Rescaling methods in complex analysis
François Berteloot
Maı̂trise de Mathématiques de l’Université Paul Sabatier
Second semestre 2014
1
2
Contents
1 Basics in one complex variable
1.1 Power series, the exponential function and the number π . .
1.1.1 Functions defined as sums of power series . . . . . . .
1.1.2 The exponential function . . . . . . . . . . . . . . . .
1.2 Holomorphic, conformal and quasi-conformal transformations
1.2.1 Linear maps of the complex plane . . . . . . . . . . .
1.2.2 Holomorphic v.s. C 1 -maps . . . . . . . . . . . . . . .
1.3 The Cauchy formula and its consequences . . . . . . . . . .
1.3.1 The Cauchy-Pompeiu formula . . . . . . . . . . . . .
1.3.2 Basic properties of holomorphic functions . . . . . . .
1.3.3 Remarks on one-to-one holomorphic maps . . . . . .
1.4 Conformal equivalence of domains . . . . . . . . . . . . . . .
1.4.1 The Riemann mapping theorem . . . . . . . . . . . .
1.4.2 The Caratheodory theorem . . . . . . . . . . . . . .
1.5 The Riemann sphere . . . . . . . . . . . . . . . . . . . . . .
1.5.1 C,
I! S 2 and P1 . . . . . . . . . . . . . . . . . . . . . .
1.5.2 Spherical derivatives . . . . . . . . . . . . . . . . . .
2 The Zalcman rescaling method for discs in the
2.1 Normal families . . . . . . . . . . . . . . . . . .
2.1.1 Ascoli’s theorem . . . . . . . . . . . . .
2.1.2 Holomorphic normal families . . . . . . .
2.2 The Bloch principle . . . . . . . . . . . . . . . .
2.2.1 The Zalcman renormalization lemma . .
2.2.2 The Bloch-Zalcman principle . . . . . . .
2.3 The Picard-Montel theory . . . . . . . . . . . .
2.3.1 Linearization of entire maps . . . . . . .
2.3.2 Picard’s and Montel’s theorems . . . . .
2.4 Starting Fatou-Julia theory . . . . . . . . . . .
2.4.1 Fatou and Julia sets . . . . . . . . . . .
2.4.2 The density of repelling cycles . . . . . .
2.5 The Ahlfors five islands theorem . . . . . . . . .
2.5.1 A theorem of Nevanlinna . . . . . . . . .
3
Riemann
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2.5.2
Ahlfors five discs theorem . . . . . . . . . . . . . . . . . . . . 57
4
Chapter 1
Basics in one complex variable
1.1
1.1.1
Power series, the exponential function and
the number π
Functions defined as sums of power series
To every sequence of complex numbers (an )n∈IN and any complex number z one may
associate a series of the form
"
n
n≥0 an z .
Such series are called power series and may define functions which generalize the
polynomials. The convergence of such a series obviously depends on the choice of
z. It turns out that the region of the complex plane C
I where a given power series is
converging is quite easy to describe.
Theorem 1.1.1 For any power series
0 such that:
"
n≥0 an z
n
there exists a number +∞ ≥ R ≥
1) the series converge absolutely for |z| < R and uniformly on every disc ∆ρ for
ρ < R,
"
2) when |z| > R then n≥0 |an z n | + ∞.
The number R is called the convergence radius of the series and is given by Hadamard’s
formula:
1
R
#
$1
= lim sup |an | n .
Proof. Let ρ# < ρ < R where R is given by Hadamard’s formula and is supposed to
#
$1
1
be strictly positive. Then ρ1 > lim sup |an | n and therefore |an | n < ρ1 for n ≥ n0 .
For |z| < ρ# , we thus have:
5
"
"
#n
n≥n0 |an |ρ ≤
n
n≥n0 |an z | ≤
"
n≥n0
# ρ! $n
ρ
< +∞.
This proves proves 1).
Let ρ > R where R is given by Hadamard’s formula and is supposed to be finite.
#
$1
Then ρ1 < lim sup |an | n and therefore there exists a subsequence (ani )i such that
1
|ani | ni >
1
ρ
for all i ∈ IN. For |z| > ρ we thus have
"
n≥0 |an z
n
|≥
This proves 2).
"
i≥0 |ani ||z|
ni
≥
"
i≥0
# |z| $ni
ρ
= +∞.
Exercise 1.1.2 What is the convergence radius of the series
&
%
"
zn
n≥0 n! ?
"
n
Exercise 1.1.3 Let f (z) :=
n≥0 an z where the power series has infinite conradius. Show that f is constant if it is bounded on C.
I Hint: compute
%vergence
2π
iθ −inθ
f
(re
)e
dθ
for
all
n
∈
IN
and
all
r
>
0.
0
Exercise 1.1.4 Show that the convergence radius of the series
"∞
n=k
does not depend on k.
n(n − 1) · · · (n − k + 1)an z n−k
Our aim now is to investigate the differentiability of a function defined as the sum
of a power series. Let us start by the very simple case of the monomial p(z) := z n .
For every z0 , h ∈ C
I we have
p(z0 + h) − p(z0 ) = (z0 + h)n − z0n =
"n
k=1
Cnk hk z0n−k = nz0n−1 + h$(h)
where limh→0 $(h) = 0. This means that p is differentiable at z0 and that its differential Dp(z0 ) is a C-linear
I
map given by the formula
#
$
Dp(z0 ) · h = nz0n−1 h.
We will now extend this computation to the case of a power series.
"
Proposition 1.1.5 Let f (z) := n≥0 an z n where the power series has positive convergence radius R. Then f is of class C 1 on ∆R and for any z0# ∈
R the differential
$
"∆
∞
n−1
Df (z0 ) of f at z0 is the C-linear
h.
I
map given by: Df (z0 ) · h =
n=1 nan z0
Proof. We have
6
f (z0 + h) − f (z0 ) = lim
lim
N →+∞
N
'
an
n=1
n
'
Cnk hk z0n−k = lim
N →+∞
k=1
)
∞
'
&
N →+∞
h
N
'
& N
'
n=0
n
an (z0 + h) −
nan z0n−1 +
n=1
nan z0n−1
n=1
*
N
'
an
n=2
h + lim
N →+∞
)
N
'
n=2
N
'
an (z0 )
n=0
n
'
(
=
(
Cnk hk z0n−k =
k=2
an
n
n
'
Cnk hk z0n−k
k=2
*
"
"
n−1
= ∞
na z n−1 comes from exercise
where the equality limN →+∞ N
n=1 nan z0
# "N n=1 "nn 0 k k n−k $
1.1.4. This show that the limit limN →+∞
exists. To
n=2 an
k=2 Cn h z0
compute it, we shall invert the order of sommation. This will be possible because
this double series is actually absolutely converging for |h| < $ when |z0 | + $ < R.
Indeed, for such h we have:
∞
'
n=2
|an |
n
'
k=2
Cnk |h|k |z0 |n−k
≤
∞
'
n=2
∞
#
$n '
#
$n
|an | |h| + |z0 | ≤
|an | |z0 | + $ < +∞.
n=2
Permuting the order of sommation we get
limN →+∞
# "N
n=2 an
"n
k=2
$ "
#"
$
"
n−k
k
k
k
Cnk hk z0n−k = ∞
=: ∞
n≥k Cn an z0
k=2 h
k=2 bk h .
Then, for |h| < R − |z0 |, we have
$
# "∞
"
n−1
k−2
h + h2 ∞
f (z0 + h) − f (z0 ) =
n=1 nan z0
k=2 bk h
&
%
and the proof is finished.
1.1.2
The exponential function
"
n
As we have already seen (exercise 1.1.2) the series n≥0 zn! defines a function on
C.
I This function is called the exponential function and is, in some sense, the most
important function in mathematics:
exp(z) :=
∞
'
zn
n=0
n!
.
According to proposition 1.1.5, the differential of exp is given by
"∞ z (n−1) "∞ z n
n=1 (n−1)! =
n=0 n! and thus:
Dexp(z0 ) · h = exp(z0 ) · h ;
7
∀z0 ∈ C,
I ∀h ∈ C.
I
"∞
n=1
nz
(n−1)
n!
=
The key property of exp is to be a morphism between the groups (IC, +) and
(IC , ·), in other words, for any complex numbers a and b we have:
∗
exp(a + b) = exp(a) · exp(b).
This follows directly from the Cauchy product formula for two series and the
binomial formula:
∞
∞
∞
∞
n
'
ak ' bp ' ' ak bp ' ' ak bn−p
·
=
=
=
k! p=0 p! n=0
k!p!
k!(n
−
p)!
n=0
k=0
k+p=n
∞
n
' '
n=0
1
n!
k=0
Cnk ak bn−p =
k=0
∞
'
(a + b)n
n=0
n!
.
Let us observe that this implies that exp does not vanish on C.
I Indeed, 1 =
exp(0) = exp(z − z) = exp(z) · exp(−z) for every z ∈ C.
I
As exp(z + w) = exp(z) · exp(w) one customary replace the notation exp(z) by
the shorter expression ez where e is the positive real number defined by e := exp(1).
Proposition 1.1.6 The exponential function restricted to the real and imaginary
axis satisfies the following properties:
1) the function IR ) x *→ ex is positive increasing, moreover limx→+∞ ex = +∞
and limx→−∞ ex = 0.
2) ∀t ∈ IR : |eit | = 1.
Proof. 1) That ex is positive, increasing on IR+ and tends to +∞ at +∞ is clear
since the power series defining exp has positive coefficients. The other assertions
then follow from e−x = e1x .
" 1 n
2)Observe that exp(z) =
z̄ = exp(z̄). Then, we have in particular
n!
|eit |2 = eit · eit = eit · e−it = e0 = 1.
&
%
Our goal now is to define the number π and establish the following fundamental
formula:
π
ei 2 = i.
To this purpose we shall first define the trigonometric functions cos and sin.
8
Definition 1.1.7 The trigonometric functions are defined by:
cos t := Re eit ,
sin t := Im eit ; ∀t ∈ IR
or equivalently by the so called Euler’s identity:
eit = cos t + i sin t.
Let us observe that
cos2 t + sin2 t = |eit | = 1.
Differentiating Euler’s identity one gets cos# t + i sin# t = Dexp(it) ·
iexp(it) = i cos t − sin t and thus:
d(it)
dt
=
cos# = −sin and sin# = cos.
#
$
2
4
6
8
The function cos is real analytic: cos t = 12 eit + e−it = 1 − t2! + t4! − t6! + t8! + · · ·.
#
4$
6
8
4
one sees
As cos 0 = 1 and cos 2 = 1 − 2!2 + 24! − 26! + 28! + · · · ≤ 1 − 2!2 + 24! = −1
3
that cos vanishes somewhere between 0 and 2. This leads to the definition of π:
π := 2 Inf{t ≥ 0/cos t = 0}.
π
We are now ready to prove the fundamental formula. As ei 2 = cos π2 + i sin π2 =
i sin π2 it suffices to show that sin π2 = 1. Since sin# = cos is positive on [0, π2 ] and
sin 0 = 0 one sees that sin is positive on [0, π2 ]. In particular sin π2 ≥ 0 and, as
sin2 π2 = 1 − cos2 π2 = 1, we have sin π2 = 1.
Exercise 1.1.8 Show that ez = 1 if and only if z = 2ikπ for some k ∈ ZZ.
We end this subsection with the following important property of the exponential
map.
# $
Proposition 1.1.9 exp C
I =C
I∗
Proof. We have to solve the equation ez = w for w ,= 0. As ez−iν0 = ez e−iν0 the
existence of a solution is not affected if we multiply w by eiν0 . Multiplying w by
π
3π
(−1), i or −i (that is eiπ , ei 2 or ei 2 ) we may assume that Re w and Im w are
both positive. It follows immediately from the first assertion of proposition 1.1.6
that |w| = ex for some x ∈ IR+ . Let us now write w = |w|w0 = ex |w0 | where
|w0 | = 1. It remains to show that w0 = eiθ0 for some θ0 ∈ IR. Since Re w0 and
Im w0 are positive we have 0 ≤ Re w0 , Im w0 ≤ 1 and thus Re w0 = cos θ0 for
some 0 ≤ θ0 ≤ π2 (remember that cos 0 = 1 and cos π2 = 0). As we know that
√
√
the function sin is positive on [0, π2 ], we get sin θ0 = sin2 θ0 = 1 − cos2 θ0 =
+
+
1 − (Re w0 )2 = (Im w0 )2 = Im w0 . We have shown that w0 = eiθ0 .
&
%
9
1.2
Holomorphic, conformal and quasi-conformal
transformations
1.2.1
Linear maps of the complex plane
The complex plane C
I is both a field and a two-dimensional real vector space which
is identified with IR2 via the decomposition of a complex number z in its real and
imaginary parts: z = x + iy. Linear transformations on C
I are thus first understood
2
as IR-linear transformations on the underlying IR but, among such maps, some
might be C-linear.
I
The important, and extremely elementary fact, is that every
linear transformation of the complex plane is decomposed as the sum of a C-linear
I
and a C-antilinear
I
one.
Definition 1.2.1 A linear transformation T : C
I →C
I is said to be C-linear
I
(resp.
C-antilinear)
I
if and only if T (uz) = uT (z) (resp. T (uz) = ūT (z)) for every u ∈ C
I
and every ∈ C.
I
The C-linear
I
transformations of C
I are of the form: z *→ T (z) = α.z for some fixed
α ∈ C.
I Indeed T (z) = T (z.1) = zT (1) = zα where α = T (1). The same argument
shows that the C-antilinear
I
transformations are of the form T (z) = αz̄. A C-linear
I
transformation T (z) = αz is the composition of a rotation of angle Arg α with a
dilation of rate |α|:
T =D◦R=R◦D
where R(z) = eiArg α z and D(z) = |α|z.
Proposition 1.2.2 Every IR-linear transformation of the complex plane C
I may be
written
T (z) = az + bz̄
where a, b are complex numbers given by
a=
(α0 +δ0 )+i(γ0 −β0 )
2
and b =
(α0 −δ0 )+i(γ0 +β0 )
.
2
Moreover
||T || = |a| + |b| and det T = |a|2 − |b|2 .
Proof. • Let z := x + iy. As T is IR-linear, we have:
T (z) = T (x, y) = (α0 x + β0 y, γ0x + δ0 y)
10
where α0 , β0 , γ0 and δ0 are real numbers.
Using x = 12 (z + z̄) and y = 2i1 (z − z̄) we get
T (z) = (α0 x + β0 y) + i(γ0 x + δ0 y) = (α0 + iγ0 )
where a =
(α0 +δ0 )+i(γ0 −β0 )
2
z + z̄
z − z̄
+ (β0 + iδ0 )
= az + bz̄
2
2i
and b =
(α0 −δ0 )+i(γ0 +β0 )
.
2
• Let us now set a := |a|eiα e b := |b|eiβ and compute T (u) for u := reiθ .
#
$ α+β
α−β
β−α
T (u) = au + bū = (|a|eiα eiθ + |b|eiβ e−iθ )r = |a|ei( 2 +θ) + |b|ei( 2 −θ) rei 2 =
#
$ α+β
α−β
α−β
= (|a| + |b|)cos(
+ θ) + i(|a| − |b|)sin(
+ θ) rei 2 .
2
2
This shows that
Sup θ (|a| +
||T ||2 = Sup |u|=1 |T (u)|2 =
+ θ) + (|a| − |b|)2 sin2 ( α−β
+ θ) = (|a| + |b|)2 .
2
|b|)2 cos2 ( α−β
2
Finally, to compute the determinant, we may write T as
T = R α+β ◦ S ◦ R α−β
2
2
where R α+β and R α−β are rotations while S is the linear map defined by S(z) :=
2
2
|a|z + |b|z̄ or equivalently
#
$
S(|u|eiθ ) := (|a| + |b|)cos(θ) + i(|a| − |b|)sin(θ) |u|.
It then comes: det T = det S = |a|2 − |b|2 .
&
%
Let us assume that |a| > |b|. The geometrical meaning of the above computation
is that the pre-image T −1 (S 1 ) of the unit circle S 1 := {z ∈ C;
I / |z| = 1} is the ellipse
of equation
,
,
1 = |T (u)| = r ,(|a| + |b|)cos( α−β
+ θ) + i(|a| − |b|)sin( α−β
+ θ),.
2
2
Yet another direct consequence of what we just have seen is the following characterization of C
I linear maps.
Definition 1.2.3 A linear map on the complex plane is C-linear
I
(resp. C-antilinear)
I
if and only if it maps circles centered at the origin to circles centered at the origin
and preserves the orientation (resp. reverse the orientation).
11
1
|a|−|b|
1
|a|+|b|
1
β−α
2
0
T z = az + bz
Figure 1.1: The ellipse T −1 (S 1 ).
1.2.2
Holomorphic v.s. C 1 -maps
Let Ω be an open subset of C
I and f : Ω → C
I be a C 1 map. Let us write:
f = u + iv
,
where u and v are real valued. To say that f is C 1 means that the functions ∂u
∂x
are well defined and continuous on the set Ω seen as an open subset of IR2 .
∂u ∂v ∂v
, ,
∂y ∂x ∂y
Let us now look for a representation of the differential Df (z0 ) which takes into
account the complex structure. As a linear transformation of IR2 , the map Df (z0 )
is given by the jacobian matrix:

∂u
(z )
 ∂x 0
Df (z0 ) =  ∂v
(z0 )
∂x

∂u
(z0 )

∂y

∂v
(z0 )
∂y
As we saw in Proposition 1.2.2 , Df (z0 ) can be written
Df (z0 ) : z *→ az + bz̄
where
$
# ∂v
$
#
∂v
∂u
(z
)
+
(z
)
+
i
(z
)
−
(z
)
2a = ∂u
0
0
0
0
∂y
∂x
∂y
$
# ∂v
$
# ∂x
∂v
∂u
(z
)
−
(z
)
+
i
(z
)
+
(z
)
.
2b = ∂u
0
0
0
0
∂x
∂y
∂x
∂y
12
Since f = u + iv, these expressions become:
#
$
(z0 ) − i ∂f
(z0 )
a = 12 ∂f
∂x
∂y
#
$
∂f
(z
)
+
i
(z
)
.
b = 12 ∂f
0
0
∂x
∂y
This computation justify the introduction of the following very important notations:
Definition 1.2.4 For every f : Ω → C
I of class C 1 one sets:
#
$
∂f
∂f
1 ∂f
:=
−
i
∂z
2# ∂x
∂y$
∂f
∂f
1 ∂f
.
:=
+
i
∂ z̄
2 ∂x
∂y
Using these notations we get:
Proposition 1.2.5 Let f : Ω → C
I be a C 1 map which is defined on some open
subset Ω of C.
I Then the differential Df (z0 ) of f is given by
Df (z0 ) : u *→
∂f
(z ).u
∂z 0
+
∂f
(z )ū.
∂ z̄ 0
for any z0 ∈ Ω.
Exercise 1.2.6 Check that
∂ f¯
∂z
=
∂f
.
∂ z̄
We are now ready to define holomorphic mappings. Let us stress that for us an
holomorphic map will be a special kind of C 1 map defined on an open subset of the
complex plane with values in the complex plane.
Definition 1.2.7 Let f : Ω → C
I be a C 1 map which is defined on some open subset
Ω of C.
I The map f is said to be holomorphic if and only if its differential Df (z0 )
is C-linear
I
for every z0 ∈ Ω.
The class of holomorphic maps from Ω to C
I will be denoted O(Ω, C)
I or sometimes
O(Ω).
Remark 1.2.8 One usually speaks of holomorphic function, we prefer here to use
the word map to emphasize geometrical aspects. Nevertheless, we shall sometimes
use the word function in order to underline that the target space is the field C.
I
The geometrical meaning of this definition is clear: f is holomorphic if and only
if the differentials Df (z0 ) either vanish or preserve both the orientation and the set
of circles centered at the origin.
Our previous analysis also yields the following important characterization:
13
Proposition 1.2.9 (The Cauchy-Riemann conditions) Let f : Ω → C
I be a C 1
map. Let us set f = u + iv that is u = Re f and v = Im f . Then f is holomorphic
on Ω if and only if f satisfies one of the following equivalent conditions:
1)
∂f
∂ z̄
= 0 on Ω
2)
∂f
∂x
= −i ∂f
on Ω
∂y
3)
∂u
∂x
=
∂v
∂y
and
∂v
∂x
= − ∂u
on Ω.
∂y
Proposition 1.2.10 If f : Ω → IR is holomorphic then f is locally constant.
&
%
Proof. By the Cauchy-Riemann conditions the differential of f vanishes.
Proposition 1.2.11 For every holomorphic map f : Ω → C
I and every z0 ∈ Ω one
has
∂f
(z )
∂z 0
(z0 )
= limh→0 f (z0 +h)−f
.
h
Proof. As f is of class C 1 we have
f (z0 + h) − f (z0 ) = Df (z0 ).h + |h|$(h)
where limh→0 $(h) = 0. By holomorphicity Df (z0 ).h =
f (z0 +h)−f (z0 )
h
=
∂f
(z )
∂z 0
+
∂f
(z )h
∂z 0
and therefore
|h|
$(h).
h
&
%
The above proposition leads to the following definition:
Definition 1.2.12 For every f : Ω → C
I which is holomorphic, one sets
∂f
(z )
∂z 0
=: f # (z0 )
and says that f # is the complex derivative of f .
Exercise 1.2.13 Using the Cauchy-Riemann conditions, show that
1
z
∈ O(IC ∗ , C).
I
(z)
and check that f # (z) =
Compute the derivative: f # (z) = limh→0 f (z+h)−f
h
Holomorphic functions enjoy very strong stability properties.
Proposition 1.2.14 (Stability)
• O(Ω, C)
I is an algebra.
14
−1
.
z2
• If f ∈ O(Ω1 , C)
I an g ∈ O(Ω2 , C)
I with f (Ω1 ) ⊂ Ω2 then g ◦ f ∈ O(Ω1 , C)
I and
#
#
#
(g ◦ f ) = (g ◦ f )f .
# $#
!
I and f1 = −f
.
• If f ∈ O(Ω, C)
I and f ,= 0 on Ω then f1 ∈ O(Ω, C)
f2
Exercise 1.2.15 Prove proposition 1.2.14 using differential calculus and exercise
1.2.13.
We are now able to give the first examples of holomorphic maps.
Proposition 1.2.16 Every rational function
P (z)
Q(z)
where P and Q are polynomials
# P $# P ! Q−Q!P
.
is holomorphic on {z ∈ C/Q(z)
I
,= 0} and satisfies Q =
Q
I and satisfies p# (z) = nz n−1 :
Proof. It suffices to check that z n is holomorphic on C
this has been done just before proposition 1.1.5. The remaining follows from proposition 1.2.14.
&
%
Proposition 1.1.5 immediately implies that:
"
n
Proposition 1.2.17 Let "
n≥0 an z a power series with convergence radius R > 0.
Then the function f (z) := n≥0 an z n is holomorphic on ∆R and satisfies f # (z) =
"
n−1
.
n≥1 nan z
Observing that the inverse of a C-linear
I
map is again C-linear,
I
one sees that
the inverse of any holomorphic diffeomorphism is holomorphic. This leads to the
following definition.
Definition 1.2.18 A map f : Ω1 → Ω2 between two open subsets of the complex
plane is called a biholomorphism if it is diffeomorphic and holomorphic.
With the same observation, one immediately deduces an holomorphic version of
the inverse mapping theorem from the classical one.
Proposition 1.2.19 Let f : Ω → C
I be holomorphic and such that f # (z0 ) ,= 0 at
some point z0 ∈ Ω. Then f induces a biholomorphism from a neighbourhood U0 of
z0 to some neighbourhood V0 of f (z0 ).
Using the above proposition and the exercise 1.1.8, it is not very hard to establish
the following important property of exp:
Proposition 1.2.20 — The map exp : C
I →C
I ∗ is an holomorphic cover, that is:
for every w0 ∈ C
I ∗ there exists an open neighbourhood V0 of w0 and a collection of
disjoint open sets Uk := U0 + 2ikπ; k ∈ ZZ such that exp induces a biholomorphism
between Uk and V0 for every k ∈ ZZ and exp−1 (V0 ) = ∪k∈ZZ Uk .
Proposition 1.2.21 If f ∈ O(Ω) has constant modulus then f is locally constant.
15
Proof. Write f = aeiu(z) where a > 0 and u(z) ∈ IR. If a = 0 then f = 0. Otherwise,
using the fact that exp is locally invertible, one sees that u is holomorphic and the
conclusion follows from Proposition 1.2.10.
&
%
The geometric interpretation of holomorphicity leads to call conformal maps
the biholomorphisms. It is actually extremely fruitfull to consider a larger class of
transformations.
Definition 1.2.22 Let f : Ω1 → Ω2 be a C 1 -diffeomorphism between two open
subsets of C.
I Let K ≥ 1. The map f is said to be K-quasiconformal if and only if
, #
,
, ∂f
$,
, (z0 ), ≤ K−1 , ∂f (z0 ),
∂ z̄
K+1
∂z
for every z0 ∈ Ω1 .
Let us consider a K-quasiconformal map f on an open subset Ω1 of C.
I If K = 1
then f is nothing but a conformal map, that is a diffeomorphism whose differentials
both preserve the orientation and the set of circles centered at the origin.
In order to understand what is going on when K > 1, let us return to our analysis
of linear maps on C
I and write a := ∂f
(z ) and b := ∂f
(z ) for a fixed point z0 ∈ Ω1 .
∂z 0
∂ z̄ 0
Then Df (z0 ) is given by Df (z0 ) · u = au + bū and maps an ellipse whose excentricity
# K−1 $
then, since
is |a|+|b|
to
the
unit
circle
(see
the
figure
1.2.1).
Let
us
set
k
:=
|a|−|b|
K+1
by assumption |b| ≤ k|a|, one gets:
|a|+|b|
|a|−|b|
≤
1+k
1−k
= K.
We have thus seen that a K-quasiconformal diffeomorphism preserves the orientation and maps ”infinitesimal” ellipse of excentricity bounded by K to ”infinitesimal” circles.
1.3
1.3.1
The Cauchy formula and its consequences
The Cauchy-Pompeiu formula
Let us recall a few facts about differential forms on C.
I
Let f : D → C
I be a C 1 - function defined on some open subset D of the complex
plane. Its differential may be written as
df =
∂f
dx
∂x
+
∂f
dy.
∂y
Taking in particular f = z or f = z̄ one gets
dz = dx + idy and dz̄ = dx − idy
16
which leads to
dx =
1
2
#
dz + dz̄
$
1
2i
and dy =
#
$
dz − dz̄ .
Replacing dx and dy by these quantities one obtains the following expression for
df :
df =
#
1 ∂f
2 ∂x
$
#
$
dz + 12 ∂f
dz̄ =
− i ∂f
+ i ∂f
∂y
∂x
∂y
∂f
dz
∂z
+
∂f
dz̄.
∂ z̄
More generally, one sees in the same way that any differential 1-form (resp. 2form) on D ⊂ C
I may be written as αdz + βdz̄ (resp. λdz ∧ dz̄) where α, β and λ
are complex valued functions.
The Cauchy formula is the key tool in the study of holomorphic functions; we
shall prove here a slightly more general formula which is known as Cauchy-Pompeiu
formula.
Theorem 1.3.1 Let D be a relatively compact subdomain of C
I with C 1 boundary.
Let f be a function of class C 1 which is defined on some neighbourhood of D. Then:
(Cauchy-Pompeiu) ∀z ∈ D : f (z) =
1
2iπ
%
f (η)
dη
bD η−z
−
If, moreover, the function f is holomorpic in D then:
(Cauchy) ∀z ∈ D : f (z) =
1
2iπ
1
2iπ
%
∂f dη̄∧dη
.
D ∂ η̄ η−z
%
f (η)
dη.
bD η−z
Proof.
The idea is simply to apply the Stokes formula to the differential form ω :=
on the domain D+ := D \ ∆(z, $) and then make $ → 0.
Let us compute dω. Using the fact that
1
η−z
is holomorphic on D+ one gets:
# ∂ # f (η) $
# f (η) $
∂ # f (η) $ $
∧ dη =
dη̄ +
dη ∧ dη =
η−z
∂ η̄ η − z
∂η η − z
∂f
# ∂ # f (η) $ $
∂ η̄
dη̄ ∧ dη =
dη̄ ∧ dη.
∂ η̄ η − z
η−z
%
%
Now the Stokes formula D$ dω = bD$ ω becomes:
dω = d
%
D$
∂f
∂ η̄
dη̄ ∧ dη =
η−z
%
f (η)
dη
bD$ η−z
=
17
%
f (η)
dη
η−z
f (η)
dη
bD η−z
−
%
f (η)
dη.
b∆(z,+) η−z
1
As η−z
is locally integrable for the Lebesgue measure on C,
I the left hand side
% ∂f dη̄∧dη
in the above equality tends to D ∂ η̄ η−z . The last term of the right hand side is
% 2π
% 2π
it )
i$eit dt = i 0 f (z + $eit )dt which, by continuity of f ,
actually equal to 0 f (z++e
+eit
tends to 2iπf (z).
If the function f is holomorphic on D and C 1 -smooth on some neighbourhood of
D then, as ∂f
identically vanishes on D, the Cauchy-Pompeiu formula becomes the
∂ η̄
% f (η)
1
classical Cauchy formula: f (z) = 2iπ
dη.
&
%
bD η−z
Remark 1.3.2 It immediately follows from the Cauchy formula that the modulus
function |f | of an holomorphic function f ∈ O(Ω) enjoys the submean value property:
3 2π
1
|f (z0 )| ≤
|f (z0 + reiθ )|dθ, ∀z0 ∈ Ω, ∀0 ≤ r < d(z0 , bΩ).
2π 0
1.3.2
Basic properties of holomorphic functions
In the following proposition, we list some of the most fundamental properties of
holomorphic functions.
Proposition 1.3.3
1- Let f ∈ O(Ω). Then f is analytic on Ω that is, f is the
sum of a power series on every disc ∆(z0 , R) which is relatively compact in Ω.
"
2- If f ∈ O(Ω) then f (k) ∈ O(Ω) for every k ∈ IN. If f = n≥0 an (z − z0 )n on
∆(z0 , R) then an =
f (n) (z0 )
.
n!
3- The zeros of any non constant holomorphic function f ∈ O(Ω) are isolated.
4- Analytic continuation property. Let Ω be a connected subdomain of C
I and let
f, g ∈ O(Ω). If f and g coincides on some neighbourhood of z0 ∈ Ω then f = g
on Ω.
5- Cauchy inequalities. If f ∈ O(Ω) and K is relatively compact in Ω then
supΩ |f |
supK |f (k) | ≤ k! d(K,bΩ)
k for any k ∈ IN.
6- Maximum modulus principle. Let Ω be a connected subdomain of C
I and let
f ∈ O(Ω). If supz∈Ω |f (z)| ≤ |f (z0 )| for some z0 ∈ Ω then is f constant on Ω.
7- Let fn ∈ O(Ω). If fn is locally uniformly converging to g then g ∈ O(Ω) and
(k)
fn is locally uniformly converging to g (k) for all k ∈ IN.
8- Hurwitz lemma. Let fn ∈ O(Ω). Assume that fn is locally uniformly converging to g and that g is vanishing at some point z0 ∈ Ω but does not vanish
identically on Ω. Then, for any neighbourhood U0 of z0 , there exists n0 ∈ IN
such that the functions fn have a zero in U0 for n ≥ n0 .
18
9- Openess. Any non-constant holomorphic function fn ∈ O(Ω) is an open map
from Ω to C
I
Proof.
1- By the Cauchy formula we have
f (z) =
3
|η−z0 |=r
f (η)
dη =
η−z
It then suffices to develop
1
z−z
1− η−z0
3
f (η)
dη =
|η−z0 |=r (η − z0 ) − (z − z0 )
3
$
f (η) #
1
z−z0 dη.
|η−z0 |=r (η − z0 ) 1 − η−z0
in power series and integrate term by term.
0
2- This is a direct consequence of the analiticity of holomorphic functions (former
point) and of properties (in particular the holomorphy) of functions defined
by power series (see proposition 1.2.17).
3- This follows easily from the analyticity. If f# is not constant
then, for z small
$
enough, one has f (z0 + z) = f (z0 ) + ak0 z k0 1 + g(z) where ak0 ,= 0 and g is
holomorphic and vanishes at 0. If f (z0 ) = 0, this implies immediately that, in
a sufficently small neighbourhood of z0 , the only zero of f is z0 .
4- The set Ω∗ := {z0 ∈ Ω / f and g coincide on some neighbourhood of z0 } is
clearly open in Ω. By analyticity, {z0 ∈ Ω / f (k) (z0 ) = g (k)(z0 ) ∀ k ∈ IN} = Ω∗
and thus Ω∗ is closed in Ω. If Ω is connected then either Ω∗ is empty or
coincides with Ω.
"
5- Let r < d(K, bΩ) and z0 ∈ K. Then f (z) = n≥0 an (z − z0 )n for z ∈ D(z0 , r).
% 2π #
$ −ikθ
1
iθ
On one hand we have ak r k = 2π
f
z
+
re
dθ and therefore |ak | ≤
e
0
0
1
−k
(k)
r supΩ |f |. On the other hand we have |ak | = k! |f (z0 )|. This implies that
|f (k) (z0 )| ≤ suprΩk |f | .
6- The set Ω∗ := {z ∈ Ω / |f (z)| = |f (z0 )|} is clearly closed in Ω. Using the
submean value property (see remark 1.3.2) on sees that it is also open. If Ω is
connected then either Ω∗ is empty or coincides with Ω. To conclude one uses
the Proposition 1.2.21 and the Analytic continuation property (point 4-).
7- That the limit g is again holomorphic follows from the uniform convergence
and the integral representation given by the Cauchy formula. The uniform
convergence of the derivatives then follows from the Cauchy inequalities applied to g − fn .
19
8- As the zeros of g are isolated, we may find a small disc D centered at z0 and
relatively compact in U0 whose boundary does not contain any zero of g. We
now argue by contradiction. Assume that the functions fn do not vanish on D
and consider the functions O(D) ) ϕn := f1n . As ϕn is uniformly converging
to g1 on bD, one has |ϕn | ≤ M < +∞ on bD for n big enough. On the other
hand, as g(z0 ) = 0, the sequence |ϕn (z0 )| is converging to +∞. For n big
enough, the function ϕn is thus violating the maximum modulus principle.
9- If f (Ω) would’nt contain a neighbourhood of f (a) then we could find a sequence ηn converging to f (a) such that (f − ηn ) ,= 0 on Ω. As (f − ηn )n
uniformly converges to f − f (a), this contradicts Hurwitz lemma.
%
&
We shall now nicely combine some of these properties for proving the following
important extension result:
Theorem 1.3.4 (Riemann extension) Any bounded holomorphic function on the
punctured disc ∆∗ extends to an holomorphic function on ∆.
Proof. Let f ∈ O(∆∗ ) such that Sup∆∗ |f | =: M < +∞. Let us define two continuous
functions g and h on ∆ by setting
g(z) := zf (z) for z ∈ ∆∗ and g(0) := 0
h(z) := zg(z) for z ∈ ∆.
Let us show that h is holomorphic on ∆. To this purpose we will check that h
I
at any point.
is C 1 on ∆ and that its differential is C-linear
Since g is continuous and vanishes at 0, the identity h(z) − h(0) = zg(z) shows that
h is differentiable at 0 and Dh(0) ≡ 0. As h = z 2 f (z) on ∆∗ , it is holomorphic there
and its differential at any point z0 ∈ ∆∗ is given by Dh(z0 ) · u = h# (z0 )u. Moreover:
h# (z0 ) = 2z0 f (z0 ) + z02 f # (z0 ) = 2g(z0 ) + z02 f # (z0 ).
Now comes the crucial point: since the function f is bounded by M and holomorphic
on the disc ∆(z0 , |z0 |) for |z0 | < 12 , the Cauchy inequality gives
|f # (z0 )| ≤
M
|z0 |
for |z0 | < 12 .
We thus get |h# (z0 )| ≤ 2|g(z0 )| + |z0 |M for |z0 | small enough and therefore z *→
Dh(z) is continuous on ∆. We have shown that h ∈ O(∆) and we may thus write
it as the sum of a convergent power series on ∆, as h(0) = h# (0) = 0 one has
h(z) = a2 z 2 + a3 z 3 + · · · = z 2 ϕ(z)
where ϕ is holomorphic on ∆. Since h(z) = z 2 f (z) for z ,= 0, this implies that ϕ
coincides with f on ∆∗ .
%
&
20
1.3.3
Remarks on one-to-one holomorphic maps
It is a special feature of holomorphic mappings that injectivity implies local invertibility. Using Riemann extension theorem we will prove this and a little bit more.
Proposition 1.3.5 Let Ω be an open subset of the complex plane and f : Ω → C
I
be a one-to-one holomorphic map. Then f : Ω → f (Ω) is a biholomorphism. In
particular, f # does not vanish on Ω.
Proof. As f is not constant, f (Ω) =: W is open and the set Cf of zeros of f # is
discrete in Ω. Let us call g : W → Ω the inverse map of f ; this is a continuous map
since f is open. We have to show that g is holomorphic on W . According to proposition 1.2.19, g is holomorphic on W \ f (Cf ). As g is continuous, the conclusion
follows from the Riemann extension theorem.
&
%
Uniform limits of one-to-one holomorphic maps are either constants or one-toone. This is an interesting consequence of Hurwitz lemma.
Proposition 1.3.6 Let fn : Ω1 → Ω2 be a sequence of one-to-one holomorphic
mappings which is locally uniformly converging to f . Then f is either one-to-one or
constant
Proof. If f is not one-to-one, we find two distinct points a and b in Ω1 such that
f (z) − f (b) vanishes at a and b. If f is not constant, Hurwitz lemma implies that
fn − fn (b) vanishes at two distinct points for n big enough. This contradicts the fact
that fn is one-to-one.
&
%
It might also be useful to know if the injectivity of limn fn forces that of the fn .
The following proposition answers this question.
Proposition 1.3.7 Let fn : Ω1 → Ω2 be a sequence of holomorphic maps which
converges locally uniformly to some one-to-one map f : Ω1 → Ω2 . Then, for any
relatively compact subdomain ω1 ⊂ Ω1 , fn is one-to-one on ω1 for n big enough.
The proof will use a simple rescaling lemma.
Lemma 1.3.8 Let fn be a sequence of holomorphic map on the unit disc ∆ such
that fn (0) = 0. If fn converges locally uniformly to f then +1n fn ($n z) converges
locally uniformly to f # (0)z for any sequence of complex numbers $n which tends to
0.
Proof. Let us write fn as a power series fn (z) = a1,n z + a2,n z 2 + · · · + ak,n z k + · · ·.
Then +1n fn ($n z) = a1,n z + $n a2,n z 2 + · · · + $nk−1 ak,n z k + · · · and
"
| +1n fn ($n z) − a1,n z| ≤ |$n | k≥2 |ak,n ||z|k .
21
By Cauchy inequalities and local uniform convergence, for n big enough, we have
#
$
|ak,n | ≤ R−k sup|z|≤R |fn | ≤ R−k sup|z|≤R |f | + 1 .
Thus | +1n fn ($n z) − a1,n z| ≤ K|$n |
"
k≥2
Let us now prove Proposition 1.3.7.
# r $k
R
for |z| ≤ r < R.
&
%
Proof. Assume to the contrary that, after taking a subsequence, fn is not one-toone on ω1 . Then there exists an , bn ∈ ω1 such that fn (an ) = fn (bn ) and an ,= bn .
After taking subsequences, we may assume that an → a ∈ Ω1 , bn → b ∈ Ω1 and
fn (an ) = fn (bn ) → f (a) = f (b). As f is one-to-one we must have a = b.
Let us set ϕn (z) := fn (an + z) − fn (an ). The maps ϕn are defined on a disc ∆ρ
centered at 0 and of radius ρ > 0 for n big enough. Moreover, ϕn (0)# = 0 and
$ ϕn con2
verges locally uniformly to f (a+z)−f (a) on ∆ρ . Let us set $n := ρ bn −an . As a =
b, we have limn $n = 0 and thus, according to Lemma 1.3.8, the sequence +1n ϕn ($n z)
converges locally uniformly to f # (a)z on ∆ρ . In particular limn +1n ϕn ($n 2ρ ) = f # (a) ρ2 .
Since ϕn ($n ρ2 ) = fn (bn ) − fn (an ) = 0 this implies that f # (a) = 0 which is impossible
because f is one-to-one (see Proposition 1.3.5).
&
%
1.4
1.4.1
Conformal equivalence of domains
The Riemann mapping theorem
The Riemann mapping theorem is a remarkable result whose interest goes far beyond
complex analysis. For instance, it characterizes the proper open subsets of the plane
which are homeomorphic to a disc as being the simply connected ones. It illustrates
the power of complex analytic methods.
Theorem 1.4.1 Let Ω ⊂ C
I be a simply connected domain. If Ω ,= C
I then Ω is
biholomorphic to the unit euclidean disc.
The three main ingredients in the proof of Riemann theorem are a compacity
result known as Montel’s theorem, the Schwarz lemma and the existence of lifts for
the exponential map. We shall first present these important tools.
• The Montel’s theorem
Montel’s theorem is a compacity statement which is actually a corollary of Ascoli’s theorem. It asserts that any family of uniformly bounded holomorphic functions is relatively compact for the topology of local uniform convergence. Here is a
concrete version of this theorem.
22
Theorem 1.4.2 (Montel) Let (fn )n∈IN be a sequence of holomorphic functions on
O(Ω). If Supn SupΩ |f | < +∞ then there exists a subsequence (fni )i∈IN which is
converging locally uniformly on Ω.
Proof. The main point is to show that for any fixed compact K ⊂ Ω, there exists a
subsequence of (fn )n which is uniformly converging on K. For this one uses Cauchy
inequalities to see that supn SupK |fn# | ≤ M < +∞ and that the family (fn )n∈IN is
uniformly equicontinuous on K. Then one applies Ascoli’s Theorem. To end the
proof it suffices to exhaust Ω by compact sets and use a diagonal process.
&
%
• The Schwarz lemma.
This simple lemma is a powerful tool which essentially characterizes the rotations
as being the holomorphic maps f : ∆ → ∆ fixing the origin and maximizing |f # (0)|.
Proposition 1.4.3 (Schwarz lemma) Let f ∈ O(∆, ∆). If f (0) = 0 then |f # (0)| ≤
1 and the equality occurs if and only if f is a rotation: f (z) = eiθ0 z.
"
"
Proof. Let f (z) = n≥1 an z n . The function g(z) := n≥1 an z n−1 is holomorphic on
on the punctured disc
the unit disc ∆ and g(0) = f # (0). As g coincides with f (z)
z
1
∆ \ {0} one sees that sup|z|=1−+ |g(z)| ≤ 1−+ . It then suffices to apply the maximum
1
modulus principle to the function g to get |f # (0)| = |g(0)| ≤ 1−+
and make $ → 0.%
&
• The existence of lifts for the exponential map.
Any holomorphic function f which does not vanish on some simply conncted
domain can be written on the form eg for some holomorphic function g.
Ω
I
!" C
!!
!
!
!!
g !!!
exp
!
!!
!!
!
!
!!
!
!!
#C
I∗
f
The existence of g is a standard consequence of the fact that exp : C
I →C
I ∗ is a
cover (see exercise 1.2.20) which only relies on topological properties of covers. The
holomorphicity of g follows from that of f and exp.
Proposition 1.4.4 Let Ω be a simply connected subdomain of C.
I Then:
I such that f = eg .
∀f ∈ O(Ω, C
I ∗ ), ∃g ∈ O(Ω, C)
I such that ew0 = f (z0 ) the map g can be
Moreover, for any z0 ∈ Ω and any w0 ∈ C
taken satisfying g(z0 ) = w0 . In that case the choice is unique.
23
We now start the proof of the Riemann theorem. We shall proceed in four steps.
Step 1: reduction to the case where Ω is bounded.
Let a ∈ C
I \ Ω. As the function (z − a) does not vanish on the simply connected
domain Ω, we may find ha ∈ O(Ω) such that eha (z) = (z − a). Then the function
#
$2
1
ψa (z) := e 2 ha (z) satisfies ψa (z) = z − a. The map ψa is clearly one-to-one on
Ω and therefore induces a biholomorphism
between Ω and ψa (Ω). On the other
#
$
hand, one easily checks that − ψa (Ω) ⊂ C
I \ ψa (Ω) which implies that C
I \ ψa (Ω)
1
contains an open disc ∆(b, $). Then the map H defined by H(z) = z−b
maps
1
biholomorphically ψa (Ω) onto a domain contained in ∆(0, + ) and therefore, after
replacing Ω by (H ◦ ψa )(Ω), we may assume that Ω is bounded. Translating and
rescaling Ω finally yields to the following situation:
0 ∈ Ω ⊂ ∆.
Step 2: an extremal problem.
Let us consider the following family of maps:
A(Ω) := {f ∈ O (Ω, ∆) /f is 1-to-1, f (0) = 0, and |f #(0)| ≥ 1}.
Observe that the identity map Id(z) = z belongs to A(Ω) (we recall that 0 ∈
Ω ⊂ ∆). Assume that σ : Ω → ∆ is a biholomophism which fixes the origin.
Applying the Schwarz lemma to σ −1 one sees that σ ∈ A(Ω) and then, applying it
to f ◦ σ −1 , one sees that |σ # (0)| ≥ |f # (0)| for any f ∈ A(Ω). In other words, the
biholomorphism σ we are looking for must be a solution to the following extremal
problem:
# $
E : σ ∈ A(Ω) and |σ # (0)| = Supf ∈A(Ω) |f # (0)|.
# $
Step 3: any solution σ : Ω → ∆ to the problem E is a biholomorphism.
# $
Let σ be a solution of E ; we only have to prove that σ is onto. Let us proceed
by contradiction and assume that there exists a ∈ ∆ \ σ(Ω).
• Let τa be an automorphism of ∆ satisfying τa (a) = 0 and τa (0) = |a| (τa is
z−a
given by τa (z) = e−i(π+Arg a) 1−āz
). Then, as τa ◦ σ does not vanish on the simply
connected domain Ω, we may find a map ha : Ω → H := {Re w < 0} such that
ha (0) = ln |a| =: x0 and exp ◦ ha = τa ◦ σ.
Let us now consider the biholomorphism ψ : H → ∆ defined by ψ(z) :=
and set ga := ψ ◦ ha so that:
24
x0 −z
x0 +z
ga ∈ O (Ω, ∆) and ga (0) = 0.
Our construction may be summarized by the following commuting diagram:
ψ−1
*
#% H
"""
"
"
""
"""
ha """
ga
exp
""
"""
"
"
""
"""
!
"
"
τa
"
σ
#∆
#∆
Ω
∆$
*
one-to-one
We aim to reach a contradiction by showing that ga ∈ A(Ω) and |ga# (0)| >
|σ (0)| = Supf ∈A(Ω) |f # (0)|.
#
• Let us set ϕ := τa−1 ◦ exp ◦ ψ −1 . As the above diagram shows we have ϕ ◦ ga = σ
and thus:
ga : Ω
# → ∆$ is# one-to-one
#
σ (0) = ϕ ga (0) ga (0) = ϕ# (0)ga# (0).
#
As ϕ is holomorphic from ∆ to ∆, fixes the origin and is NOT one-to-one (exp
is not), the Schwarz lemma implies that |ϕ# (0)| < 1. It then follows that:
|ga# (0)| > |σ # (0)| = Supf ∈A(Ω) |f # (0)|.
This ends to show that ga ∈ A(Ω) (we now have |ga# (0) ≥ 1) and contradicts the
fact that σ is a solution to the extremal problem (E).
# $
Step 4: solution of the extremal problem E .
Let σn be a sequence in A(Ω) such that limn |σn# (0)| = Supf ∈A(Ω) |f # (0)|. By
Montel’s theorem, this sequence admits a subsequence σnk which is locally uniformly
converging to some limit σ which is holomorphic on Ω. Since |σ # (0)| = limnk |σn# k (0)|,
all we have to do is to show that σ ∈ A(Ω). Let us first observe that |σ # (0)| ≥ 1
and therefore σ is non-constant. A priori σ takes its values in ∆, but the maximum
modulus principle shows that σ(Ω) ⊂ ∆. It remains to prove that σ is one-toone, this follows from the fact that the σn are one-to-one via Hurwitz theorem (see
exercise 1.3.6).
&
%
Exercise 1.4.5 Show that there exists a constant K > 0 such that |f # (0)| ≤ K$ for
every $ > 0 and every holomorphic map f : ∆ → {Re w < $} satifying f (0) = −$.
Exercise 1.4.6 Let Ω be a bounded domain in C
I and Aut(Ω) be the group of holomorphic automorphisms of Ω. let ϕn ∈ Aut(Ω) such that ϕn (z0 ) → η0 for some
points z0 ∈ ω and η0 ∈ bΩ.
25
∆
ϕ−1
ϕ
Ω
Figure 1.2: ϕ extends continuously but ϕ−1 does’nt.
1) Show that ϕn is locally uniformly converging to η0 .
2) Show that if bΩ is C 1 -smooth near η0 then Ω is biholomorphic to ∆.
Exercise 1.4.7 Let Aut(∆) denote the group of holomorphic automorphisms of the
unit disc ∆.
1) Check that τa :=
z−a
1−āz
∈ Aut(∆) for every a ∈ ∆.
2) Describe Aut(∆). Hint: use the Schwarz lemma.
1.4.2
The Caratheodory theorem
Theorem 1.4.8 Let Ω be a bounded domain in the complex plane. If the boundary
of Ω is locally connected then any biholomorphism ϕ : ∆ → Ω between the unit disc
and Ω extends continuously to ∆.
Let us consider a simple example. Let Ω := ∆ \ [0, 1], this is a simply connected
domain whose boundary S 1 ∪ [0, 1] is locally connected. According to the Riemann
and Caratheodory theorems, there exists a biholomorphism ϕ : ∆ → Ω which
extends continuously to ∆.
However, the inverse map ϕ−1 : Ω → ∆ does not extends continuously to Ω.
Indeed, in that case ϕ would induce an homeomorphism between S 1 and S 1 ∪ [0, 1]
and also between S 1 \ {ϕ−1 (1)}, which is connected, and (S 1 \ {1}) ∪ [0, 1[, which is
not.
The proof of Caratheodory’s theorem combines a topological argument based on
Jordan’s theorem and a lenght/area argument.
Proof.
Let ϕ : ∆ → Ω be a biholomorphism and a be a point in b∆. For 0 < ρ < 1, we
set
26
∆
ϕ
Ω
4
Γρn
a
Uρn
α−
n
α
Vn
Vα
α+
n
γn
# $
ϕ Γρn
Figure 1.3: The construction of Jn .
Γρ := {z/ |z − a| = ρ} ∩ ∆ = {a + ρeiθ / θρ− < θ < θρ+ }
Uρ := {z/ |z − a| ≤ ρ} ∩ ∆.
Let us denote by λ(ρ) the lenght of the curve ϕ (Γρ ). We shall first establish the
following
Fact 1.4.9 If there exists a sequence 0 < ρn converging to 0 such that lim λ(ρn ) = 0
then ϕ extends continuously to a. More precisely, there exists α ∈ bΩ such that for
any neighbourhood Vα of α one has ϕ (Uρ ) ⊂ Vα for ρ small enough.
Since λ(ρn ) < ∞, the map ϕ extends continuously to b∆ along Γρn . This is only
a way to say that the following limits exist:
#
$
limθ→θρ− ϕ a + ρeiθ =: αn− ∈ bΩ
$
#
limθ→θρ+ ϕ a + ρeiθ =: αn+ ∈ bΩ.
As lim supn |αn− − αn+ | ≤ limn λ(ρn ) = 0 and bΩ is bounded we find, after taking
subsequences, a point α ∈ bΩ such that αn− , αn+ → α. Let us then take an open disc
Vα centered at α such that ϕ(0) ∈
/ Vα . We may now pick n sufficently big so that
−
ϕ (Γn ) ⊂ Vα and αn is connected to αn+ by an arc γn contained in Vα ∩ bΩ. To see
this one uses the fact that limn λρn = 0 and the assumption of local connectedness
of bΩ. The closed curve Jn := ϕ (Γn ) ∪ γn is a Jordan curve which, by construction,
is contained in Vα .
According to Jordan’s theorem, the complement C
I \ Jn of the curve Jn splits into
two disjoint open connected subsets Vn and Vn# which are respectively bounded and
unbounded:
C
I = Vn ∪ Jn ∪ Vn# .
27
Moreover, any open set meeting the curve Jn also meets the open sets Vn and Vn# .
In particular we have:
Vn ∩ Ω ,= ∅.
c
The open set Vα is connected, unbounded and does not meet Jn . It must
thus be contained in Vn# . This also implies that Vn ⊂ (Vn# )c ⊂ Vα and actually, as
bVn ⊂ Jn ⊂ Vα , that Vn ⊂ Vα . Let us summarize these observations:
c
Vα ⊂ Vn# and Vn ⊂ Vα .
To end the proof of the Fact, it remains to show that ϕ (Uρn ) ⊂ Vn since then we
will have ϕ (Uρ ) ⊂ Vα for ρ < ρn .
$
#
The open connected set ϕ ∆ \ Uρn does not meet Jn but, as ϕ(0) ∈
/ Vα , meets
$
#
#
#
Vn . We thus have ϕ ∆ \ Uρn ⊂ Vn . One sees similarly that either ϕ (Uρn ) ⊂ Vn or
ϕ (Uρn ) ⊂ Vn# . If ϕ (Uρn ) would be contained in Vn# we would have
Ω = ϕ (∆) ⊂ Vn# ⊂ (IC \ Vn )
which is absurd since Vn ∩ Ω ,= ∅. We thus have ϕ (Uρn ) ⊂ Vn . The Fact is proved.
We will now use a length/area argument for showing the existence of a sequence
ρn converging to 0 such that limn λ(ρn ) = 0. This will end the proof of the theorem.
By the holomorphic change of variables formula ( see proposition ??) we have:
%
%ρ
% θ+
Area (ϕ (Uρ0 )) = Uρ |ϕ# (z)|2 dm = 0 0 dρ θ−ρ ρ|ϕ# (ρeiθ )|2 dθ
ρ
0
which, setting A(ρ) :=
%
θρ+
θρ−
ρ|ϕ# (ρeiθ )|2 dθ, becomes:
Area (ϕ (Uρ0 )) =
3
ρ0
A(ρ) dρ.
(1.4.1)
0
On the other hand, we have:
% θ+ d
% θ+
λ(ρ) = θ−ρ | dθ
ϕ(ρeiθ )| dθ = θ−ρ ρ|ϕ# (ρeiθ )| dθ
ρ
ρ
and therefore, by the Cauchy-Schwarz inequality, we get:
#
$
(λ(ρ))2 ≤ θρ+ − θρ− ρA(ρ) ≤ 2πρA(ρ).
It follows from 1.4.1 and 1.4.2 that
%ρ
% ρ0 λ(ρ)2
dρ ≤ 0 0 A(ρ) dρ = Area (ϕ (Uρ0 )) ≤ Area (Ω) < ∞.
2πρ
0
This implies that lim inf ρ→0 λ(ρ) = 0.
28
(1.4.2)
&
%
1.5
The Riemann sphere
1.5.1
C,
I! S 2 and P1
To construct the Riemann sphere C
I! we will show that the Alexandroff compactification of C
I is homeomorphic to the euclidean sphere S 2 and then define a structure of
one-dimensional complex manifold on it. We will also show that the one-dimensional
complex projective space P1 is holomorphically equivalent to C.
I!
One defines a new space C
I! := C
I ∪ {∞} by adding a point, denoted ∞, to the
complex plane. It is easy to check that the following collection of subsets defines a
topology τ on C:
I!
∅ and C
I!
all the open sets of C
I
all sets of the form K c ∪ {∞} where K is compact in C.
I
Let us observe that a neighbourhood basis of ∞ is therefore given by the open
sets
I |z| > n} ∪ {∞} n ∈ IN
Vn := {z ∈ C/
I! if and only if |zk |
and, consequently, that a sequence C
I ) zk is converging to ∞ in C
tends to +∞:
C
I ) zk → ∞ in C
I! ⇔ limk |zk | = +∞.
Let us also observe that topology induced by τ on C
I coincides with the usual one.
!
It is standard, and easy to prove,4 that5C
I satisfies the Borel-Lebesgue property and
!
I τ is the Alexandroff compactification of C.
I
is therefore compact. The space C,
5
4
I! τ is compact.
Proposition 1.5.1 The topological space C,
Let S 2∗ denote the unit sphere S 2 in IR3 with the north pole N deleted. The
stereographical projection πN from S 2∗ onto the equatorial plane (thought as C)
I
2∗
maps a point p ∈ S to the point z(p) which is defned as the intersection of the line
(Np) with the equatorial plane. It is a continuous, one-to-one and onto map which
I! by setting
obviously can be extended to some homeomorphism between S 2 and C
2
πN (N) = ∞. This identification of C
I! with S explains why C
I! is called the Riemann
!
sphere. It also shows that C
I is simply connected.
We will now define a complex manifold structure on C.
I!
Let us consider the following two open subsets of the Riemann sphere : Ω0 := C
I
and Ω∞ := {0}c ∪ {∞} = C
I! \ {0}. We may write C
I! as the union of these two open
sets:
29
N
p
y
z(p)
x
Figure 1.4: The stereographical projection.
C
I! = Ω0 ∪ Ω∞
and then consider the two following maps
I
χ0 : Ω0 → C
χ∞ : Ω∞ → C
I
I ∗.
which are respectively defined by χ0 (z) = z and χ∞ (∞) = 0, χ∞ (z) = 1z , for z ∈ C
One easily checks that χ0 and χ∞ are homeomorphisms. Moreover, the maps
I ∗ →C
I and χ∞ ◦ (χ0 )−1 : C
I ∗ →C
I
χ0 ◦ (χ∞ )−1 : C
are holomorphic functions on C
I ∗ : χ0 ◦ (χ∞ )−1 (z) = χ∞ ◦ (χ0 )−1 (z) = 1z .
In other words, the charts (Ω0 , χ0 ), (Ω∞ , χ∞ ) form an atlas of C
I! and changes of
−1
−1
charts χ0 ◦ (χ∞ ) and χ∞ ◦ (χ0 ) are holomorphic. We have thus found a natural
structure of (one-dimensional) complex manifold on C
I! which is therefore a compact
Riemann surface.
For any open subset Ω of the Riemann sphere C,
I! we shall denote by O(Ω, C)
I! the
set of holomorphic mappings from Ω to C.
I!
Exercise 1.5.2 Show that any polynomial P extends to some holomorphic map from
the Riemann sphere to itself by setting P (∞) = ∞. More generally, establish the
following proposition.
Proposition 1.5.3 Every rational function
P
Q
extends continuously to some holo-
morphic endomorphism ϕ of the Riemann sphere. This means that ϕ(z0 ) =
P (z0 )
Q(z0 )
P (z)
when Q(z0 ) ,= 0, ϕ(z0 ) = ∞ when z0 is a pole of PQ and ϕ(∞) = lim|z|→+∞ Q(z)
.
All holomorphic endomorphisms of the Riemann sphere are of this form and the
automorphisms are those induced by homographies.
30
We now transpose to the context of holomorphic maps to the Riemann sphere two
basic results which we have encountered for holomorphic functions. These results
will play an important role when we will study the Picard-Montel theory.
Theorem 1.5.4 (Liouville) Let S 1 be the unit circle seen in the Riemann sphere.
Then O(IC, C
I! \ S 1 ) ⊂ {constants maps}.
Proof. Let us consider ∆ as a subset of the Riemann sphere whose boundary is S 1 .
As f (IC) is connected we have either f (IC) ⊂ ∆ or f1 (IC) ⊂ ∆. It then suffices to
I and see that all non constants terms in the
expand f (or f1 )in power series on C
series must be constant (see exercise 1.5.4).
&
%
Theorem 1.5.5 (Riemann extension) Let V0 be any non empty open subset of
C.
I! Then O(∆∗ , C
I! \ V0 ) ⊂ O(∆, C)
I!
1
Proof. Let w0 ∈ V0 , composing f with w−w
we get a map which avoids a neighbour0
hood of ∞. This map may be considered as a bounded function on ∆∗ and therefore
extends holomorphically to ∆ (see theorem 1.3.4).
&
%
To end this subsection, we will show that the Riemann sphere can be identified
to the one-dimensional complex projective space.
Let us say that two points (z1 , z2 ) and (z1# , z2# ) in C
I 2 \ {(0, 0)} are equivalent if
they belong to the same complex line passing through the origin. In other words, if
P denotes this equivalence relation, then:
(z1 , z2 )P(z1# , z2# ) ⇔ ∃λ ∈ C
I ∗ such that (z1# , z2# ) = λ(z1 , z2 ).
The complex projective space P1 is defined as being the quotient of C
I 2 \ {(0, 0)} by
this equivalence relation:
# 2
$
P1 := C
I \ {(0, 0)} /P
and can be thought as being the space of all complex lines passing through the
origin.
The class of any (z1 , z2 ) ∈ C
I 2 \ {(0, 0)} is denoted by [z1 : z2 ] and the canonical
projection associated to P is therefore given by:
π :C
I 2 \ {(0, 0)} → P1
(z1 , z2 ) *→ [z1 : z2 ].
We endow P1 with the quotient topology, which means that ω ⊂ P1 is open if and
only if π −1 (ω) is an open subset of C
I 2 \ {(0, 0)}. In particular, the canonical projection π is continuous and, as its restriction to the unit sphere of C
I 2 is clearly onto,
one sees that P1 is compact.
We may now identify, as topological spaces, the Riemann sphere C
I! and P1 .
31
Proposition 1.5.6 The map J : P1 → C
I! defined by J ([z1 : z2 ]) =
phic.
z1
z2
is homeomor-
Proof. The map J is clearly well defined (let us stress that J ([z1 : 0]) = ∞) and, as
I! are both compact, it suffices
J ◦ π is continuous, it is continuous. Since P1 and C
to check that J is one-to-one and onto.
We have J ([1 : t]) = t for every t ∈ C
I and J ([1 : 0]) = ∞ which shows that J is
#
#
onto. If J ([z1 : z2 ]) = J ([z1 : z2 ]) = t ∈ C
I then [z1 : z2 ] = [z1# : z2# ] = [1 : t] and
[z1 : z2 ] = [z1# : z2# ] = [1 : 0] if J ([z1 : z2 ]) = J ([z1# : z2# ]) = ∞; the map J is one-toone.
&
%
A structure of Riemann surface can be defined on P1 exactly in the same way
than for C.
I! The open subsets
Ω1 := {[z1 : z2 ] ∈ P1 / z1 ,= 0}
Ω2 := {[z1 : z2 ] ∈ P1 / z2 ,= 0}
cover P1 and the maps
χ1 : Ω1 → C
I
χ2 : Ω2 → C
I
which are respectively defined by χ1 ([z1 : z2 ]) =
holomorphic atlas of P1 .
z2
z1
and χ2 ([z1 : z2 ]) =
z1
,
z2
form a
We then get the following identification.
I! defined by J ([z1 : z2 ]) =
Proposition 1.5.7 The map J : P1 → C
morphism.
1.5.2
z1
z2
is a biholo-
Spherical derivatives
The aim of this section is to define a notion of derivative for the holomorphic mappings between open sets of the complex plane and the Riemann sphere. To this end
we first define a distance on C.
I!
Definition 1.5.8 The chordal distance χ on C
I! is defined by
−1
−1
χ(a, b) = 8πN
(a) − πN
(b)8
where πN : S → C
I! is the homeomorphism obtained by extending the stereographical
projection to the north pole N (πN (N) = ∞) and 8 8 is the euclidean norm in IR3 .
Clearly, χ is a distance which is bounded by 2 and induces the topology τ on C.
I!
A direct computation shows that χ enjoys the following properties:
Proposition 1.5.9
I one has:
1) For every z1 , z2 ∈ C
32
χ(z1 , z2 ) = √
2|z1 −z2 |
1+|z1 |2
√
1+|z2 |2
and χ(z1 , ∞) = √
2
.
1+|z1 |2
2) For every z1 , z2 ∈ C
I one has: χ(z1 , z2 ) = χ( z11 , z12 ).
(z),f (z0 ))
when z tends to z0 for some holomorphic
We now would like to study χ(f |z−z
0|
map f : Ω → C
I! defined on some open subset of C.
I Let us assume that f (z0 ) ,= ∞.
Then, as
χ(f (z),f (z0 ))
|z−z0 |
=
2|f (z)−f (z0 )|
|z−z0 |
· √
1+|f (z)|2
1
√
1+|f (z0 )|2
one sees that the limit exists and
limz→z0
χ(f (z),f (z0 ))
|z−z0 |
=
2|f ! (z0 )|
.
1+|f (z0 )|2
If f (z0 ) = ∞, the limit will exist too since χ(f (z1 ), f (z2 )) = χ( f (z11 ) , f (z12 ) ). This
leads to the following definition:
Definition 1.5.10 Let f : Ω → C
I! be an holomorphic map defined on some open
subset of C.
I The spherical derivative of f at z0 ∈ Ω is defined by |f # (z0 )|σ :=
χ(f (z),f (z0 ))
limz→z0 |z−z0 | .
The above discussion shows that |f #(z0 )|σ = 2|f #(z0 )| 1+|f1(z0 )|2 when f (z0 ) ,= ∞
and that |f # |σ = |( f1 )# |σ .
The next lemma will allow us to “integrate” spherical derivatives in order to get
chordal distance estimates.
I! such that |f # |σ ≤ A < ∞ on ∆(z0 , r0 ).
Lemma 1.5.11 Let f ∈ O(∆(z0 , r0 ), C)
Then
χ(f (z), f (z0 )) ≤ A|z − z0 |,
∀z ∈ ∆(z0 , r0 ).
Proof. Fix θ0 ∈ IR and set zt := z0 + teiθ0 . Since we may assume that f is not
identically equal to f (z0 ), the sets f −1 {∞} and f −1 {f (z0 )} are discrete in Ω. We
may therefore find a subdivision
0 = t0 < t1 < · · · < tN −1 < tN = r < r0
such that f (zt ) ∈
/ {∞, f (z0)} for any tj < t < tj+1 and any 0 ≤ j ≤ N − 1.
This implies that the real function
α(t) := χ(f (z0 ), f (zt )) = √
33
2|f (zt )−f (z0 )|
1+|f (zt )|2
√
1+|f (z0 )|2
is differentiable on each segment tj < t < tj+1 . Let us check that α# (t) ≤ A:
#
$
#
$
χ f (zt+u ), f (z0 ) − χ f (zt ), f (z0 )
α(t + u) − α(t)
α (t) = lim
= lim
≤
u→0
u→0
u
|zt+u − zt |
#
$
χ f (zt+u ), f (zt )
= |f # (zt )|σ ≤ A.
≤ lim
u→0
|zt+u − zt |
#
The desired estimate now follows by integration. Let z := z0 + reiθ0 .
−1
'
#
#
$
#
$ N
$
χ f (z0 ), f (z) = χ f (z0 ), f (ztN ) =
α(tj+1 ) − α(tj ) =
j=0
=
N
−1 3
'
j=0
tj+1
tj
α# (t) dt ≤ A
N
−1
'
j=0
(tj − tj+1 ) = Ar = A|z − z0 |.
&
%
34
Chapter 2
The Zalcman rescaling method for
discs in the Riemann sphere
2.1
Normal families
For any open subset Ω of the complex plane, M(Ω, C)
I! will denote the space of all
mappings from Ω to the Riemann sphere and, as C
I! is a complex manifold we will
also consider the following subspaces of M(Ω, C)
I! :
C(Ω, C)
I! := {f ∈ M(Ω, C)/
I! f is continuous on Ω}
O(Ω, C)
I! := {f ∈ M(Ω, C)/
I! f is holomorphic on Ω}.
The aim of this section is to provide simple sufficient conditions for compactness
in O(Ω, C)
I! endowed with the topology of local uniform convergence. This is the very
beginning of the classical ”normal families” theory.
2.1.1
Ascoli’s theorem
Using the chordal distance on C,
I! one easily describes the notion of uniform convergence. Let us recall the following basic definitions.
For any subset A of Ω, one says that a sequence (fn )n in M(Ω, C)
I! is uniformly
converging on A to some f ∈ M(Ω, C)
I! if and only if:
#
$
∀$ > 0, ∃n0 ∈ IN such that n ≥ n0 ⇒ χ fn (z), f (z) < $, ∀z ∈ A.
As (C,
I! χ) is a complete metric space (it is actually compact), the uniform convergence
on A is equivalent to the following:
#
$
∀$ > 0, ∃n0 ∈ IN such that m ≥ n ≥ n0 ⇒ χ fm (z), fn (z) < $, ∀z ∈ A.
When (fn )n is uniformly converging on any compact set contained in Ω one says
I! is
that (fn )n is locally uniformly converging on Ω. Let us also recall that M(Ω, C)
a complete metric space for the topology of local uniform convergence.
35
Proposition 2.1.1 Let (Kp ) be an exhaustion of Ω by compacts sets. Let
"
d(f, g) := p 21p Supz∈Kp χ(f (z), g(z)).
Then d is a distance on M(Ω, C)
I! and d(fn , g) tends to 0 if and only if (fn )n is locally
uniformly converging to g.
We are now ready to define the notion of normality.
Definition 2.1.2 Let F ⊂ M(Ω, C).
I! One says that F is normal if F is compact
for the topology of local uniform convergence. This is equivalent to say that every
sequence (fn )n of elements of F admits a subsequence which is locally uniformly
converging.
We shall say that F is normal at some point z0 ∈ Ω if there exists a neighbourhood
V0 ⊂ Ω of z0 such that the restricted family F0 := {f |V0 /f ∈ F } is normal.
It will be useful to keep in mind that normality is a “smallness” notion for subsets of M(Ω, C).
I!
The fundamental result about normal families is a characterization in terms of
equicontinuity. Let us recall the basic definitions concerning this concept.
Definition 2.1.3 Let F ⊂ M(Ω, C).
I! One says that F is equicontinuous at some
point z0 ∈ Ω if and only if:
#
$
∀$ > 0, ∃η > 0 such that: |z − z0 | < η ⇒ χ f (z), f (z0 ) < $, ∀f ∈ F .
We say that F is equicontinuous on A ⊂ Ω if it is equicontinuous at every point of
A and that F is uniformly equicontinuous on A ⊂ Ω if and only if:
#
$
∀$ > 0, ∃η > 0 such that: |z − z # | < η ⇒ χ f (z), f (z # ) < $, ∀z, z # ∈ A, ∀f ∈ F .
Let us also recall the simple fact (known as Heine’s lemma) that F ⊂ M(Ω, C)
I! is
equicontinuous on Ω if and only of it is uniformly equicontinuous on each compact
set K contained in Ω.
We may now state the above mentionned characterization.
Theorem 2.1.4 (Ascoli) Let F ⊂ M(Ω, C).
I! Then F is normal if and only if it is
equicontinuous on Ω.
Proof. It is easy to show that every normal family F ⊂ M(Ω, C)
I! is equicontinuous
on Ω. Let us prove that if F is equicontinous on Ω then it is normal.
36
• Let (fn )n be a sequence of elements of F . Let K be an arbitrary compact set
contained in Ω. It suffices to show that a subsequence of (fn )n is uniformly converging on K. Then, exhausting Ω by compact subsets and using a diagonal process, we
get a subsequence which is locally uniformly converging on Ω.
I! is
• Let us consider a countable
#
$ dense subset S = {zj , j ∈ IN} in Ω. As C
compact, each sequence fn (zj ) n admits a converging subsequence. We may replace
(f$n )n by some subsequence obtained by a diagonal process and assume that
#
fn (zj ) n converges for every point zj ∈ S.
• We shall now use the uniform equicontinuity of F on K to propagate the
uniform convergence from S to S ∩ K = K. Let us pick $ > 0. As F is uniformly
equicontinuous on K there exists η > 0 such that:
#
$
z, z # ∈ K and |z − z # | < η ⇒ χ fn (z), fn (z # ) < 3+ , ∀n ∈ IN.
Since K is compact and S is dense, we may cover K by a finite number of discs
centered at points of S and of radius η:
K ⊂ ∪N
k=1 ∆(zjk , η).
By the convergence of (fn )n on S, there exists n0 ∈ IN such that:
#
$
m ≥ n ≥ n0 ⇒ χ fm (zjk ), fn (zjk ) < 3+ , ∀1 ≤ k ≤ N.
For every k ∈ {1, ..., N} and any point z ∈ ∆(zjk , η) we thus have:
#
$
#
$
#
$
#
$
χ fm (z), fn (z) ≤ χ fm (z), fm (zjk ) + χ fm (zjk ), fn (zjk ) + χ fn (zjk ), fn (z)
#
$
2$
+ χ fm (zjk ), fn (zjk )
<
3
and thus, if m ≥ n ≥ n0 :
#
$
χ fm (z), fn (z) ≤ 2+3 + 3+ = $.
#
$
We have shown that SupK χ fm (z), fn (z) < $ for m ≥ n ≥ n0 .
2.1.2
&
%
Holomorphic normal families
Combining Ascoli’s theorem with lemma 1.5.11 one immediately obtains the following sufficent condition of normality for holomorphic families. This result, which is
sometimes called Marty’s theorem, is actually a version of Montel’s theorem (see
theorem 1.4.2) for O(Ω, C).
I!
Theorem 2.1.5 Let Ω be an open subset of C
I and F ⊂ O(Ω, C).
I! If |f # (z)|σ ≤ A <
∞ for every z ∈ Ω and every f ∈ F then the family F is normal.
37
Proof. Let z0 ∈ Ω and r > 0 such that ∆(z0 , r) ⊂ Ω. According to lemma 1.5.11 we
have
χ(f (z), f (z0 )) ≤ A|z − z0 |, ∀f ∈ F , ∀z ∈ ∆(z0 , r).
In other words, F is equicontinuous at every point z0 in Ω. It then suffices to apply
Ascoli’s theorem 2.1.4.
&
%
The following immediate corollary is very useful:
Corollary 2.1.6 Let Ω be an open subset of C
I and F ⊂ O(Ω, C).
I! Let (An )n be any
sequence of positive numbers such that limn An = +∞. If F is not normal at some
point z0 ∈ Ω then there exists a sequence Ω ) zn → z0 and a sequence of maps
fn ∈ F such that:
|fn# (zn )|σ ≥ An
and, in particular, limn |fn# (zn )|σ = +∞.
As expected, holomorphy is stable by local uniform convergence.
Proposition 2.1.7 Let Ω be an open subset of C
I and (fn )n be a sequence of holomorphic mappings from Ω to C.
I! If (fn )n is locally uniformly converging to f on Ω
#
!
then f ∈ O(Ω, C)
I and |fn |σ is locally uniformly converging to |f # |σ .
Proof. Let us start by observing that f is continuous on Ω. Let us now pick z0 ∈ Ω
and show that f is holomorphic on some neighbourhood of z0 .
We first assume that f (z0 ) ,= ∞.
By continuity of f and local uniform convergence we find r > 0 and n0 ∈ IN such
that
#
$
fn ∆(z0 , r) ⊂ ∆(f (z0 ), 1) ⊂ C,
I ∀n ≥ n0 .
We may thus consider (fn )n as a sequence of maps from ∆(z0 , r) to the complex
plane C
I and therefore f is holomorphic on ∆(z0 , r) and fn# is uniformly converging
2|fn! |
#
#
to f on, say, ∆(z0 , 2r ) (see proposition 1.3.2). As |fn# |σ = 1+|f
2 one sees that |fn |σ
n|
is uniformly converging to |f #|σ on ∆(z0 , r2 ).
If f (z0 ) = ∞ we simply apply what we just have done to f1 and use the fact that
# $#
&
%
| f1 |σ = |f # |σ .
Finally we have the following transposition of Hurwitz lemma
Lemma 2.1.8 (Hurwitz) Let Ω be an open subset of C
I and (fn )n be a sequence of
!
holomorphic mappings from Ω to C
I which is locally uniformly converging to f on Ω.
Then:
1) If f is not constant then: w0 ∈ f (Ω) ⇒ w0 ∈ fn (Ω), for n big enough.
2) If the maps fn are one-to-one and f is not constant then f is one-to-one.
The proof is easily obtained from the classical version of Hurwitz lemma (see
proposition 1.3.2 and exercise 1.3.6) using charts.
38
2.2
2.2.1
The Bloch principle
The Zalcman renormalization lemma
The following lemma is the cornerstone of this chapter. It is a powerful tool which
allows to extract informations from non-normal holomorphic families.
#
$
Lemma 2.2.1 (Zalcman) Let (fn )n be a sequence of elements of O ∆, C
I! . If (fn )n
is not normal at some point z0 ∈ ∆ then there exists sequences (zn )n and (ρn )n
satisfying
(i) ∆ ) zn → z0
(ii) 0 < ρn → 0
and such that, after taking a subsequence, the renormalized sequence
gn (z) := fn (zn + ρn z)
converges locally uniformly to a non-constant limit g : C
I → C.
I! Moreover, the spher#
ical derivative of g satisfies the following estimate: |g |σ ≤ |g #(0)|σ = 1.
The original proof of Zalcman’s lemma is totally elementary but somewhat tricky.
The proof which we shall present here is based on a simple metric space lemma and
is maybe more transparent.
Lemma 2.2.2 Let (X, d) be a complete metric space and ϕ : X → IR+ be a locally
bounded function. Let $ > 0 and τ > 1. Then, for every a ∈ X such that ϕ(a) > 0
there exists ã ∈ X such that:
(i) d(a, ã) ≤
τ
+ϕ(a)(τ −1)
(ii) ϕ(ã) ≥ ϕ(a)
(iii) ϕ(x) ≤ τ ϕ(ã) if d(x, ã) ≤
1
.
+ϕ(ã)
To understand the meaning of this lemma, let us consider the case τ = 2 and
$ = 1. The general case is just a refined version which allows to make clever choices
of the parameters τ and $. Assume that the function ϕ takes a positive value (in
practice very big) at point a. Then, close to a, we may find a new point ã where ϕ
takes an even bigger value but is bounded by 2ϕ(ã) on a ball centered at ã and of
1
radius ϕ(ã)
(see figure 2.2.1).
It should be stressed that we cannot usually control the position of the “good
1
1
ball” B(ã, ϕ(ã)
). Indeed, the bound on d(a, ã) is much bigger than ϕ(ã)
if ϕ(ã) is
much bigger than ϕ(a).
Proof. We proceed by contradiction and assume that for any point in X one of the
three properties (i), (ii), (iii) fails. As a obvioulsy satisfies the two first conditions,
the third one must fail:
39
∃a1 ∈ X such that d(a, a1 ) ≤
1
+ϕ(a)
and ϕ(a1 ) ≥ τ ϕ(a).
Let us assume to have constructed a1 , a2 , · · ·, ak such that:
$# 1 $
# 1 $
#
1
τ
≤ τ −1
d(ak , a) ≤ 1 + τ1 + · · · + τ k−1
+ϕ(a)
+ϕ(a)
k
ϕ(ak ) ≥ τ ϕ(a).
Then ak satisfies the properties #(i) and
$ (ii). Thus (iii) must fail and we may
1
1
find ak+1 such that d(ak , ak+1) ≤ τ k +ϕ(a) and ϕ(ak+1 ) ≥ τ ϕ(ak ). We thus have
#
d(ak+1 , a) ≤ d(ak , a) + d(ak , ak+1) ≤ 1 +
ϕ(ak+1 ) ≥ τ k+1 ϕ(a).
1
τ
+···+
1
τk
$#
1
+ϕ(a)
$
By induction,# we $have constructed a sequence (ak )k which is Cauchy since
1
d(ak , ak+1 ) ≤ τ1k +ϕ(a)
and along which ϕ is not bounded: ϕ(ak ) ≥ τ k ϕ(a). This
contradicts the assumptions of the lemma.
&
%
ϕ ≤ 2ϕ(ã)
ã
a
1
ϕ(ã)
≤
2
ϕ(a)
Figure 2.1: Le lemme 2.2.2.
Proof of Zalcman lemma. According to corollary 2.1.6 there exists a sequence of
points an which tends to z0 and a sequence of integers (kn )n such that |fk# n (an )|σ ≥
n3 . After re-indexing we have:
an → z0 and |fn# (an )|σ ≥ n3 .
Pick 0 < r < 1 such that |an | ≤ r for all n ∈ IN. Let us now apply lemma 2.2.2
with X = ∆(0, r), ϕ = |fn# |σ , a = an , $ = n1 and τ = 1 + n1 . We get a new point
∆(0, r) ) ã =: zn such that:
40
(i) |zn − an | ≤
τ
+ϕ(a)(τ −1)
≤
2n2
|fn! (an )|σ
≤
2
n
(ii) |fn# (zn )|σ = ϕ(ã) ≥ ϕ(a) = |fn# (an )|σ ≥ n3
$
#
(iii) |fn# (z)|σ = ϕ(z) ≤ τ ϕ(ã) = 1 + n1 |fn# (zn )|σ for |z − zn | ≤
1
+ϕ(ã)
=
n
.
|fn! (zn )|σ
We then set:
ρn :=
1
|fn! (zn )|σ
and gn (z) := fn (zn + ρn z)
and rewrite (iii) as:
#
$
(iii’) |fn# (z)|σ ≤ 1 + n1 ρ1n for |z − zn | ≤ nρn .
By (i) one sees that zn → z0 and by (ii) that nρn ≤ n12 . This shows that gn is well
defined on ∆n for n big enough. Moreover, (iii’) implies that:
$
#
|gn# (z)|σ = ρn |fn# (zn + ρn z)|σ ≤ 1 + n1 on ∆n .
Thus, using theorem 2.1.5 and a diagonal process one sees that, after taking
a subsequence, gn is locally uniformly converging to g : C
I → C.
I! According to
proposition 2.1.7 we have |g # (0)|σ = limn |gn# (0)|σ = 1 which
# shows
$ that g is not
constant. Similarly, we have |g # (z)|σ = limn |gn# (z)|σ ≤ limn 1 + n1 = 1.
&
%
Remark 2.2.3 All the interest of the lemma relies on the fact that the limit g is
non constant and the holomorphicity has only been used to establish this point.
To illustrate the power of this renormalization technique we shall give a proof of
Koebe distorsion theorem. This well-known theorem is a very useful result which
allows to control the distorsion of one-to-one holomorphic images of discs. The
precise statement is as follows:
Theorem 2.2.4 Let Let S := {f ∈ O(∆, C)/f
I
is 1-to-1, f (0) = 0, f #(0) = 1}.
There exists a contant 0 < κ ≤ 14 such that ∆κ ⊂ f (∆) for every f ∈ S.
The proof is obtained by combining the two following simple lemmas with a
renormalization argument.
Lemma 2.2.5 A one-to-one holomorphic map from C
I to C
I is necessarily onto.
Proof. Assume that f : C
I →C
I is one-to-one. The image Ω of f is a subdomain of
C
I which is biholomorphic to C.
I In particular, Ω is simply conected and therefore,
according to Riemann mapping theorem 1.4.1; is either equal to C
I or biholomorphic
to the unic disc ∆. By Liouville theorem ∆ and C
I are not holomorphically equivalent, thus Ω = C.
I
&
%
41
Lemma 2.2.6 The class S := {f ∈ O(∆, C)/f
I
is 1-to-1, f (0) = 0, f # (0) = 1} is
closed for the topology of local uniform convergence on ∆.
Proof. If fn ∈ S and (fn )n is locally uniformly converging to f on ∆ then f (0) = 0
and f # (0) = limn fn# (0) = 1. Moreover, by Hurwitz lemma, f is one-to-one and
therefore f ∈ S
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%
Proof of Koebe theorem. We argue by contradiction. Assume that there exists a
sequence (fn )n such that fn ∈ S and an ∈
/ fn (∆) for some sequence of points (an )n
converging to 0. If the sequence (fn )n was normal then, after taking a subsequence,
it would converge to some f ∈ O(∆, C)
I which, according to lemma 2.2.6, would be
in S. This contradicts Hurwitz lemma since fn − an does not vanish on ∆ but is
locally uniformly converging to f which vanishes only at 0.
Thus (fn )n is not normal and therefore, by Zalcman lemma, we may find sequences
zn → z0 ∈ ∆ and 0 < ρn → 0 such that, after taking a subsequence, fn (zn + ρn z)
#
$
is locally uniformly converging to a non constant map ϕ ∈ O C,
I C
I! . According to
Hurwitz lemma, ϕ is actually a one-to-one holomorphic map from C
I to C.
I By lemma
2.2.1 ϕ is onto and therefore vanishes precisely in one point. We may now repeat
the above argument with fn − an replaced by fn (zn + ρn z) − an to reach again a
contradiction.
It remains to show that 14 is the best possible constant. To this purpose we simply
z
−1
observe that the Bieberbach function (1−z)
2 is in S but does not take the value 4 .
&
%
Remark 2.2.7 It turns out that κ = 14 . This is the Koebe-Bieberbach one-quarter
theorem. Its proof is more delicate and is provided by the problem 2.2.8.
Exercise 2.2.8 The Koebe-Bieberbach one-quarter theorem
I. "
Let f ∆ → C
I! be a one-to-one holomorphic map defined by f (z) =
n
n≥1 bn z .
1
z
+ b0 +
1) For 0 < r < 1 we set Ωr := C
I \ f (∆r ). After checking that Ωr is a bounded
domain of C
I show that
3
i¯
Area(Ωr ) =
f (z)f # (z)dz.
2
|z|=r
2) Deduce from the above formula that
'
n≥0
n|bn |2 ≤ 1.
42
II. "
Let S := {f ∈ O(∆, C)/f
I
is 1-to-1, f (0) = 0, f # (0) = 1}. Let f (z) = z +
n
n≥2 an z ∈ S.
1) Using a map g of the form √1
z
where f (z 2 ) = z 2 f0 (z) and the result I.2)
f0 (z)
show that |a2 | ≤ 2.
2) Let f ∈ S and c ∈ C
I \ f (∆). Applying the result II.1) to f and
1
that |c| ≤ 4. This implies that
cf (z)
,
c−f (z)
show
∆ 1 ⊂ f (∆)
4
which is the celebrated (and important) Koebe one-quarter theorem.
3) Show that
4) Using
2.2.2
1
4
z
(1−z)2
≤ d(0, bf (∆)) ≤ 1.
show that the estimate
1
4
is sharp.
The Bloch-Zalcman principle
The Bloch principle is an heuristic principle, discovered by Bloch around 1920, which
can be informally staded as follows:
if there are no non constants entire curves (elements of O(IC, C))
I! which satisfy a
certain property, then there are very few holomorphic discs (elements of O(∆, C))
I!
satisfying the same property.
The language of normal families allows to give a precise sense to this principle.
Let P be a property which can be satisfied by any holomorphic map from the
complex plane C
I to C
I! or from a subdomain Ω to C.
I! The Bloch principle asserts that
the follwing implication should be true:
{f ∈ O(IC, C)
I! s.t. f satisfies P} ⊂ {cst} ⇒ {f ∈ O(∆, C)
I! s.t. f satisfies P} is normal.
A simple, and good to keep in mind, example of property to which the Bloch
principle could be applied is the following:
A map f satisfies the property Ppic if and only if f avoids the values 0, 1 and ∞.
There are of course properties for which this principle fails. However, as we shall
see, an actual principle follows immediately from Zalcman renormalization lemma;
we shall call it the Bloch-Zalcman principle.
Theorem 2.2.9 (Bloch-Zalcman) Let P be a property which makes sense for
holomorphic mappings from subdomains of C
I (or C
I itself ) to the Riemann sphere C.
I!
Assume that P is both stable under local uniform convergence to non constant limits
and under affine change of variables at the source. Then:
{f ∈ O(IC, C)
I! s.t. f satisfies P} ⊂ {cst} ⇒ {f ∈ O(∆, C)
I! s.t. f satisfies P} is normal.
In other words, the Bloch principle is valid for the proprety P.
43
To be precise, the stability assumptions on P are as follows. If f ∈ O(Ω, C)
I! satI! also satisfies P for any
isfies P then the map f ◦ A (which belongs to O(A−1 (Ω), C))
affine transformation A. If fn ∈ O(Ωn , C)
I! is a sequence of maps satisfying P which
locally uniformly converges to some non constant map f ∈ O(Ω, C)
I! (we assume that
any compact K ⊂ Ω is contained in Ωn for n big enough) then f also satifies P.
As the Hurwitz theorem 2.1.8 shows, the property Ppic satisfies the assumptions
of theorem 2.2.9. In that case, the Bloch-Zalcman principle shows that Montel’s
general theorem follows from Picard’s first theorem:
#
$
#
$
O C,
I C
I! \ {0, 1, ∞} ⊂ {cst} ⇒ O ∆, C
I! \ {0, 1, ∞} is normal.
In the next section we will investigate the Picard-Montel theory from the BlochZalcman principle view point.
2.3
2.3.1
The Picard-Montel theory
Linearization of entire maps
We shall establish the following “linearization” result from which, in the next subsection, we will easily deduce Picard’s first theorem.
Theorem 2.3.1 For any non constant f ∈ O(IC, C
I! \ {0, ∞}) there exists α0 ∈ C
I ∗,
β0 ∈ C
I and two sequences of complex numbers (Ak )k and (Bk )k such that f (Ak +Bk z)
is locally uniformly converging to exp(α0 z + β0 ) on C.
I
Proof. • Without any loss of generality we may assume that |f # (0)|σ ,= 0. Considering
f as a non vanishing holomorphic function on C
I we find gk such that
f = (gk )k .
I!
∞}) or as a non vanishing holoWe will consider gk either as an element of O(IC, C\{0,
morphic function on C.
I Let us pick a sequence 0 < Rk → +∞ such that (gk (Rk z))k
is not normal at 0. This is possible because |(gk (Rk z))# (0)|σ = Rk |(gk )# (0)|σ and
|(gk )# (0)|σ ,= 0.
• Applying the Zalcman renormalization lemma to (gk (Rk z))k we get a sequence
of complex numbers (ak )k and a sequence of positive numbers (ρk )k such that
hk (z) := gk (ak + ρk z) → ϕ
where the convergence is locally uniform on C
I and ϕ ∈ O(IC, C)
I! is not constant.
According to the Hurwitz lemma, ϕ actually belongs to O(IC, C
I! \ {0, ∞}) and we
may therefore also consider it as a non vanishing holomorphic function on C.
I
44
• By Liouville’s theorem (see theorem 1.5.4), ϕ(IC) must meet the unit circle S 1 .
As ϕ is open and the zeros of ϕ# are isolated, one actually sees that:
I such that ϕ(z0 ) = eiθ0 for some θ0 ∈ IR and ϕ# (z0 ) ,= 0.
∃z0 ∈ C
Then, Hurwitz lemma shows that:
∃zk → z0 such that hk (zk ) = eiθ0 = ϕ(z0 ).
• We now observe that, by construction, there are complex numbers Ak and Bk
such that:
# #
$$k
hk zk + uk
= f (Ak + Bk u) ∀u ∈ C.
I
We will use the behaviour of ϕ near z0 to compute the limit of this sequence.
Let us write the power series expansions as:
#
$
ϕ(z0 + u) = eiθ0# 1 + α0 u + u2 τ (u) $
hk (zk + u) = eiθ0 1 + αk u + u2 τk (u) .
As hk is locally uniformly converging to ϕ, we have αk → α0 . Let us check that τk
is uniformly converging to τ on ∆ 1 . By the Cauchy formula we have
2
3
hk (zk + η)
1
hk (zk + η)
dη =
dη =
η−u
2iπ b∆1 η(1 − uη )
b∆1
3
#
1
u u2
u $ dη
hk (zk + η) 1 + + 2 (1 − )−1
=
2iπ b∆1
η η
η
η
3
dη
u
hk (zk + η)(1 − )−1 3 .
hk (zk ) + h#k (zk )u + u2
η
η
b∆1
%
This shows that τk (u) = e−iθ0 b∆1 hk (zk + η)(1 − uη )−1 dη
is uniformly converging to
η3
%
dη
e−iθ0 b∆1 ϕ(z0 + η)(1 − uη )−1 η3 = τ (u) on ∆ 1 .
1
hk (zk + u) =
2iπ
3
2
To end the proof we write
# #
$$k
#
hk zk + uk
= eikθ0 1 + αk uk +
2
2
$k
u2
τ (u) .
k2 k k
Then, as k uk2 τk ( uk ) = uk τk ( uk ) is locally uniformly converging to 0, the next lemma
immediately yields to the conclusion.
&
%
# $
Lemma 2.3.2 Let σk k be a sequence of holomorphic functions on C
I such that kσk
is locally uniformly converging to 0 and let (αk )k be a sequence of complex numbers
converging to α0 ∈ C
I ∗ . Then:
#
$k
limk 1 + αk uk + σk (u) = eα0 u
45
and the convergence is locally uniform.
This lemma is a slight generalization of Euler’s famous formula:
#
$k
limk 1 + uk = eu .
Proof. Let us consider an holomorphic local inverse ln of exp defined on some neighbourhood U1 of 1 and such that ln(1) = 0 (see exercise 1.2.19 or 1.2.20). For $ > 0
small enough we have
ln(1 + z) = z + zr(z) on ∆+
where r is holomorphic on ∆+ and r(0) = 0.
Let us fix R > 0 and take k0 ∈ IN such that |αk uk + σk ( uk )| < $ on ∆R for k ≥ k0 .
Then, for k ≥ k0 and u ∈ ∆R we have:
#
$
#
u
u $k
k ln 1+αk u
+σk ( u
)
k
k
1 + αk + σk ( ) = e
=
k$
#
$#
#k
$ #
$
u
u
u
u
u
u
u
ek αk k +σk ( k ) 1+r(αk k +σk ( k )) = e αk u+kσk ( k ) e 1+r(αk k +σk ( k )) .
$k
#
%
&
which shows that 1 + αk uk + σk ( uk ) is uniformly converging to eα0 u on ∆R .
2.3.2
Picard’s and Montel’s theorems
We start by showing how Picard’s first theorem can be deduced from the linearization
theorem 2.3.1.
#
$
Theorem 2.3.3 (Picard’s first theorem) O C,
I C\{0,
I!
1, ∞} ⊂ {cst} or, in other
words, any non constant holomorphic map f : C
I →C
I! avoids at most two points of
the Riemann sphere.
Proof. Assume that f : C
I →C
I! is holomorphic, non constant, and avoids three distinct points of C.
I! After composing
f with a suitable
homography we may assume
#
$
that the map f belongs to O C,
I C
I! \ {0, 1, ∞} . Applying theorem 2.3.1 we get a
sequence f (Ak + Bk z) which is locally uniformly converging to some non constant
exponential map eα0 z+β0 . By Hurwitz lemma this implies that exp avoids the value
1, which is absurd (see Proposition 1.1.9).
&
%
As we already explained in subsection 2.2.2, Montel’s general theorem is a consequence of Picard’s first theorem. We will repeat here the argument and get a slight
improvement which is very useful in applications.
Theorem 2.3.4 (Montel’s general theorem) Let E1 , E2 and E3 be three closed
subsets of the Riemann sphere which are mutually disjoint. Then the family
46
is normal.
#
$
ME := {f ∈ O C,
I C
I! / f (∆) ∩ Ej ,= Ej for j = 1, 2, 3}
# $
Proof. Assume that some sequence fk k in ME is not normal, say at some point
#
$
z0 ∈ ∆. By definition, fk ∈ O ∆, C
I! \ {a1,k , a2,k , a3,k } where aj,k ∈ Ej for j = 1, 2, 3.
After taking a subsequence we have limk (aj,k ) = aj ∈ Ej for j = 1, 2, 3.
By the Zalcman lemma there exists sequences ∆ ) zk → z0 and 0 < ρk → 0 such
that, after taking a subsequence, fk (zk + ρk z) locally uniformly converges to some
non constant entire map ϕ : C
I → C.
I!
Using Hurwitz’s lemma one sees that ϕ(IC) ⊂ C
I! \ {a1 , a2 , a3 }, which contradicts Picard’s first theorem.
&
%
Exercise 2.3.5 (The λ-lemma) Let E ⊂ C
I! be a subset of the Riemann sphere.
An holomorphic motion of E is a map
φ : E × ∆ ) (z, λ) *→ φ(z, λ) =: φλ (z) ∈ C
I!
which enjoys the following properties:
i) φ0 = Id|E
ii) E ) z *→ φλ (z) ∈ C
I! is one-to-one for every λ ∈ ∆
iii) ∆ ) λ *→ φλ (z) ∈ C
I! is holomorphic for every z ∈ E.
I! such that φ̃ = φ
1) Show that there exists an holomorphic motion φ̃ : E × ∆ → C
on E × ∆.
I! is continuous.
2) Show that φ̃ : E × ∆ → C
Exercise 2.3.6 Show that S := {f ∈ O(∆, C)/f
I
is 1-to-1, f (0) = 0, f #(0) = 1} is
compact for the topology of local uniform convergence on ∆.
We finally show how Montels theorem implies Picard’s big theorem. This second
theorem of Picard is a strong and, as the proof will reveal, natural generalization of
the Riemann extension theorem.
$
#
$
#
I! \ {0, 1, ∞} ⊂ O ∆, C
I! or, in
Theorem 2.3.7 (Picard’s big theorem) O ∆∗ , C
other words, every holomorphic map from the punctured disc ∆∗ to the Riemann
sphere with three points delated extends holomorphically to the disc ∆.
$
#
I! \ {0, 1, ∞} . Let us consider the corona T := { 14 < |z| < 1}
Proof. Let f ∈ O ∆∗ , C
#
$
and the sequence of maps fn ∈ O T, C
I! \ {0, 1, ∞} defined by
47
fn (z) = f ( 2zn ).
According to Montel’s general theorem, this sequence is normal and thus, after
taking a subsequence we may assume that fn is uniformly converging to ϕ on a
slightly smaller corona T̃ . By the Hurwitz lemma, either ϕ is constant equal to ∞
or ϕ is valued in C.
I Replacing f by f1 allows us to assume that ϕ(T̃ ) ⊂ C.
I Then for
n big enough we have
Sup|z|=
1
2(n+1)
|f | ≤ Sup|z|= 1 |ϕ| + 1
2
which, by the maximum modulus principle, implies that |f | is bounded by some
constant C on { 21n ≤ |z| ≤ 12 } for every sufficently big n ∈ IN. We may thus apply
Riemann extension theorem 1.3.4 to f .
&
%
2.4
2.4.1
Starting Fatou-Julia theory
Fatou and Julia sets
Our aim in this section is to define the Julia and Fatou sets of a rational map and
collect their first properties.
#
$
A rational map f is an element of O C,
I! C
I! . We want to study the behaviour of
the sequence of iterates (f ◦n )n which we will simply denote (f n )n .
Definition 2.4.1 The Julia set Jf of a rational map f is the set of points of the
Riemann sphere around which the sequence of iterates (f n )n is not normal:
Jf := {z0 ∈ C
I! such that (f n )n is not normal at z0 }.
The complement of the Julia set is called the Fatou set of f and is denoted Ff :
# $c
Ff := Jf .
The interpretation of these sets is rather clearly related to the behaviour of orbits
under perturbation. The orbit of some point z0 is simply the set
O + (z0 ) := {f n (z0 ) n ≥ 0}.
A point z0 belongs to Ff if and only if the orbit O + (z0# ) of any nearby point z0# stays
close to the orbit of z0 (this is what equicontinuity means). To the contrary, z0
belongs to Jf if there are arbitrarly close points z0# for which this fails to be true. In
other words, z0 ∈ Jf means that the iteration of f is sensitive to initial data near z0 .
48
The Julia set of f is not empty as soon as the degree of f is bigger than two.
This is essentially due
# to$ the compacity of the Riemann sphere. Let us recall that
the degree of f ∈ O C,
I! C
I! , which will be denoted deg(f ), coincides with the number
of preimages of a generic point of C.
I!
#
$
Theorem 2.4.2 Let f ∈ O C,
I! C
I! . If deg(f ) ≥ 2 then Jf ,= ∅.
Proof. Assume that Jf = ∅. Then, any subsequence of the sequence of iterates
admits a subsequence which is uniformly converging on the Riemann sphere. Let us
consider a sequence of integers (nk )k such that limk nk = limk nk+1 − nk = +∞ and
set νk := nk+1 − nk . We may assume that f nk and f νk are respectively uniformly
νk
nk
nk+1
converging to h and g on C.
I!# From
one gets h ◦ g = g.
$ f ◦f =f
!
!
Both h and g belongs to O C,
I C
I and are onto. Indeed, for any w0 ∈ C
I! there exists
I! such that f nk (zk ) = w0 . Since C
I! is compact, zk is converging to some z0 after
zk ∈ C
taking a subsequence and therefore g(z0 ) = w0 .
It thus follows from h◦g = g that h = Id and we have shown that (f νk )k is uniformly
converging to Id on C.
I! It remains to see that this is impossible when deg(f ) ≥ 2.
I! which admits two distinct preimages by f . For every k there exPick a point w0 in C
#
ists ak ,= ak such that f νk (ak ) = f νk (a#k ) = w0 . After taking a subsequence, ak → a
and a#k → a# and, as (f νk )k is uniformly converging to Id, we have a = a# = w0 . This
is impossible for k big enough.
&
%
The following properties are easily deduced from the definition, we first recall a
Definition
2.4.3# A$ subset E ⊂ C
I! is said to be totally invariant by f if and only if
# $
f E = E = f −1 E .
#
$
Proposition 2.4.4 Let f ∈ O C,
I! C
I! . Then:
1) Ff is open and Jf is closed
2) both Ff and Jf are totally invarinat by f
3) Jfk = Jf for every k ∈ IN∗ .
Proof. 1)That Ff is open
# $
# $ and Jf closed is clear from the definitions.
that
f −1 Ff ⊂ Ff and this
2)One clearly has f Ff ⊂ Ff . #It is #also$$obvious
#
$
implies, as f is onto, that Ff = f f −1 Ff ⊂ f Ff . We have shown that Ff is
totally invariant
one sees that the same occurs for Jf .
# kn $ by f . Taking complements
# n$
3)Since f n is a subseqence of f n one has Ff ⊂ Ffk . Let us establish the op# $
# $
posite inclusion. Let z0 ∈ Ffk and let f ni i be an arbitrary subsequence of f n n .
# $
We have to show that, after taking a subsequence, f ni i is normal at z0 . Let us
write ni = pi k + ri where 0 ≤ ri < k, after taking# a subsequence
we may assume
$p
that ri = r for all i ∈ IN. Then f ni = f pik+r = f r ◦ f k i is clearly normal at z0 . %
&
49
Exercise 2.4.5 For a polynomial p the basin of attraction Ωp (∞) at ∞ is defined
by
Ωp (∞) := {z ∈ C
I! such that pn (z) → ∞}.
Show that Ωp (∞) is connected and that Jp = bΩp (∞).
We may consider that the goal of our introduction to the Fatou-Julia theory is
to establish the following basic property of Julia sets.
#
$
Proposition 2.4.6 Let f ∈ O C,
I! C
I! such that deg(f ) ≥ 2. For any open set U
such that U ∩ Jf ,= ∅ there exists N ∈ IN such that Jf ⊂ f N (U).
This property is sometimes called “zoom effect” and is a first step towards an
understanding of the self-similarity properties of Julia sets. We will be able to prove
it at the end of next subsection. Right now we may make a first step towards it.
#
$
Lemma 2.4.7 Let f ∈ O C,
I! C
I! be such that deg(f ) ≥ 2.
1) Every finite set which is totally invariant by f is contained in Ff .
2) If U is open in C
I! and U ∩ Jf ,= ∅ then Jf ⊂ ∪n f n (U)
the orbit of z0 must
Proof. 1) Let I be a finite totally invariant set.
# nIf z0 $∈ I then
k
n
be captured by a cycle which means that f f (z0 ) = f (z0 ) for some inetegers
k, n ∈ IN. Let us set a = f n (z0 ) and g = f k . If g −1{a} ,= {a} then one constructs by induction an infinite sequence of points (a−q )q≥1 such that a0 = a and
g(a−q ) = a−q+1 . This contradicts the fact that I is finite and totally invariant by
f . Thus g −1 {a} = {a} and therefore a is both fixed by g and critical which implies
that a ∈ Fg = Ff and z0 ∈ Ff .
I! \ E. By Montel’s theorem E contains at most two
2)Let us define E by ∪f n (U) = C
points: Card E ≤ 2. On the other hand one clearly has f −1 (E) ⊂ E. From this two
facts it is easy to deduce that E is totally invariant. Thus, by 1), we have E ⊂ Ff . %
&
Let us observe that this implies that a Julia set has empty interior unless it
coincides with the full Riemann sphere.
#
$
# $
Proposition 2.4.8 Let f ∈ O C,
I! C
I! be such that deg(f ) ≥ 2. If Int Jf ,= ∅ then
I!
Jf = C.
Proof. If U ⊂ Jf and U is a non empty open set then Jf ⊂ ∪n f n (U) ⊂ Jf and
I! The last equality is, as before, a consequence of
therefore Jf = ∪n f n (U) = C.
Montel’s theorem.
&
%
The so-called Lattès examples are rational maps whose Julia sets coincide with
the Riemann sphere, one says that these maps are chaotic. Surprinsingly there
50
exists actually plenty of other chaotic maps. The original construction of Lattès is
as follows.
Let us consider the complex torus C/Γ
I
where Γ is the lattice ZZ ⊕ iZZ. The
multiplication by 2 induces an expanding holomorphic self-map D on C/Γ
I
(such a
map is called isogeny). The Weierstrass elliptic function p is actually an holomorphic
map from C/Γ
I
to C.
I! As the fiber p−1 {w} of any w ∈ C
I! is of the form {z, −z}, one
!
!
sees that D induces a map f : C
I → C:
I
p ◦ D = f ◦ p.
I!
It is not difficult to check that f is holomorphic and Jf = C.
2.4.2
The density of repelling cycles
Our aim in this subsection
# is$ to use the renormalization method for showing that
the Julia set of f ∈ O C,
I! C
I! is a perfect compact subset of the Riemann sphere
which coincides with the closure of the the set of repelling cycles of f . This is a
fundamental result in the theory of iteration of rational maps. We shall combine it
with lemma 2.4.7 to explain the ”zomm effect”.
#
$
Theorem 2.4.9 The Julia set of any f ∈ O C,
I! C
I! whose degree is bigger than two
is a compact perfect set.
Proof. Let a ∈ Jf , we may assume that a ,= ∞ and consider f n as defined on some
neighbourhood of a in C
I with values in C.
I! By Zalcman lemma there exists ak → a
and 0 < ρk → 0 such that f nk (ak + ρk z) → ϕ where ϕ : C
I →C
I! is not constant.
As Jf is not finite, Picard’s first theorem shows that there exists z0 , z0# ∈ C
I such
#
that ϕ(z0 ) and ϕ(z0 ) are two distinct points of Jf . Using Hurwitz lemma, one
finds two sequences zk → z0 and zk# → z0# such that f nk (ak + ρk zk ) = ϕ(z0 ) and
f nk (ak + ρk zk# ) = ϕ(z0# ) . Since J#f is totally$ invariant,
ak $+ ρk zk and ak + ρk zk# are
#
in Jf . One of the two sequences ak + ρk zk k , ak + ρk zk# k is not stationary (both
cannot be equal to a for k big enough) and thus a is not isolated in Jf .
&
%
A n-cycle of f is a set of n points {a0 , a1 , · · ·, an−1 } such that f (aj ) = aj+1 for
every 0 ≤ j ≤ n − 1 and f (an−1 ) = a0 . Any point aj of such a cycle is a fixed point
of the n − th-iterate f n of f .
$#
#
The multiplier of the cycle is, by definition, the derivative χ◦f n ◦χ−1 (0) where
χ is any holomorphic chart at some point aj of the cycle such that χ(aj ) = 0. One
easily checks that this number depends only on the cycle; it does not depend on of
the chart χ or the point aj in the cycle.
A cycle is said to be repelling if its multiplier has a modulus strictly bigger than 1.
For any point a belonging to some n-cycle of f , the repellingity of the cycle implies
51
the existence of some neighbourhood U of a such that for any other neigbourhood
V ⊂ U one has U ⊂ f kn (V ) for k big enough. In particular the repelling cycles of f
belong to its Julia set.
The following theorem shows that the distribution of these cycles actually “determines” the shape and the properties of the Julia set.
#
$
Theorem 2.4.10 Let f ∈ O C,
I! C
I! such that deg(f ) ≥ 2. Then the Julia set of f
coincides with the closure of the set of repelling cycles of f .
Proof. Let us consider the critical set Cf and the post-critical set Cf+ of f . The first
of these sets is simply the set of the critical points of f , it contains at most 2d − 2
points where d is the degree of f . The second set is the union of orbits of critical
points, Cf+ := ∪j≥0 f j (Cf ). It is a countable subset of C.
I!
As Jf is a compact perfect set, Jf \ Cf+ is dense in Jf and it therefore suffices to
approximate points which are not in Cf+ .
Let us take a ∈ Jf \Cf+ ∪{∞}. We will consider f n as defined on some neighbourhood
of a ∈ C
I with values in C.
I! According to Zalcman lemma, there exists ak → a and
0 < ρk → 0 such that f nk (ak + ρk z) is locally uniformly converging on C
I to some
!
non constant holomorphic map ϕ : C
I → C.
I
By Picard’d first theorem, ϕ(IC) avoids at most two points of C
I! and therefore ϕ(U) ∩
Jf ,= ∅ for some open subset U of C.
I Shrinking U if necessary we will assume that
#
∞∈
/ ϕ(U) and that
# ϕ $does not vanish on U.
As Jf ⊂ ∪q≥1 f q ϕ(IC) (see lemma 2.4.7) we may find q ∈ IN and z0 ∈ U so that
f q (ϕ(z0 )) = a. Applying Hurwitz lemma to f q ◦ f nk (ak + ρk z), we find zk → z0 such
that:
f q ◦ f nk (ak + ρk zk ) = ak + ρk zk .
Since (ak + ρk zk ) → a, it remains to check that ak + ρk zk , as a fixed point of f nk +q ,
is repelling. This is why the assumption a ∈
/ Cf+ was made for. For k big enough,
we may consider f nk +q (ak + ρk z) as an holomorphic function defined near z0 . Then
by the chain rule one gets:
#
$#
#
$#
ρk f q+nk (ak + ρk z) = (f q )# ◦ f nk (ak + ρk z) · f nk (ak + ρk z)
and in particular
#
$#
#
$#
ρk f q+nk (ak + ρk zk ) = (f q )# (f nk (ak + ρk zk )) · f nk (ak + ρk z) (zk ).
The righthand side of the above identity converges to (f q )# (ϕ(z0 )) · (ϕ# (z0 )). By
assumption f q ◦ ϕ(z0 ) = a is not in the post critical set of f and therefore ϕ(z0 ) is
not a critical point of f q (observe that f q (Cf q ) ⊂ Cf+ ), moreover ϕ# (z0 ) ,= 0. Thus,
#
$#
as ρk → 0, one has | f q+nk (ak + ρk zk )| → +∞ which means that the fixed point
52
&
%
ak + ρk zk is repelling for k big enough.
We may now prove Proposition 2.4.6.
Proof. According to theorem 2.4.10, U meets some n0 -cycle of f which is# repelling.
$
Shrinking U if necessary we may assume that U ⊂ f n0 (U). The sequence f kn0 (U) k
is thus an increasing sequence of open sets and, since its union contains Jj by lemma
2.4.7, the conclusion follows from the compacity of Jf .
&
%
f q+nk
D(ak , ρk )
f nk
Jf
a
fq
D(a, $)
Figure 2.2: The creation of a repelling cycle.
2.5
The Ahlfors five islands theorem
Ahlfors five islands theorem is one of the most famous result in geometric function
theory. Let Ω1 , · · ·, Ω5 be a collection of five simply connected subdomains of the
Riemann sphere with piecewise real-analytic boundaries and disjoint closures. Then
Ahlfors theorem says that any non-constant holomorphic map f : C
I →C
I! admits
an inverse branch ϕj0 : Ωj0 → C
I above at least one of the fixed subdomains Ωj .
In other words, for some 1 ≤ j0 ≤ 5, there exists a subdomain ωj0 ⊂ C
I such that
f : ωj0 → Ωj0 is a biholomorphism. One then says Ωj0 is an island for f .
This result is already extremely significant when the five subdomains are discs:
Ωj := D(aj , $j ). In that case, when the radii $j tend to 0 one may hope to rely
this result to a simpler one, due to Nevanlina, which says that any holomorphic
map f : C
I →C
I! which has more than 4 totally ramified values is constant. This is
actually possible by using the Zalcman rescaling technique.
Our aim here is to use this approach to obtain a version of Ahlfors theorem
for discs after giving an elementary proof of Nevanlinna’s theorem. We follow the
53
exposition of W. Bergweiler.
2.5.1
A theorem of Nevanlinna
Let us start by a formal definition.
Definition 2.5.1 Let D be a subdomain of C
I and f ∈ O(D, C).
I! One says that a ∈ C
I!
is a totally ramified value of f if |f # |σ = 0 for all z ∈ f −1 {a}.
For any finite subset S of C
I! one defines FN (D, S) by:
FN (D, S) := {f ∈ O(D, C)
I! / a is a totally ramified value of f, ∀a ∈ S}
Nevanlinna’s theorem may now be stated as follows.
I! of 5 distincts points the family
Theorem
2.5.2
#
$ For any collection {a1 , · · ·, a5 } ⊂ C
5
FN C,
I {aj }j=1 reduces to constants.
It should be noted that the number five is sharp : the Weierstrass P function
has four totally ramified values.
To prove this theorem we shall need a few lemmas. The first one is a very basic
stability statement for the property of having a minimal number of totally ramified
values.
Lemma 2.5.3 The property of being totally ramified over more than N points is
stable by affine change of variable at the source and local uniform convergence to
non-constant limits. More precisely, if fn locally uniformly converges to some nonconstant map f : Ω → C
I! where Ωn → Ω ⊂ C
I and fn ∈ FN (Ωn , S) then f ∈
FN (Ω, S).
Proof. The stability by affine change of variables at the source is obvious. Let us
deal with the stability under local uniform convergence. Let z ∈ f −1 {a} for some
a ∈ S. By Hurwitz theorem, there exists zn → z such that fn (zn ) = a for n big
enough. Since by assumption |fn# (zn )|σ = 0 one gets |f # (z)|σ = 0.
&
%
We shall also use the following Schwarz style lemma which is due to Nehari.
Lemma 2.5.4 Let F ∈ O(∆, ∆). If all zeros of F are multiple then
#
$#
$2
|F #(z)|2 ≤ 4|F (z)| 1 − |F (z)|2 1 − |z|2
and in particular |F #(0)|2 ≤ 4|F (0)| (1 − |F (0)|2 ).
Proof. Replacing F by F (rz) and then making r → 1 allows to assume that F is
holomorphic on some neighbourhood of ∆ and does not vanish on b∆.
Let us then introduce the following two real valued functions on ∆:
54
u(z) := ln √
2
|F ! (z)|
|F (z)|(1−|F (z)|)
1
and v(z) := ln 1−|z|
2.
The function v is smooth on ∆ and tends to +∞ when |z| → 1. The function
u is clearly smooth on ∆ \ {F = 0} although it is equal to −∞. In particular
u is continuous on b∆. Let us examine u near a zero z0 of F . We may write
F (z) = (z − z0 )k0 G(z) where G is a non vanishing holomorphic function near z0
and, by assumption, k0 ≥ 2. A straightforward computation then reveals that u is
smooth near z0 when k0 = 2 and tends to −∞ at z0 when k0 ≥ 3. All of this shows
that the function (u − v) takes its maximum on ∆ at some point where it is smooth.
In particular we thus have:
∃ξ0 ∈ ∆ such that ∆(u − v)(ξ0 ) ≤ 0 and(u − v) ≤ (u − v)(ξ0 ).
To get the conclusion we now use the identity ∆(u − v) = 4e2u − 4e2v which
will be established at the end of the proof. It implies that (u − v)(ξ0 ) ≤ 0 and thus
!
1
u(z)−v(z) ≤ (u−v)(ξ0) ≤ 0 which actually means that ln √ |F (z)|
≤ ln 1−|z|
2.
2
As promised we now check that
|F (z)|(1−|F (z)|)
∆v = 4e2v and ∆u = 4e2u
where u is smooth. The first of these two identities is obtained by a direct computation. To get the second one it suffices to check it near a point where F and
F # are not vanishing
√ and then argue by continuity. Near such a point we may
consider that f := F is a well-defined non-vanishing holomorphic function and
observe that u = ln |f #| + v ◦ f where the function ln |f # | is harmonic. We then get
!
∆u = ∆(v ◦ f ) = |f # |2 (∆v) ◦ f = 4e2 ln |f | e2v◦f = 4e2u .
&
%
Our last lemma is a purely technical one which will be used for proving Nevanlinna’s theorem.
Lemma #2.5.5 Let ${a1 , · · ·, a5 } be a collection of 5 distincts points in C
I and let
I {aj }5j=1 . If f is non-constant and has bounded spherical derivatives
f ∈ FN C,
then the function gf defined by
gf (z) :=
f ! (z)2
j=1 (f (z)−aj )
!5
is entire and limn gf (zn ) = 0 when lim f (zn ) = ∞.
I \ f −1 {aj , ∞}. To examine the
Proof. The function gf is clearly holomorphic on C
behaviour of gf near a preimage zj of aj we write f (z) − aj = (z − zj )k G(z) where
G is a non-vanishing holomorphic function on some neighbourhood of zj and, by
assumption, k ≥ 2. A straightforward computation then shows that gf is actually
holomorphic near zj .
55
2
If |f # |σ ≤ M then |f # | ≤ M (1 + |f |2 ) and therefore |gf (z)| ≤
M 2 (1+|f (z)|2 )
!5
j=1 (|f (z)−aj |)
when
f (z) ,= ∞. This shows that limn gf (zn ) = 0 when lim f (zn ) = ∞ which, by Riemann
extension theorem 1.3.4, implies that gf extends holomorphically through any point
z0 ∈ f −1 {∞}.
&
%
Remark 2.5.6 It is precisely for proving the above lemma that the assumptions of
Nevanlinna’s theorem are used. The fact that f is totally ramified above {aj } implies
that gf is holomorphic on C
I \ f −1 {∞}. The comparison between gf and f when f
is big (and thus the holomorphy on all C)
I is based on the fact that 5 > 22 !
We will now prove Nevanlinna’s theorem.
#
$
I {aj }5j=1
Proof. We proceed by contradiction and assume that there exists f ∈ FN C,
which is non constant. Replacing f by T ◦ f for a suitable homography T allows to
assume that aj ,= ∞ for 1 ≤ j ≤ 5.
Since f is non constant the sequence f (nz) =: ϕn (z) is non normal at 0. Applying Zalcman renormalization lemma 2.2.1 one gets a new map which has
# bounded
$
I {aj }5j=1 .
spherical derivatives and, according to lemma 2.5.3, still stays in FN C,
From now on we may assume that
|f #(z)|σ ≤ M < +∞ for all z ∈ C.
I
Let gf be the entire function associated to f by lemma 2.5.5. Let us prove the
following
I such that gf (zn ) → ∞ and (after taking
Fact: there exists a sequence (zn )n in C
a subsequence) f (z + zn ) is locally uniformly converging to one of the constants aj .
Since gf is non-constant and is small when f is big, there exists a sequence (zn )n
in C
I such that
gf (zn ) → ∞ and supn |f # (zn )| < ∞.
!
I which are
Let us then consider the sequence of holomorphic map hn ∈ O(IC, C)
defined by: hn (z) := f (z + zn ).
Since |h#n |σ ≤ |f #|σ ≤ M < +∞ for all n, there exists some h ∈ O(IC, C)
I! to which,
after taking a subsequence, hn is locally uniformly converging. If h would not be
! (z)2
constant then, using lemmas 2.5.3 and 2.5.5 one would see that gh (z) := !5 h(h(z)−a
)
j=1
j
is an entire function and in particular that gh (z) ,= ∞ for all z ∈ C.
I This would
contradict the fact that gf (zn ) → ∞ since, as one easily checks, gf (z + zn ) is locally
uniformly converging to gh . This shows that h is constant equal to a. Then, to have
56
gf (zn ) → ∞ we must have a ∈ {a1 , · · ·, a5 }. The Fact is proved.
We are now ready to finish the proof. According to the above Fact, we may
assume that f (z + zn ) is locally uniformly converging to a1 and gf (zn ) → ∞.
Then, (hn − a1 ) ∈ O(∆, ∆) for n big enough. Moreover, since f is totally ramified over a1 , all zeros of (hn − a1 ) are multiple. Lemma 2.5.4 thus implies that
4
. This is absurd
|f # (zn )|2 ≤ 4|f (zn ) − a1 | and therefore |gf (zn )| ≤ !
2≤j≤5 |f (zn )−aj |
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since f (zn ) → a1 and gf (zn ) → ∞.
To end this subsection we would like to observe that one may#apply the Bloch$
Zalcman theorem 2.2.9 to deduce the normality of any family FN D, {aj }5j=1 from
#
$
the fact that FN C,
I {aj }5j=1 reduces to constants.
Indeed, as lemma 2.5.3 shows, the stability assumptions required to use this
principle are satisfied. Thus combining theorem 2.5.2 with theorem 2.2.9 one immediately gets the following normality statement.
Theorem 2.5.7 For any collection of five distinct #points {a1 , ·$· ·, a5 } on the Riemann sphere and any domain D ⊂ C
I the family FN D, {aj }5j=1 is normal.
2.5.2
Ahlfors five discs theorem
We start with definitions.
Definition 2.5.8 Let D ⊂ C
I be a domain in the complex plane. An open subdomain
Ω of the Riemann sphere is called an island for a non-constant holomorphic map
f : D →C
I! if there exists a subdomain ω ⊂ D such that f : ω → Ω is a biholomorphism.
For any finite collection S of open subsets of C
I! one defines FA (D, S) by:
I! / none of the Ω ∈ S is an island for f }.
FA (D, S) := {f ∈ O(D, C)
Ahlfors five island theorem may then be stated as follows.
Theorem 2.5.9 Let Ω1 , · · ·, Ω5 be a collection of five simply connected subdomains
of C
I!# with piecewise
real-analytic boundaries and pairwise disjoint closures. Then
$
5
FA C,
I {Ωj }j=1 reduces to constants.
Our aim here is to prove the following weaker version of Ahlfors theorem. We
denote by D(a, $) the open disc centered at a and of radius $.
Theorem 2.5.10 Let {a1 , · · ·, a5 } be a collection
of five distinct
points of the Rie#
$
5
mann sphere. Then for $ small enough FA C,
I {D(aj , $)}j=1 reduces to constants.
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The idea of the proof is to deduce Ahlfors theorem from Nevanlinna’s one by
making $ → 0. The tool is the Zalcman renormalization lemma.
We need a couple of simple lemmas. The first one is a stability statement.
Lemma 2.5.11 Let {D(aj , $# )}5j=1 be a collection of five discs in C
I! with pairwise
#
$
disjoint closures. Let 0 < $ < $# . Let fn ∈ FA Dn , {D(aj , $)}5j=1 . If the sequence
of domains Dn converges to a subdomain D of C
I and# fn uniformly converges
on
$
# 5
!
compact subsets of D to some f : D → C
I then f ∈ FA D, {D(aj , $ )}j=1 .
Proof. Assume to the contrary that f induces a biholomorphism f :#ω # → D(a$j , $# )
onto some subdomain ω # of D. Let us pick $ < $## < $# and ω ## := f −1 D(aj , $## ) ∩ ω #
so that f : ω ## → D(aj , $## ) is a biholomorphim and ω ## is relatively compact in D.
According to Proposition 1.3.7, fn is one-to-one on ω ## for n big enough. To get a
contradiction, it remains to check that D(aj , $) ⊂ fn (ω ## ) for n big enough. This is
an easy consequence of Hurwitz lemma.
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%
Now comes the key point, we use here the rescaling argument.
Lemma 2.5.12 Let {D(aj , $# )}5j=1 be a collection of five discs #on the Riemann$
I {D(aj , $)}5j=1
sphere with pairwise disjoint closures. Let 0 < $ < $# . If FA C,
#
$
does not reduce to constants then there exists f ∈ FA C,
I {D(aj , $# )}5j=1 such that
|f # (z)|σ ≤ |f #(0)|σ = 1 for all z ∈ C.
I
#
$
Proof. Assume that f ∈ FA C,
I {D(aj , $)}5j=1 is not constant. Then f (nz) is nonnormal at 0. Applying Zalcman renormalization lemma to f (nz) we get f : C
I →C
I!
#
#
1 for all z ∈ C
I and which, according to lemma 2.5.11,
such that |f (z)|
σ =$
# σ ≤ |f (0)|
# 5
belongs to F C,
I {D(aj , $ )}j=1 .
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Our last lemma connects Ahlfors and Nevanlinna’s problems.
Lemma 2.5.13 Let {D(aj , $)}5j=1 be a collection of five discs on the Riemann sphere
#
$
I {D(aj , $n )}5j=1 where $ >
with pairwise disjoint closures. Assume that fn ∈ FA C,
#
$
I →C
I! then f ∈ FN C,
I {aj }5j=1 .
$n → 0. If fn locally uniformly converges to f : C
Proof. Assume to the contrary that f is not totally ramified above aj . Then we
may find a relatively compact subdomain ω of C
I and a disc D(aj , r) such that
f : ω → D(aj , r) is a biholomorphism.
Arguing exactly like in the proof of Lemma 2.5.11, one sees that after shrinking
D(aj , r) is an island for fn if n is big enough. This is a contradiction when $n < r.%
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We are now ready to prove Theorem 2.5.10.
58
Proof. We proceed by contradiction. If the conclusion of the theorem is wrong then
there exists a sequence $k which tends to 0 and a sequence of non-constants holomorphic maps fk : C
I →C
I! such that
# none of the5 D(a
$ j , $k ) (1 ≤ j ≤ 5) is an island
for fk . This means that fk ∈ FA C,
I {D(aj , $k )}j=1 .
Using Lemma 2.5.12 we may assume that |fk# (z)|σ ≤ |fk# (0)|σ = 1 for all z ∈ C
I
and all k ∈ IN. Then, according to Marty’s lemma 2.1.5, the sequence fk is
normal and thus after taking a subsequence we may assume
# that5 fk$ locally uniformly converges to f : C
I → C.
I! By lemma 2.5.13, f ∈ FN C,
I {aj }j=1 . But, since
#
#
|f (0)|σ = limk |fk (0)|σ = 1, the map f is not constant and this contradicts Nevanlinna’s theorem 2.5.2.
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59