noncontractible loop $\beta$ in $X_{1}$ with
$\ell_{1}([\beta])<\ell_{1}(X_{1})+\epsilon$. Then $f\circ\beta$ is
noncontractible in $X_{2}$, since $f_{*}$ is injective. Then
$\ell_{1}([f\circ\beta])\geq\ell_{1}(X_{2})$. The distance nonincreasing
property of $f$ implies that

$\ell_{1}(X_{2})\leq\ell_{1}([f\circ\beta])\leq\ell_{1}([\beta])<\ell_{1}(X_{1})+\epsilon.$

Since this holds for any $\epsilon>0$, the assertion follows immediately. ∎

Distance nonincreasing/length nonincreasing maps arise naturally in complex
analysis. The classical Schwarz Lemma is equivalent to the fact that a
holomorphic function $f$ from the unit disc
$\Delta=\\{z\in\mathbb{C}\colon|z|<1\\}$ into itself with $f(0)=0$ has the
property that $d(0,f(z))\leq d(0,z)$, where $d=$ the Poincaré distance. Since
the action by holomorphic isometries of $\Delta$ to itself relative to the
Poincaré metric are transitive on $\Delta$, what is known as the Schwarz-Pick
Lemma follows immediately:  If $f\colon\Delta\to\Delta$ is holomorphic, then
$d(f(z_{1}),f(z_{2}))\leq d(z_{1},z_{2})$ for all $z_{1},z_{2}\in\Delta$,
where $d$ is the Poincaré distance.

This can be extended to Riemann surfaces as follows: If $M$ is a Riemann
surface that is holomorphically covered by the unit disc $\Delta$. say
$\pi\colon\Delta\to M$ is a holomorphic covering map, then the covering
transformation of the covering $\pi$ are holomorphic isometries for the
Poincaré metric. It follows that $M$ has a unique Riemannian (indeed
Hermitian) metric for which $\pi$ is locally isometric. Let us call this the
canonical metric for $M$.

If $M_{1}$ and $M_{2}$ are two such Riemann surfaces, with canonical
(Riemannian) metrics $g_{1}$ and $g_{2}$ respectively, and if $F\colon
M_{1}\to M_{2}$ is a holomorphic map, then the pull back $F^{*}g_{2}$ to
$M_{1}$ of the metric $g_{2}$, is less than or equal to $g_{1}$, i.e.,

$F^{*}g_{2}|_{p}\leq g_{1}|_{p}$

for each $p\in M_{1}$, where

$F^{*}g_{2}|_{p}(v,w)=g_{2}|_{F(p)}(dF_{p}(v),dF_{p}(w)),$

for all $v,w$ in the real tangent space of $M_{1}$ at $p$, so $dF_{p}(v)$ and
$dF_{p}(w)$ are in the real tangent space of $M_{2}$ at $F(p)$.

To see this distance nonincreasing property, note that if
$\pi_{1}\colon\Delta\to M_{1}$ and $\pi_{2}\colon\Delta\to M_{2}$ are
holomorphic covering maps, then $F\colon M_{1}\to M_{2}$ can be lifted to a
holomorphic map $\hat{F}\colon\Delta\to\Delta$ in such a way that the diagram

$\begin{CD}\Delta @>{\hat{F}}>{}>\Delta\\\ @V{\pi_{1}}V{}V@V{}V{\pi_{2}}V\\\
M_{1}@>{}>{F}>M_{2}\end{CD}$

commutes. The map $\hat{F}$ is nonincreasing for the Poincaré metric by the
Schwarz-Pick Lemma. (cf. [14])

It follows, since $\pi_{1}$ and $\pi_{2}$ are (local) isometries, in the sense
indicated of the inequality on the pull-back metric, that $F\colon M_{1}\to
M_{2}$ is length nonincreasing/distance nonincreasing for the canonical
metrics of $M_{1}$ and of $M_{2}$, respectively.

##  3\. Holomorphic maps of Riemann surfaces of general type

In the case that
$M=\\{z\in\mathbb{C}\colon\frac{1}{\sqrt{R}}<|z|<\sqrt{R}\\}$, $R>1$, which is
covered by the unit disc $\Delta$, the canonical metric on $M$ is
straightforward to compute. The function $F(z)=e^{iz}$ defines a covering map
of the “strip” $\\{x+iy\colon-\ln R<y<\ln R\\}$ onto $M$, while the strip is
biholomorphic to the upper half plane via the composition of a linear map,
exponentiation, and a linear fractional transformation. Tracing through gives
the formula

$\frac{\big{(}\pi/(2\ln R)^{2}\big{)}}{r^{2}\cos^{2}\big{(}\pi\ln r/(2\ln
R)\big{)}}dr^{2}+\frac{\big{(}\pi/(2\ln R)\big{)}^{2}}{\cos^{2}\big{(}\pi\ln
r/(2\ln R)\big{)}}d\theta^{2}$

for the canonical metric on $M$ in $(r,\theta)$ polar coordinates. (Cf. [8],
p. 39.)

The free homotopy classes of closed curves in $M$ are characterized by winding
number (around $0$) and are nontrivial for all winding numbers except $0$.
Since winding number is $\frac{1}{2\pi}$ times the total change in polar angle
$\theta$ around the curve, it follows easily that, for any rectifiable curve
$\gamma$ in $M$, the length of the curve in the canonical metric is
$\geq\pi^{2}/\ln R$. Thus the $\ell_{1}$-invariant of $M$ in its canonical
metric is $\geq\pi^{2}/\ln R$ and indeed

(1) $\ell_{1}(M)=\frac{\pi^{2}}{\ln R},$

with the infimum realized by once-around the curve $r=1$ ($\theta$ goes from
$0$ to $2\pi$), either clockwise or counterclockwise. (Throughout, we are
using the Poincaré metric on the unit disc $\Delta$ to be
$4(1-z\bar{z})^{-2}dz\ d\bar{z}$ so that the curvature $\equiv-1$.)

Since $\\{z\in\mathbb{C}\colon A<|z|<B\\}$ is linearly biholomorphic to
$\\{z\colon\sqrt{A/B}<|z|<\sqrt{B/A}\\}$, the $\ell_{1}$-invariant of
$\\{z\in\mathbb{C}\colon A<|z|<B\\}$ is equal to $\pi^{2}/\ln(B/A)$. The
$\ell_{1}$-invariant is preserved by biholomorphic mapping, so the classical
result follows:

######  Theorem 3.1 (Hadamard).

The region $\\{z\colon A_{1}<|z|<B_{1}\\}$ is biholomorphic to $\\{z\colon
A_{2}<|z|<B_{2}\\}$ if, and only if, $B_{1}/A_{1}=B_{2}/A_{2}$.

This result, originally proved by Hadamard (Cf. [17]), is usually proved by
non-metric methods, e.g., Schwarz Reflection, [9, 1, 25] et al.

The nonincreasing property of the $\ell_{1}$-invariant gives an extension of
this result:

######  Theorem 3.2.

If $0<A_{1}<B_{1}$, $0<A_{2}<B_{2}$ and $B_{1}/A_{1}>B_{2}/A_{2}$, then every
holomorphic mapping $f\colon\\{z\colon A_{1}<|z|<B_{1}\\}\to\\{z\colon
A_{2}<|z|<B_{2}\\}$ is homotopically trivial, that is, it is homotopic to a
constant map.

Proof. The map $f$ is homotopically trivial if and only if the $f$-image of
the curve $\gamma(t)=\frac{1}{2}(A_{1}+B_{1})(\cos t,\sin
t),~{}(t\in[0,2\pi])$ is homotopic to a constant curve: this is an immediate
consequence of covering space theory. If $f$ is not homotopically trivial,
then the free homotopy class of the $f$-image indicated is nontrivial. Then
Proposition 2.1 gives that

$\ell_{1}(\\{z\colon A_{2}<|z|<B_{2}\\})\leq\ell_{1}(\\{z\colon
A_{1}<|z|<B_{1}\\}),$

and the result follows from the formula (1) for the $\ell_{1}$-invariant.
$\Box$

This Theorem 3.2 implies in particular the historical result:

######  Theorem 3.3 (de Possel [3]).

There is a 1-1 holomorphic mapping $g\colon\\{z\colon
0<A_{1}<|z|<B_{1}\\}\to\\{z\colon 0<A_{2}<|z|<B_{2}\\}$ whose image separates
the boundary components of $\\{z\colon A_{2}<|z|<B_{2}\\}$ if and only if
$B_{1}/A_{1}\leq B_{2}/A_{2}$.

Notice that the condition on $g$ implies that $g$ is injective on homotopy;
For instance the $g$-image of the curve $\gamma$ in the preceding proof will
be a closed curve in $\\{z\colon A_{2}<|z|<B_{2}\\}$ that goes around the
origin exactly once. Notice also that the “if” part of this theorem is
obviously true by a complex linear map. (Cf. [6, 11] for the original proof.)

A conclusion by the same argument holds for multiply connected domains and
Riemann surfaces as well.

######  Theorem 3.4.

Let $X$ and $Y$ be Riemann surfaces holomorphically covered by the unit disc
with $\ell_{1}(X)<\ell_{1}(Y)$. If $f\colon X\to Y$ is a holomorphic map, then
it cannot be injective on homotopy. In particular, $f$ cannot be a homotopy
equivalence.

##  4\. Non-Riemannian Kobayashi hyperbolic case

The only properties of the “canonical metric” on Riemann surfaces covered by
the disc that were crucial here were that holomorphic maps were distance
nonincreasing and that the canonical metric was locally comparable (in both
directions) with any metric derived from local coordinates. In this context,
it is clear that the whole viewpoint has an immediate extension to complex
manifolds which are “hyperbolic” in the sense introduced by S. Kobayashi. In
this section, we consider only complex manifolds (or complex spaces) that are
hyperbolic in the sense of Kobayashi, namely, those for which the Kobayashi
pseudodistance is an actual distance function. [15, 16].

In this setting, one can define again the $\ell_{1}$-invariant $\ell_{1}(M)$
of a hyperbolic manifold to be the infimum of the lengths of closed curves
that are not freely homotopic to 0 (assuming $M$ is not simply connected).
Then following the pattern of before one gets the results:

######  Theorem 4.1.

If $f\colon M_{1}\to M_{2}$ is a holomorphic map of hyperbolic manifolds, and
if $f$ is injective on free homotopy classes of closed curves (in the sense
that if $\gamma(t)$ is a closed curve in $M_{1}$ not freely homotopic to a
constant, then $f(\gamma(t))$ is not freely homotopic to a constant), then

$\ell_{1}(M_{2})\leq\ell_{1}(M_{1}).$

Thus the results discussed in the concrete instances of Riemann surfaces
covered by the unit disc and of annular regions in $\mathbb{C}$ in particular,
can be extended to far more general settings. The idea of the
$\ell_{1}$-invariant via free homotopy classes of closed curves can also be
extended to higher dimensional homotopy classes of maps of the $k$-sphere into
complex hyperbolic manifolds, and to some extent, into general length spaces.
These methods will be explored in subsequent sections.

##  5\. Holomorphic mappings of tubular domains

The previous discussion of annular regions in $\mathbb{C}$ has a
straightforward extension to tubular domains in $\mathbb{C}^{n}$, $n>1$. For
this, define, for $0<r<1$,

$T^{n}(r)=\\{\vec{z}=(z_{1},\ldots,z_{n})\in\mathbb{C}^{n}\colon|z_{1}|=1,\|\vec{z}-(z_{1},0,\ldots,0)\|<r\\},$

where $\|\cdot\|$ is the Euclidean norm. Set
$A(r_{1},r_{2})=\\{z\in\mathbb{C}\colon r_{1}<|z|<r_{2}\\}$ where
$0<r_{1}<r_{2}$. The projection map $P_{n}(z_{1},\ldots,z_{n})=z_{1}$ takes
$T^{n}(r)$ onto $A(1-r,1+r)$ and there is a natural injection
$J_{n}(z)=(z,0,\ldots,0)$ mapping $A(1-r,1+r)$ into $T^{n}(r)$. These maps are
homotopy equivalences: $J_{n}$ and $P_{n}$ are homotopy inverses to each
other. This observation together with Theorem 3.2 gives rise to a comparison
result on these tubular domains (meaning tubular neighborhoods of a circle,
where the dimensions need not be equal):

######  Theorem 5.1.

If $f\colon T^{n}(r)\to T^{m}(s)$ with $r,s$ between $0$ and $1$, is
holomorphic and if $s<r$, then $f$ is homotopic to a constant map.

Proof. The map $f$ is homotopic to a constant map if and only if the $f$-image
of the closed curve $\Gamma(t)=(e^{it},0,\ldots,0),~{}t\in[0,2\pi]$ in
$T^{n}(r)$ is homotopic to a constant curve in $T^{m}(s)$. This is again by
the standard covering space theory. But, this happens if and only if the
holomorphic map $F\colon A(1-r,1+r)\to A(1-s,1+s)$ defined by $F:=P_{m}\circ
f\circ J_{n}$ is homotopic to a constant. This last follows from the homotopy
equivalences already noted. Now Theorem 3.2 implies that, if $s<r$, this map
$F$ is, and consequently $f$ is also, homotopic to a constant. $\Box$

##  6\. Higher homotopy invariants

It is natural to consider extending the previous introduced ideas about curve
lengths in free homotopy classes of closed curves to higher dimensional
homotopy classes. In particular, if $(X,d)$ is a metric space (which we assume
arc-wise connected for convenience), then consider continuous maps $f\colon
S^{k}\to X$, $k>1$, $S^{k}=\\{(x_{0},\cdots,x_{k})\in\mathbb{R}^{k+1}\colon
x_{0}^{2}+\ldots+x_{k}^{2}=1\\}$ and define two such $f_{0},f_{1}\colon
S^{k}\to X$ to be free-homotopic if there is a continuous function $H\colon
S^{k}\times[0,1]\to X$ such that, for all $p$, $H(p,0)=f_{0}(p)$ and
$H(p,1)=f_{1}(p)$. If one has a situation where, for suitably restricted $f$,
one can define the measure of (the image of) such $f\colon S^{k}\to X$, then
one could imitate the definition of the $\ell_{1}$-invariant corresponding to
the $k=1$ case. But certain difficulties arise: One needs an idea of such a
measure and one would want to know that each free homotopy class of continuous
maps $S^{k}\to X$ would contain at least one map $f$ with the associated
measure of the image of $f$ being finite. This would correspond to rectifiable
curves and their lengths, and free-homotopy classes of curves, as in earlier
sections.

In the case where $(X,d)$ is a Riemannian manifold and $d$ its Riemannian
metric distance, the right idea of measure is, for smooth maps $f\colon
S^{k}\to X$, the measure of $f|_{A}$, $A\subset S^{k}$, $A=$ the set of points
of $S^{k}$ where the differential $df$ has rank $k$, and the measure of
$f|_{A}$ is the usual Riemann-metric induced measure on $k$-dimensional
submanifolds. This measure is

$\mu(f)=\int_{S^{k}}|\textsl{volume form}|$

where the $|\textsl{volume form}|$ on $(v_{1},\ldots,v_{k})$, where
$v_{1},\ldots,v_{k}\in T_{p}S^{k}$, is equal to the absolute value of the
$k$-dimensional Riemannian volume at $f(p)$ of $f_{*}(v_{1})\wedge\cdots\wedge
f_{*}(v_{k})$. It is easy to show (and well-known) that every free homotopy
class of continuous maps $S^{k}\to X$ in this case ($X$ is a Riemannian
manifold) contains a smooth (not necessarily everywhere nonsingular) map: If
$f$ is a continuous map in the free homotopy class then a sufficiently good
smooth approximation of $f$ will be in the same free homotopy class (by
deformation along minimal geodesics). So every free homotopy class contains
finite $k$-measure maps and hence one can define the infimum of measures over
finite-measure (smooth, e.g.) maps in the class and also an $\ell_{k}$
invariant (= infimum over all nontrivial free homotopy classes).

In the case of more general metric spaces, it is not in general clear that any
useful concept of $k$-dimensional measure exists. But in the case of metrics
which are locally equivalent to Riemannian metrics, the familiar general idea
of Hausdorff $k$-dimensional measure, can be used.

######  Definition 6.1 (Cf. [16], p. 343).

Given a metric space $(X,d)$, for a nonnegative real number $k$ and a subset
$A$, the $k$-dimensional Hausdorff measure $\mu^{k}_{X,d}(A)$ is defined as

$\mu^{k}_{X,d}(A)=\sup_{\epsilon>0}\inf\Big{\\{}\sum_{i=1}^{\infty}\big{(}\delta(A_{i})\big{)}^{k}\colon
A=\bigcup_{i=1}^{\infty}A_{i},\delta(A_{i})<\epsilon\Big{\\}},$

where $\delta(A_{i})$ is the diameter of $A_{i}$ defined by

$\delta(A_{i})=\sup_{p,q\in A_{i}}d(p,q).$

For a smooth map $f\colon S^{k}\to M$ of the $k$-sphere $S^{k}$ into a
Riemannian manifold $M$, this $\mu^{k}_{M}(f(S^{k}))$ gives essentially the
same concept of the $k$-dimensional measure as the Riemannian measure already
defined. But this new notion of measure has the advantage that smoothness is
not involved.

Rather than exploring these matters further in generality, we restrict our
attention now to the specific situation of the Kobayashi metric for a bounded
connected open set $U$ in $\mathbb{C}^{n}$. The open set $U$ has its Kobayashi
metric in the infinitesimal form, the Kobayashi-Royden metric, as it is
usually defined by

$F^{\textrm{Kob}}_{U}(p,v)=\inf\Big{\\{}r>0\colon
f\in\mathcal{O}(\Delta,U),f(0)=p,df(0)=\frac{v}{r}\Big{\\}}$

where $\Delta=\\{z\in\mathbb{C}\colon|z|<1\\}$ and where
$\mathcal{O}(\Delta,U)$ denotes the set of holomorphic maps from $\Delta$ to
$U$. As is well-known [24, 23], the Kobayashi distance from $p$ to $q$ for
$p,q\in U$ is the infimum of the length of the $C^{1}$ curves from $p$ to $q$
in $U$ where the length is taken to be the integral
$\int_{\gamma}F^{Kob}_{U}(z,dz)$. (The existence of this integral was also
shown in [Op. cit.]). Moreover, it was shown in [2] that $F^{Kob}_{U}(p,v)$ is
locally comparable to the standard Euclidean norm of $v$ in $\mathbb{C}^{n}$.
This leads easily to the fact that for every free homotopy class of maps
$S^{k}\to U$, there is a smooth map $f\colon S^{k}\to U$ (as already noted)
and the smooth map will have finite Hausdorff $k$-measure relative to the
Kobayashi metric, denoted by $\mu^{k}_{\textrm{Kob},U}(f(S^{k}))$. We call
such a map $k$-rectifiable (meaning: Hausdorff $k$-measure is finite), and the
$k$-measure just constructed will be called, in this paper, the Hausdorff-
Kobayashi $k$-measure.

Thus, one can define an invariant

$\displaystyle\ell_{k}(U)=$ infimum of the Hausdorff-Kobayashi $k$-measures
over all the Hausdorff-Kobayashi $k$-rectifiable representatives of the
nontrivial free homotopy classes in $U$

######  Lemma 6.1.

If $F\colon U\to V$ is a holomorphic mapping from a bounded domain in
$\mathbb{C}^{n}$ to a bounded domain in $\mathbb{C}^{m}$ and if it is
injective on free homotopy in the sense that the $F$-image of every nontrivial
$k$-class in $U$ is nontrivial in $V$, then $\ell_{k}(V)\leq\ell_{k}(U)$.

Proof. This follows immediately from the fact that $F$ is distance
nonincreasing and hence diameter nonincreasing with respect to the Kobayashi
metrics of $U$ and $V$, so the Hausdorff sums are not increased by composition
with $F$. $\Box$

##  7\. Illustration of the $k$-homotopy invariant

The most natural way to find a bounded connected open set in $\mathbb{C}^{n}$
which has a nontrivial $k$-homotopy class is to take a tubular neighborhood of
an embedded $k$-sphere. This is particularly interesting from the viewpoint of
complex analysis if one takes the $k$-sphere to be totally real as a
submanifold. In particular, the unit $k$-sphere in $\mathbb{R}^{k+1}$ is
totally real in $\mathbb{C}^{k+1}$. Let $T_{r}$ be the radius $r>0$ tubular
neighborhood

$\displaystyle T_{r}=\\{(x_{0},\ldots,x_{k})+\lambda\vec{u}\colon$
$\displaystyle x_{0}^{2}+\ldots+x_{k}^{2}=1,$
$\displaystyle\vec{u}\in\mathbb{C}^{k+1},~{}\|u\|=1,~{}0\leq\lambda<r\\}.$

For small values of $r$, this is a strongly pseudoconvex domain with
$C^{\infty}$ boundary (actually $0<r<\frac{1}{2}$ suffices). According to [7],
the Kobayashi-Royden metric goes to $+\infty$ near the boundary relative to
the Euclidean metric. Note that there is a natural smooth projection, say
$P\colon T_{r}\to S^{k}$, defined by $P(z)=$ the closest point to $z$ in the
Euclidean sense in the set $S^{k}$. The injection $S_{k}\hookrightarrow T_{r}$
is a homotopy equivalence with $P$ its homotopy inverse. And the identity
(injection) map $S_{k}\to T_{r}$ is homotopically nontrivial. There may be
other smooth maps, say $\Gamma\colon S^{k}\to T_{r}$, homotopic to the
injection which have the associated Hausdorff-Kobayashi $k$-measure assigned
to them being less than the measure assigned to the injection. But the infimum
will be realized among the family of maps with image lying in some set of the
form

$\\{\vec{x}+\lambda\vec{u}\colon\vec{x}\in
S_{k},\vec{u}\in\mathbb{C}^{k+1},~{}\|u\|=1,~{}0\leq\lambda\leq r_{1}\\}$

for some $r_{1}<r$: this follows easily from the estimates on the growth of
the Kobayashi metric near the boundary of $T_{r}$ compared to the Euclidean
metric. This will yield a conclusion similar to Theorem 3.1, once a technical
point about the Kobayashi/Royden infinitesimal metric is established. This
will be discussed further in the following sections.

##  8\. Monotonicity of the Kobayashi metric

To find some analogue in this situation of Theorems 3.1, one needs to have an
idea of strict monotonicity of the Kobayashi distance for the domains (open
connected subsets) in $\mathbb{C}^{n}$. Specifically, if two domains
$\Omega_{1}$ and $\Omega_{2}$ satisfy $\Omega_{1}\subset\Omega_{2}$, then of
course their respective Kobayashi-Royden metrics
$F^{\textrm{Kob}}_{\Omega_{1}},F^{\textrm{Kob}}_{\Omega_{2}}$ satisfy
$F^{\textrm{Kob}}_{\Omega_{2}}(p,v)\leq F^{\textrm{Kob}}_{\Omega_{1}}(p,v)$
for all $p\in\Omega_{1}$ and $v\in\mathbb{C}^{n}$. But we would like to obtain
a strict comparison between them.

######  Lemma 8.1.

If $\Omega_{1}$ and $\Omega_{2}$ are bounded domains in $\mathbb{C}^{n}$
satisfying $\Omega_{1}\subset\subset\Omega_{2}$ (relatively compact), then for
any compact subset $K$ of $\Omega_{1}$ there exists a constant $c_{K}$ with
$0<c_{K}<1$ such that

$F^{\textrm{Kob}}_{\Omega_{2}}(p,v)<c_{K}F^{\textrm{Kob}}_{\Omega_{1}}(p,v),$

for any $p\in K$ and any $v\in\mathbb{C}^{n}$.

Proof. Let the bounded domains $\Omega_{1}$ and $\Omega_{2}$ in
$\mathbb{C}^{n}$ satisfy $\Omega_{1}\subset\subset\Omega_{2}$, i.e.,
$\Omega_{1}$ is a relatively compact subdomain of $\Omega_{2}$.

Then there exists $\delta>0$ such that the Euclidean distance between
$\Omega_{1}$ and $\mathbb{C}^{n}\setminus\Omega_{2}$ is at least $\delta$.

Let $p\in\Omega_{1}$. Notice that there exist positive numbers $b$ and $B$
such that

$b\leq F^{\textrm{Kob}}_{\Omega_{1}}(p,v)\leq B,~{}\forall
v\in\mathbb{C}^{n}\textrm{ with }\|v\|=1,$

where $\|~{}\|$ denotes the standard Euclidean norm. Fix $p\in\Omega_{1}$ and
$v\in\mathbb{C}^{n}$ with $\|v\|=1$. Let $\epsilon>0$ be given arbitrarily,
and then choose $h\in\mathcal{O}(\Delta,\Omega_{1})$ satisfying $h(0)=p$,
$h^{\prime}(0)=v/r$ for some $r>0$ and

$F^{\textrm{Kob}}_{\Omega_{1}}(p,v)\leq
r<F^{\textrm{Kob}}_{\Omega_{1}}(p,v)+\epsilon.$

Take

$\tilde{h}(z)=h(z)+\delta zv.$

Then

$\tilde{h}(0)=h(0)=p,~{}\tilde{h}^{\prime}(0)=h^{\prime}(0)+\delta
v=\Big{(}\frac{1}{r}+\delta\Big{)}v$

and

$\tilde{h}(z)\subset\Omega_{2},\ \forall z\in\Delta,$

since $\|\tilde{h}(z)-h(z)\|=\|\delta zv\|<\delta$ for every $z\in\Delta$. So
$\tilde{h}\in\mathcal{O}(\Delta,\Omega_{2})$.

This implies

$\displaystyle F^{\textrm{Kob}}_{\Omega_{2}}(p,v)$
$\displaystyle\leq\frac{1}{1/r+\delta}$ $\displaystyle=\frac{r}{1+r\delta}$
$\displaystyle\leq\frac{F^{\textrm{Kob}}_{\Omega_{1}}(p,v)+\epsilon}{1+\delta
F^{\textrm{Kob}}_{\Omega_{1}}(p,v)}$ $\displaystyle\leq\frac{1}{1+\delta
b}(F^{\textrm{Kob}}_{\Omega_{1}}(p,v)+\epsilon).$

Since $\epsilon>0$ is arbitrary, we obtain that

$F^{\textrm{Kob}}_{\Omega_{2}}(p,v)\leq\frac{1}{1+\delta
b}F^{\textrm{Kob}}_{\Omega_{1}}(p,v)$

for any $v\in\mathbb{C}^{n}$, due to the homogeneity of the Kobayashi-Royden
metric.

Note that $b$ depends on the location of $p$. But on a compact set it stays
bounded away from zero, and the desired conclusion follows immediately. $\Box$

The restriction in Lemma 8.1 to a compact set $K$ can be removed.

######  Lemma 8.2.

If $U$ is a bounded open set in $\mathbb{C}^{n}$ with its closure contained in
another bounded open set $V$, then there is a constant $c\in(0,1)$ such that

$F^{\textrm{Kob}}_{V}(p,v)\leq cF^{\textrm{Kob}}_{U}(p,v)\quad\forall(p,v)\in
U\times\mathbb{C}^{n},$

where $F^{\textrm{Kob}}_{U}$ and $F^{\textrm{Kob}}_{V}$ are the infinitesimal
Kobayashi-Royden metrics of $U$ and $V$, respectively.

Proof. Denote by $\textrm{cl}(A)$ the closure in $\mathbb{C}^{n}$ of the
subset $A$ of $\mathbb{C}^{n}$. There is a bounded open set $W$ satisfying

$\textrm{cl}(U)\subset W\subset\textrm{cl}(W)\subset V.$

With $W$ so chosen, we have:

$F^{\textrm{Kob}}_{W}(q,v)\leq F^{\textrm{Kob}}_{U}(q,v)\quad\forall(q,v)\in
U\times\mathbb{C}^{n},$

and

$F^{\textrm{Kob}}_{V}(p,v)\leq F^{\textrm{Kob}}_{W}(p,v)\quad\forall(p,v)\in
W\times\mathbb{C}^{n}.$

Lemma 8.1 gives that there is a constant $c$ with $0<c<1$ such that

$F^{\textrm{Kob}}_{V}(q,v)\leq
cF^{\textrm{Kob}}_{W}(q,v)\quad\forall(q,v)\in\textrm{cl}(U)\times\mathbb{C}^{n}.$

In particular, this yields that

$F^{\textrm{Kob}}_{V}(p,v)\leq cF^{\textrm{Kob}}_{W}(p,v)\leq
cF^{\textrm{Kob}}_{U}(p,v)\quad\forall(p,v)\in U\times\mathbb{C}^{n},$

as desired. $\Box$

##  9\. The example from Section 7 concluded

The results of Section 6 and Section 7 together with the growth of the
Kobayashi-Royden metric established in [7] can be combined to establish the
results about the domains $T_{r}$ defined in Section 6:

First we note that the $\ell_{k}$-invariants are nonzero in this case. As
before, we assume that all $r$-values are small enough that $T_{r}$ is
strongly pseudoconvex.

######  Lemma 9.1.

With $T_{r}$ as defined in Section 6, and $\ell_{k}$ defined as before,

$\ell_{k}(T_{r})>0.$

Proof. Since $T_{r}$ has smooth strictly pseudoconvex boundary, Graham [7]
gives that there is a compact subset $K$ of $T_{r}$ such that for each $p\in
T_{r}\setminus K$ and all $v$,

$F^{\textrm{Kob}}_{T_{r}}(p,v)\geq\|v\|,$

where $\|~{}\|$ represents the usual Euclidean norm. For such a fixed compact
set $K$, there is a constant $c>0$ such that

$F^{\textrm{Kob}}_{T_{r}}(p,v)\geq c\|v\|$

for all $p\in K$ and all $v\in\mathbb{C}^{n}$.

Replace $c$ by $\min\\{c,1\\}$. It follows that $\ell_{k}(T_{r})\geq c^{k}\
L(T_{r})$, where $L(T_{r})$ represents the infimum of the Euclidean Hausdorff
$k$-measure of the image of $S^{k}$ in $T_{r}$ not homotopic to a constant.
This latter infimum is positive by elementary considerations. $\Box$

######  Theorem 9.1.

If $0<r<s$, then $\ell_{k}(T_{r})>\ell_{k}(T_{s})$. In particular, $T_{r}$ is
not biholomorphic to $T_{s}$.

Proof. Consider the inclusion map $T_{r}\hookrightarrow T_{s}$. By Graham [7],
given any $C>0$, there is a compact subset $K$ of $T_{r}$ such that
$F^{\textrm{Kob}}_{T_{r}}\geq C\cdot\textrm{(Euclidean metric)}$ at every
point $p\in T_{r}\setminus K$, since $F^{\textrm{Kob}}_{T_{r}}$ goes to
infinity at the boundary of $T_{r}$ and hence, choosing $C$ sufficiently
large, $F^{\textrm{Kob}}_{T_{r}}\geq 2F^{\textrm{Kob}}_{T_{s}}$. This follows
since $F^{\textrm{Kob}}_{T_{s}}$ is bounded by some multiple of Euclidean
metric on $T_{r}$ since $T_{r}\subset\subset T_{s}$. By Lemma 8.1 (and its
proof) there is an $\epsilon$ with $0<\epsilon<1$ such that
$F^{\textrm{Kob}}_{T_{r}}\geq(1+\epsilon)F^{\textrm{Kob}}_{T_{s}}$ at every
point of $K$. Then

$F^{\textrm{Kob}}_{T_{r}}\geq(1+\epsilon)F^{\textrm{Kob}}_{T_{s}}$

at every point of $T_{r}$. It follows that
$\ell_{k}(T_{r})\geq(1+\epsilon)^{k}\ell_{k}(T_{s})$ and, since
$\ell_{k}(T_{s})>0$,

$\ell_{k}(T_{r})>\ell_{k}(T_{s}).$

This completes the proof. $\Box$

The arguments used to prove Theorem 3.2 can be extended in a straightforward
way to prove a corresponding result for the domains of $T_{r}$ type:

######  Theorem 9.2.

If $r_{1}>r_{2}$, then every holomorphic mapping $f\colon T_{r_{1}}\to
T_{r_{2}}$ is homotopic to a constant map.

##  10\. Tubular neighborhoods in general

The analysis of tubular neighborhoods of totally real embeddings of spheres in
the previous section can be extended to more general circumstances. But this
extension involves what amounts to a shift from homotopy to homology: the role
of being homotopically nontrivial is taken over by having degree not equal to
$0$.

The first step is to define the relevant concept of degree: Suppose that $M$
is a smooth ($C^{2}$ suffices) compact connected submanifold without boundary
of a Euclidean space $\mathbb{R}^{N}$ and let

$T_{r}=\bigcup_{w\in M}\\{v\in\mathbb{R}^{N}\colon\|v-w\|<r\\},$

where $\|~{}\|$ is the usual Euclidean norm. Notice that there exists $R>0$
such that $T_{r}$, for any $r$ with $0<r<R$, there exists a natural projection
$\pi\colon T_{r}\to M$ onto $M$, defined by

$\pi(v)=\inf_{p\in M}\|p-v\|$

so that $\pi(v)$ is the closest point to $v$ among points of $M$, with respect
to the Euclidean distance. If $g\colon M\to T_{r}$ is a continuous map of $M$
into $T_{r}$, then it is natural to define the degree of $g$, denoted by $\deg
g$, to be the degree of $\pi\circ g\colon M\to M$. (If $M$ is orientable, this
is to be the usual $\mathbb{Z}$-valued degree. If $M$ is nonorientable, we
take degree to be $\mathbb{Z}_{2}$-valued.) If $G\colon T_{r_{1}}\to
T_{r_{2}}$ is a continuous map, with $r_{1},r_{2}\in(0,R)$, we define

$\deg G:=\textrm{the degree of }G\circ i\colon M\to T_{r_{2}},$

where $i\colon M\to T_{r_{1}}$ is an injection. As well known, these concepts
of degree are multiplicative: the degree of a composition is equal to the
product of the degrees.

With these definitions in sight, the analogue of Theorem 9.2 is the following:

######  Theorem 10.1.

Let $M$ be a smooth compact connected totally real submanifold of
$\mathbb{C}^{n}$. Then there is a constant $R>0$ such that, if $0<r<s<R$ and
if $F\colon T_{s}\to T_{r}$ is holomorphic then, $F$ has degree zero.

The conditions on $R$ here are such that

  * •

$T_{r}$ has smooth strongly pseudoconvex boundary and

  * •

the projection map $\pi\colon T_{r}\to M$ is well defined (and continuous in
particular),

whenever $0<r<R$.

Note that this result is in fact an extension of Theorem 9.2 since, in the
case that $M$ is a sphere, $F$ being homotopic to a constant is equivalent to
$\deg F=0$. But in general, of course, $\deg F=0$ does not imply that $F$ is
homotopic to a constant. The most obvious example may be the map of
$S^{p}\times S^{p}$ to itself, $p\geq 1$, identity on the first factor and
constant on the second.

Proof of Theorem 10.1 follows the general pattern of the proofs of previous
theorems, but requires some preparation.

First, we need

######  Definition 10.1.

Let $k=\dim M$. For $r\in(0,R)$ as above, let

$\displaystyle V_{r}^{k}:=\inf\\{\mu^{k}_{\textrm{Kob},T_{r}}(g(M))\mid
g\colon$ $\displaystyle M\to T_{r}\textrm{ continuous}$
$\displaystyle\textrm{and rectifiable with }\deg g\neq 0\\}.$

Now we need

######  Lemma 10.1.

$V^{k}_{r}>0$ for any $r\in(0,R)$.

Proof; With $g\colon M\to T_{r}$, $\deg g\neq 0$, the composition
$\hat{g}:=\pi\circ g\colon M\to M$ has nonzero degree and is hence surjective.

The projection $\pi\colon T_{R}\to M$ admits a constant $C>0$ such that
$\|\pi(x)-\pi(y)\|\leq C\|x-y\|$ with respect to the standard Euclidean norm.
Thus the Euclidean Hausdorff $k$-volume of $\pi\circ g(M)$ is at least as
large as that of $M$, bounded away from $0$.

On the other hand, for $0<r<R$, $T_{r}$ is bounded strongly pseudoconvex. So,
by [7], the Kobayashi-Royden metric $F^{\textrm{Kob}}_{T_{r}}$ of $T_{r}$ goes
to infinity compared to the Euclidean metric as the base point approaches the
boundary of $T_{r}$. Hence there is a constant $c>0$ such that
$F^{\textrm{Kob}}_{T_{r}}(q,w)\geq c\|w\|$ for any $q\in
T_{r},w\in\mathbb{C}^{n}$. Hence we obtain that

$\mu^{k}_{\textrm{Kob},T_{r}}(g(M))\geq c^{k}\mu^{k}_{Euc}(g(M)).$

Since

$\mu^{k}_{Euc}(g(M))\geq C^{-k}\mu^{k}_{Euc}(\pi(g(M)))\geq
C^{-k}\mu^{k}_{Euc}(M),$

it follows that
$\inf_{g}\mu_{\textrm{Kob},T_{r}}^{k}(g(M))\geq(c/C)^{k}\mu^{k}_{Euc}(M)>0$,
as desired. $\Box$

######  Lemma 10.2.

$V^{k}_{s}<V^{k}_{r}$ if $0<r<s<R$.

Proof: By Lemma 8.2, the strict inclusion of the closure of $T_{r}$ into
$T_{s}$ implies that there is a constant $c$ with $0<c<1$ such that
$F^{\textrm{Kob}}_{T_{s}}(z,v)\leq cF^{\textrm{Kob}}_{T_{r}}(z,v)$ for any
$z\in T_{r}$ and $v\in\mathbb{C}^{n}$. Hence for each $g\colon M\to T_{r}$ of
nonzero degree,

$\mu^{k}_{\textrm{Kob},T_{s}}(g(M))\leq
c^{k}\mu^{k}_{\textrm{Kob},T_{r}}(g(M)).$

So $V^{k}_{s}\leq c^{k}V^{k}_{r}$. Since $V_{s}>0$ by Lemma 10.1, it follows
that $V^{k}_{s}<V^{k}_{r}$. $\Box$.

Proof of Theorem 10.1. Let $0<r<s<R$ as in the hypothesis. Suppose that
$F\colon T_{s}\to T_{r}$ is holomorphic. And suppose, for a proof by
contradiction that $\deg F\neq 0$. If a continuous map $g\colon M\to T_{s}$
has a nonzero degree, then

$\deg(F\circ g)=(\deg F)(\deg g)\neq 0.$

Thus

$V^{k}_{r}\leq\mu^{k}_{\textit{Kob},T_{r}}(F\circ g(M)),$

since $F\circ g\colon M\to T_{r}$ is a continuous map with nonzero degree. But
$\mu^{k}_{\textit{Kob},T_{r}}(F\circ
g(M))\leq\mu^{k}_{\textit{Kob},T_{s}}(g(M))$, since $F$ is Kobayashi-metric
nonincreasing. Thus

$V^{k}_{r}\leq\mu^{k}_{\textit{Kob},T_{s}}(g(M)),$

which implies that

$V^{k}_{r}\leq V^{k}_{s}.$

This contradicts Lemma 10.2, and the proof is complete. $\Box$

##  11\. Contraction Mapping and Homotopy

The situation of a map $f\colon U\to V$ with $V\subset U$ with $f$ distance
nonincreasing for some metric $d_{U}$ on $U$ is a natural condition for
considering the ideas associated to iterations of contraction mappings. It is
a familiar and long-standing principle of analysis that in the case where $f$
is distance nonincreasing by a factor $0\leq\alpha<1$, then $f$ must have a
fixed point if $U$ is complete with respect to the metric $d_{U}$. This is a
natural way to prove the existence of a short-term solution of ordinary
differential equations, the local surjectivity in the Inverse Function Theorem
via Newton’s Method, and many other basic results (Cf. e.g. [5]). Such
contraction mapping ideas were used in [4] to prove a fixed point theorem for
holomorphic maps in Banach spaces, now known as the Earle-Hamilton Fixed Point
Theorem. The finite dimensional version was proved earlier in [22]. In both
papers, the Carathéodory metric rather than the Kobayashi metric was used. We
point out also that the finite dimensional version was proved even earlier in
[13] p. 83 (p. 92, in the 2nd ed.), using the fact that any compact analytic
set in the complex Euclidean space consists of finitely many points. On the
other hand, the following theorem shows that the Kobayashi metric can also be
used in the contraction mapping context in a way similar to [22] and [4].

######  Theorem 11.1.

If $U$ is a bounded domain in $\mathbb{C}^{n}$ and $f\colon U\to U$ is a
holomorphic mapping such that $f(U)$ is contained in a compact subset of $U$,
then there is a unique point $z_{0}\in U$ such that $f(z_{0})=z_{0}$.
Moreover, this $z_{0}$ is exactly the only point such that the iterates
$f^{n}(z)=f\circ\ldots\circ f(z)$, $n$-times, of $f$ converge uniformly on $U$
to the constant map at $z_{0}$ on $U$.

Proof. Choose an open set $V$ in $\mathbb{C}^{n}$ with its compact closure
$\overline{V}$ in $\mathbb{C}^{n}$ satisfying

$f(U)\subset V\subset\overline{V}\subset U.$

Let $d_{U}$ be the Kobayashi distance on $U$. Lemma 8.2 shows that there
exists $0\leq c<1$ such that

$F^{\textrm{Kob}}_{U}(f(p),df|_{p}(v))\leq c\cdot
F^{\textrm{Kob}}_{V}(f(p),df|_{p}(v)),~{}\forall(p,v)\in
U\times\mathbb{C}^{n},$

which in turn implies that

$\displaystyle
F^{\textrm{Kob}}_{U}(f\circ\gamma(t)),(f\circ\gamma)^{\prime}(t))$
$\displaystyle\leq c\cdot
F^{\textrm{Kob}}_{V}(f\circ\gamma(t)),(f\circ\gamma)^{\prime}(t))$
$\displaystyle\leq c\cdot F^{\textrm{Kob}}_{U}(\gamma(t),\gamma^{\prime}(t))$

for any $C^{1}$ curve $\gamma\colon[0,1]\to U$. So we have

$F^{\textrm{Kob}}_{U}(f\circ\gamma(t)),(f\circ\gamma)^{\prime}(t))\leq c\cdot
F^{\textrm{Kob}}_{U}(\gamma(t),\gamma^{\prime}(t))$

for any $C^{1}$ curve $\gamma\colon[0,1]\to U$, and consequently

$d_{U}(f(p),f(q))\leq c\cdot d_{U}(p,q),~{}\forall p,q\in U.$

Namely, $f$ is a strict contraction with the contraction factor $c$.

Now denote by $f^{n}$ the $n$-th iterate of $f$ defined inductively by

$f^{1}=f,~{}f^{n+1}=f\circ f^{n}\quad(n=1,2,\cdots).$

If $p\in U$, then it follows that the sequence $f^{n}(p)$ is a Cauchy sequence
with respect to $d_{U}$. Even if $U$ were not necessarily $d_{U}$-complete,
this Cauchy sequence still converges because, for $n\geq 1$, $f^{n}(p)$
belongs to $\overline{V}$, a compact set with $\overline{V}\subset U$. Thus
$z_{0}:=\lim\limits_{n\to\infty}f^{n}(p)$ exists in $\overline{V}$, and
consequently in $U$. And $f(z_{0})=z_{0}$. Of course this is the unique fixed
point: if $f(z_{1})=z_{1}$ then

$d_{U}(z_{0},z_{1})=d_{U}(f(z_{0}),f(z_{1}))\leq c\cdot d_{U}(z_{0},z_{1}),$

which implies $d_{U}(z_{0},z_{1})=0$ and $z_{0}=z_{1}$.

The uniform convergence of $f^{n}$ on $U$ follows by

$\displaystyle d_{U}(z_{0},f^{k+1}(p))$
$\displaystyle=d_{U}(f^{k+1}(z_{0}),f^{k+1}(p))$ $\displaystyle\leq
c^{k}d_{U}(f(z_{0}),f(p))$ $\displaystyle\leq c^{k}d_{U}(z_{0},f(p)),$

which implies

$\sup_{p\in U}d_{U}(z_{0},f^{k+1}(p))\leq c^{k}\sup_{f(p)\in
V}d_{U}(z_{0},f(p)).$

Note that $\sup_{f(p)\in V}d_{U}(z_{0},f^{m}(p))$ is bounded, since the
Kobayashi distance is a continuous function by [2]. Moreover, there exists a
positive integer $m$ such that for some constant $C>0$ it holds that

$\frac{1}{C}\ d_{U}(z_{0},f^{n}(p))\leq\|z_{0}-f^{n}(p)\|\leq C\
d_{U}(z_{0},f^{n}(p))$

for any $n>m$. Altogether, we see that the convergence of $f^{k}$ to the
constant function at $z_{0}$ is uniform on $U$, $\Box$

This recovers Theorem 9.2: With the notation therein, if $f\colon T_{r_{1}}\to
T_{r_{2}}$ with $r_{1}>r_{2}$ is holomorphic then, since $T_{r_{2}}$ is
relatively compact in $T_{r_{1}}$, there is a point $z_{0}\in T_{r_{1}}$ with
$f(z_{0})=z_{0}$. Consider $f^{n}$ with $n$ large. Then as mentioned before

$\deg(f^{n})=\big{(}\deg f\big{)}^{n}.$

But $f^{n}(T_{r_{1}})$ is contained in a small ball around $z_{0}$. So the
composition $P_{r_{2}}\circ f^{n}\circ i$ must have degree $0$, and the rest
of necessary arguments follows immediately.

A variant of this idea also produces this result about connected open subsets
in $\mathbb{C}^{n}$ and their images under holomorphic maps to themselves
where the image is contained in a relatively compact subset.

######  Theorem 11.2.

If $U$ is a bounded connected open set with smooth boundary in
$\mathbb{C}^{n}$, and if there is a holomorphic map $f\colon U\to U$ such that
$f(U)$ is contained in a compact subset of $U$ and that, for some $z\in U$ and
for all $k=1,2,\ldots$, the induced map
$f_{*}\colon\pi_{k}(U,z)\to\pi_{k}(U,f(z))$ is an isomorphism, then $U$ is
contractible.

The theorem is illustrated by a bounded open sets $U$ with smooth boundary
that are star-shaped around the origin with $f(z)=rz$ for a constant $r$ with
$0<r<1$.

The proof involves two lemmas:

######  Lemma 11.1.

Every bounded open set $U$ in $\mathbb{R}^{N}$ with smooth boundary has the
homotopy type of a finite CW complex.

######  Lemma 11.2.

With $U$ and $f$ as in the hypotheses of Theorem 11.2, and with $z_{0}$ the
fixed point of $f$ (which has already been shown to exist), the $k$-th
homotopy group $\pi_{k}(U)=0$ for any $k=1,2,\cdots$.

The proof of Theorem 11.2 follows from these two lemmas and the Whitehead
theorem ([26]. See also [12], Theorem 4.5 in page 346), since $f$ and the
constant map $z_{0}$ have the same action on the homotopy groups of $U$ at
$z_{0}$. [The general result of Whitehead is that, if both $X$ and $Y$ have
the homotopy type of connected finite CW complexes and if two continuous maps
$f,g\colon X\to Y$, where $f$ is a homotopy equivalence, satisfy the
properties that $f(x_{0})=g(x_{0})$ for some $x_{0}\in X$ and
$g_{*},f_{*}:\pi_{k}(X,x_{0})\to\pi_{k}(Y,f(x_{0}))$ are identical for every
$k=1,2,\cdots$, then $f$ is homotopic to $g$.] In our case, $f_{*}=0$ and $g$
is to be the constant map at $z_{0}$, so both $f_{*}$ and $g_{*}$ coincide as
the trivial map of $\pi_{k}$ for every $k$. $\Box$

######  Remark 11.1.

The open set $U$ is homotopically equivalent to a finite CW complex, but not
homeomorphic to. Note for instance that any finite CW complex is compact. On
the other hand, the fact that $U$ is of the same homotopy type with a finite
CW complex is sufficient for the preceding proof.

Proof of Lemma 11.1: Let $d(z)=\textrm{dist }(z,\mathbb{C}\setminus U)$, where
$\textrm{dist }(z,A)=\inf\\{\|z-w\|\colon w\in A\\}$. Here of course $\|~{}\|$
is the Euclidean norm of $\mathbb{C}^{n}$. Since $U$ is open, $d(z)>0$ for
every $z\in U$ and the map $z\to d(z)$ is Lipschitz continous. Also, since $U$
has smooth boundary, $d(z)$ is of class $C^{1}$ at least ($C^{\infty}$ if $U$
has $C^{\infty}$ boundary) on $\mathcal{N}_{\epsilon}:=\\{z\in U\colon
d(z)<\epsilon\\}$ for sufficiently small a constant $\epsilon>0$, and
$\|\textsl{grad }d\|=1$ at every $z\in U$ with $d(z)<\epsilon$, where
$\textsl{grad }d$ is the real Euclidean gradient.

Now set $\delta(z)=-\ln d(z)$. Then $\delta$ is smooth on
$\mathcal{N}_{\epsilon}$, and $\|\textsl{grad }\delta(z)\|\geq 1/\epsilon$ for
every $z\in\mathcal{N}_{\epsilon}$. (In fact, the inequality is $=$, but we do
not need it here.) Now, for notational convenience, let

$K_{1}=\\{z\in U\colon d(z)\geq\epsilon\\},\quad K_{m}=\\{z\in U\colon
d(z)\geq\epsilon/m\\}$

for $m=1,2,\cdots$. (We shall need only the first few of these). The sets
$K_{m}$ defined as such are compact in $\mathbb{C}^{n}$.

Now $d$ may not be smooth on $U$; indeed, $d$ cannot be smooth since $d$ has a
maximum in $U$ but $\|\textsl{grad }d\|=1$ at every point at which $d$ is of
class $C^{1}$. So $\delta$ is also nonsmooth at some points. However, by
standard convolution smoothing arguments, there is a smooth function, say
$\Delta$, such that $\Delta$ is uniformly close to $\delta$ on $K_{4}$ but
(uniformly) $C^{1}$-close to $\delta$ on $K_{4}\setminus K_{1}$. By usual
Morse theoretic considerations ([19]. Cf. Section I.6 of [18]. See also [10].)
there is a function $\Delta_{1}$ which is uniformly $C^{1}$-close to $\Delta$
on $K_{3}$ and has only nondegenerate critical points. Since $\Delta$ and
$\delta$ are $C^{1}$-close on $K_{4}\setminus K_{1}$ and $\|\textsl{grad
}\delta\|\geq 1/\epsilon$ on $K_{4}\setminus K_{1}$, it follows by the usual
partition of unity argument that $\Delta$ on the interior of $K_{3}$, and
$\delta$ on $U$ can be patched together to yield a function $\Delta_{2}$, say,
such that

$\Delta_{2}(z)=\Delta(z),\forall z\in K_{3}$

with $\|\textsl{grad }\Delta_{2}\|\geq 1/(2\epsilon)$ on $U\setminus K_{1}$,
and such that

$\Delta_{2}(z)=\delta(z),\forall z\in U\setminus K_{1}.$

Then $\Delta_{2}$ is an exhaustion function for $U$ with only nondegenerate
critical points, since $\Delta_{2}$ has no critical points in $U\setminus
K_{1}$ and hence only nondegenerate critical points in $U$, which necessarily
lie in the set where $\Delta=\Delta_{1}$.

The standard Morse theory gives now that $U$ has the homotopy type of a finite
CW complex, with cells given by the finite number of nondegenerate critical
points of $\Delta_{2}$. $\Box$

Proof of Lemma 11.2: Let $z_{0}$ be the fixed point of $f$. Suppose that
$\Gamma\colon S^{k}\to U$ is a representation of a $k$-homotopy class in
$\pi_{k}(U,z_{0})$. By Theorem 11.1, the iterates $f^{n}\circ\Gamma$ converge
uniformly on $S^{k}$ to the constant map at $z_{0}$. In particular, if a
positive constant $r$ is such that
$B^{n}(z_{0},r)=\\{z\colon\|z-z_{0}\|<r\\}\subset U$ and
$f^{n}\Gamma(S^{k})\subset B^{n}(z_{0},r)$, then $[f^{n}\circ\Gamma]=0$ in
$\pi_{k}(U,z_{0})$. But by hypothesis,
$f^{n}_{*}\colon\pi_{k}(U,z_{0})\to\pi_{k}(U,z_{0})$ is an isomorphism. So
$[\Gamma]=0$ in $\pi_{k}(U,z_{0})$. $\Box$

## Acknowledgements

After this article was written and posted in arXiv.org, T. Pacini kindly
informed us the relevance of [20]. We express our gratitude for this.

Research of the second named author (Kim) is partially supported by the NRF
Grant 4.0021348 of The Republic of Korea.

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