Chordal generators and the hydrodynamic normalization for the unit ball

0.24 MB·English

by  Sebastian Schleissinger

Chordal generators and
the hydrodynamic normalization for the unit ball
∗
Sebastian Schleißinger
January 20, 2015
5 Abstract
1
0 Let c 0 and denote by (H,c) the set of all infinitesimal generators G : H C on the
2 upper half≥-planeHsuchthatlimKsup y G(iy) c.Thisclassisrelatedto univale→ntfunctions
n f :H H with hydrodynamic normya→li∞zati·o|n and|a≤ppears in the so called chordal Loewner equa-
→
a tion.
J Inthispaper,wegeneralizetheclass (H,c)andthehydrodynamicnormalizationtotheEuclidean
9 unit ball in Cn. The generalizationisKbasedon the observationthat G (H,c) can be character-
1 ∈K
ized by an inequality for the hyperbolic length of G(z).

]
V Keywords: Semigroupsofholomorphicmappings,infinitesimalgenerators,hydrodynamicnor-
C malization, chordal Loewner equation
.
h
2010 Mathematics Subject Classification: 20M20, 32A40, 37L05.
t
a
m
Contents
[

1
1 Introduction 2
v
1.1 One-parameter semigroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
5
4 1.2 The hydrodynamic normalization in one dimension . . . . . . . . . . . . . . . . . . . 3
5
4 2 Chordal generators in higher dimensions 4
0 2.1 Invariant formulation for (D,c) and (H,c) . . . . . . . . . . . . . . . . . . . . . . 4
. K K
1 2.2 Chordal generators in the unit ball . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0
5 3 Slices 5
1
3.1 Normalized geodesics and slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
:
v 3.2 Some explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
i 3.3 Slices of generators in (H ,c) and examples . . . . . . . . . . . . . . . . . . . . . . 7
X n
K
r 4 Univalent functions with hydrodynamic normalization 8
a
4.1 Basic properties of P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
n
4.2 Semigroups with generators in (H ,c) . . . . . . . . . . . . . . . . . . . . . . . . . 10
n
K
4.3 On the behavior of iterates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
5 Appendix 13
∗Supportedby theERC grant “HEVO - Holomorphic Evolution Equations” n. 277691.
1
1 Introduction
1.1 One-parameter semigroups
Let B = z Cn z <1 be the Euclidean unit ball in Cn. In one dimension we write D:=B
n 1
{ ∈ |k k }
for the unit disc.
Definition 1.1. A continuous one-real-parameter semigroup of holomorphic functions on B is a
n
map [0, ) t Φ (B ,B ) satisfying the following conditions:
t n n
∞ ∋ 7→ ∈H
(1) Φ is the identity,
0
(2) Φ =Φ Φ for all t,s 0,
t+s t s
◦ ≥
(3) Φ tends to the identity locally uniformly in B when t tends to 0.
t n
Given such a semigroup Φ and a point z B , then the limit
t t≥0 n
{ } ∈
Φ (z) z
t
G(z):= lim −
t→0 t
exists and the vector field G : B Cn, called the infinitesimal generator1 of Φ , is a holomor-
n t
phic function (see, e.g., [Aba92]). →We denote by Inf(B ) the set of all infinitesimal generators of
n
semigroupsinB . Foranyz B ,themapw(t):=Φ (z)isthesolutionoftheinitialvalueproblem
n n t
∈
dw(t)
=G(w(t)), w(0)=z. (1.1)
dt
There are various characterizations of holomorphic functions G : B Cn that are infinitesimal
n
→
generators;see [RS05] (Section 7.3), [BCDM10] (Theorem 0.2), [BES14] (p. 193).
The set Inf(D), i.e. all infinitesimal generatorsin the unit disc, canbe characterizedcompletely by
the Berkson-Portarepresentation formula (see [BP78]):
Inf(D)= z (τ z)(1 τz)p(z) τ D, p (D,C)withRe(p(z)) 0for allz D . (1.2)
{ 7→ − − ∈ ∈H ≥ ∈ }
Remark 1.2. Let F : D D be a holomorphic self-map. Recall the Denjoy-Wolff theorem (see,
→
e.g., [RS05], Theorem 5.1): If F is not an elliptic automorphism (i.e. an automorphism with ex-
actly one fixed point in D), then there exists one point τ D (the Denjoy-Wolff point of F) such
that the iterates Fn converge locally uniformly in D to the∈constant map τ.
If Φ is a semigroup on D, then we call τ D the Denjoy-Wolff point of Φ if τ is the
t t≥0 t t≥0
{ } ∈ { }
Denjoy-Wolff point of Φ , which is equivalent to lim Φ =τ locally uniformly.
1 t→∞ t
Ifaninfinitesimalgeneratorintheunitdiscdoesnotgenerateasemigroupofellipticautomorphisms
of D, then the point τ D from formula (1.2) is exactly the Denjoy-Wolff point of the semigroup.
∈
There are two special cases of infinitesimal generatorsin D that have been studied intensively and
turned out to be quite useful in Loewner theory and its applications. The two different cases arise
from certain normalizations of the Berkson-Porta data τ and p from formula (1.2). In the radial
case, one considers those elements G Inf(D) whose Berkson-Portadata τ and p satisfy
∈
τ =0 and p(0)=1,
i.e. G(z)= zp(z).
−
This class plays a central role in studying the class S of all univalent functions f : D C with
→
f(0)=0, f′(0)=1, by the powerful tools of Loewner’s theory; see, e.g., [Pom75], Chapter 6. The
classofradialgeneratorsaswellastheclassS havebeengeneralizedinthiscontexttothepolydisc
Dn, see [Por87a, Por87b], and to the unit ballB , see [GK03] for a collectionof severalresults and
n
references.
Thesecondclass,thesetofallchordal generators2,consistsofallG Inf(D)whoseBerkson-Porta
∈
data τ and p satisfy
p(z)
τ =1 and ∠lim is finite.
z→1z 1
−
The aim of this paper is to introduce a generalizationof the chordal class for the unit ball B .
n
1Thereisnostandardconventionintheliteratureandoften−Giscalledtheinfinitesimalgeneratorofthesemigroup.
2 Note that there is no standard use of the words “radial” and “chordal” in the literature. In [CDMG10], e.g., an
element G∈Inf(D) is called radial if τ ∈D and chordal if τ ∈∂D.
2
1.2 The hydrodynamic normalization in one dimension
Instead of fixing an interior point, like in the class S, it can be of interest to investigate univalent
self-mappings of D that fix a boundary point. In this case, one usually passes from D to the upper
half-plane H= z C Im(z)>0 .
{ ∈ }
A class of such mappings that is easy to describe and that appears in several applications is the
set of all univalent mappings f : H H that fix the boundary point and have the so called
→ ∞
hydrodynamic normalization. Basicpropertiesofthisclasscanbefoundin[GB92],seealso[Bau05]
and [CDMG10]. One of its main applications is the chordal Loewner equation, see [ABCDM10],
Section 4, for further references.
A univalent function f :H H has hydrodynamic normalization (at ) if f has the expansion
→ ∞
c
f(z)=z +γ(z),
− z
where c 0, which is usually called half-plane capacity, and γ satisfies ∠lim z γ(z)=0.
z→∞
≥ ·
We denote by P the set of all these functions. Then P is a semigroup and the functional l :P
→
[0, ), l(f)=c, is additive: If f ,f P, then f f P and l(f f )=l(f )+l(f ).
1 2 1 2 1 2 1 2
∞ ∈ ◦ ∈ ◦
Remark 1.3. Let f P with l(f) = c. If we transfer f to the unit disc by conjugation by the
Cayley transform, then∈we obtain a function f˜:D D having the expansion
→
c
f˜(z)=z (z 1)3+γ˜(z),
− 4 −
where ∠lim γ˜(z) =0.
z→1 (z−1)3
If Φ is a one-real-parametersemigroup contained in P with l(Φ )=a, then it is easy to see
t t≥0 1
{ }
that l(Φ )=a t. If H is the generator of this semigroup, then we also define l(H):=a.
t
·
We will be interested in the following set of chordal generators.
Definition1.4. By (H,c)wedenotethesetofallinfinitesimalgeneratorsH ofone-realparameter
K
semigroups Φ contained in P with l(H) c.
t t≥0
{ } ≤
Remark 1.5. The set (H,c) can be characterized in various ways; see [GB92], Section 1 and
K
[Maa92], Proposition 2.2.
It is known that H (H,c) for some c 0 if and only if H maps H into H and
∈K ≥
limsupy H(iy) c. (1.3)
| |≤
y→∞
In fact, l(H)=limsup y H(iy).
y→∞ | |
Furthermore, this is equivalent to: H maps H into H and
c
H(z) (1.4)
| |≤ Im(z)
for all z H. The number l(H) is the smallest constant such that this inequality holds.
∈
Finally, it is known that this property is equivalent to the fact that G is the Cauchy transform of
a finite, non-negative Borel measure µ on R, i.e. −
µ(du)
H(z)= . (1.5)
R u z
Z −
The number l(H) can be calculated by l(H)=µ(R).
Remark 1.6. It is easy to see that the following holds: if f P with c=l(f), then H :=f id
(H,c) with l(H)=c. ∈ − ∈
K
Let C :H D, C(z)= z−i, be the Cayley map. We define (D,c) by
→ z+i K
(D,c)= C′(C−1) (H C−1) H (H,c) .3
K { · ◦ | ∈K }
Therestofthispaperisorganizedasfollows: InSection2welookforaninvariantcharacterization
of chordal generators, i.e. of the sets (H,c) and (D,c), and we introduce the class (B ,c) for
n
K K K
the higher dimensional unit ball. It will turn out to be quite useful to study “slices” of this class,
whichisdone inSection3. InSection4 weintroduce andstudythe classP ,ahigher dimensional
n
analog of the class P.
3If {Φt}t≥0 is a semigroup in H with generator H, then {C◦Φt◦C−1}t≥0 is a semigroup in D and its generator is
given by C′(C−1)·(H◦C−1).
3
2 Chordal generators in higher dimensions
2.1 Invariant formulation for (D,c) and (H,c)
K K
For R>0 we let ED(1,R) be the horodisc in D with center 1 and radius R, i.e.
1
ED(1,R)= z D <R ,
∈ | uD(z)
(cid:26) | | (cid:27)
where uD(z)= 1−|z|2 is the Poissonkernel in D with respect to 1.
−|1−z|2
ByusingtheCayleymapwedefineanalogouslyEH( ,R)=C−1(ED(1,R))= z H 1 <R .
∞ { ∈ |Im(z) }
Forz Dandatangentvectorv Cwedenoteby v D,z thehyperboliclengthofv (withcurvature
∈ ∈ | |
-1), i.e.
2v
|v|D,z := 1 | z| 2.
−| |
Furthermore, we let RD(z) be the radius R of the horodisc ED(1,R) that satisfies z ∂E(1,R);
in short RD(z) = 1 . Analogously, for z H and v C, we define RH(z) := 1/Im∈(z) and the
|uD(z)| ∈ ∈
hyperbolic length v H,z := v /Im(z).
| | | |
Accordingto(1.4)weknowthatH (H,c)ifandonlyifH mapsHintoHand H(z) c/Im(z)
for all z H. By using the Berkson∈-PKorta formula it is easy to see that we can|rephr|a≤se this to:
H (H∈,c) if and only if H Inf(H) and H(z) c/Im(z) for all z H.
∈K ∈ | |≤ ∈
The last inequality is equivalent to H(z)/Im(z) c/Im(z)2 or
| | ≤
c
|H(z)|H,z ≤ Im(z)2 =c·RH(z)2.
IfwepassfromHtoDandtransformH intoG=C′(C−1)(H C−1),thenGsatisfies G(C(z))D,C(z) =
· ◦ | |
H(z)H,z and we immediately get the following characterization.
| |
Proposition 2.1. Let G Inf(D). Then
∈
G (D,c) G(z)D,z c RD(z)2 for all z D.
∈K ⇐⇒ | | ≤ · ∈
Let H Inf(H). Then
∈
H (H,c) H(z)H,z c RH(z)2 for all z H.
∈K ⇐⇒ | | ≤ · ∈
2.2 Chordal generators in the unit ball
For n N, let u be the pluricomplex Poisson kernel in B with pole at e :=(1,0,...,0), i.e.
n n 1
∈
1 z 2
uBn,p =− 1−kzk2.
1
| − |
The level sets of uBn are exactly the boundaries of horospheres with center e1, more precisely, the
set
EBn(e1,R):={z ∈Bn||uBn(z)|−1 <R},R>0,
is the horosphere with center e and radius R.
1
Furthermore, for z ∈Bn and v ∈Cn we denote by kvkBn,z the Kobayashi-hyperbolic length of the
vector v with respect to z.
Motivated by Proposition 2.1, we make the following definition.
Definition 2.2. Letc 0. Wedefinetheclass (B ,c)tobethesetofallinfinitesimalgenerators
n
G on B such that for a≥ll z B the following iKnequality holds:
n n
∈
c
kG(z)kBn,z ≤ uB (z)2. (2.1)
n
Remark2.3. (B ,c)is acompact family: Montel’s theorem andthedefinition of (B ,c)imme-
n n
diately imply thKat it is a normal family. If a sequence (G ) (B ,c) converges loKcally uniformly
n n
to G : B Cn, then G is holomorphic and also an infinit⊂esiKmal generator which can be seen by
n
→
using the characterization given in [BCDM10], Theorem 0.2. Of course, G also satisfies (2.1) and
we conclude G (B ,c).
n
∈K
4
Just as we passed from D to H in one dimension, we can pass from the unit ball B to the Siegel
n
upper half-space H = (z ,z˜) Cn Im(z )> z˜ 2 in order to get simpler formulas:
n 1 1
{ ∈ | k k }
The Cayley map
z i 2z 2z
C :H B , C(z)=(C (z),...,C (z))= 1− , 2 ,..., n ,
n n 1 n
→ z +i z +i z +i
(cid:18) 1 1 1 (cid:19)
mapsH biholomorphicallyontoB . Itextendstoahomeomorphismfromtheone-pointcompact-
n n
ification H =H ∂H of H ∂H to the closure of Bn.
n n n n n
∪ ∪{∞} ∪
The pluricomplex Poisson kernel transforms as follows:
b
uHn(z):=uBn(C(z))=−Im(z1)+kz˜k2.
Thus, we define the horosphere EH ( ,R) with center and radius R>0 by
n ∞ ∞
1
EHn(∞,R):={z ∈Hn|Im(z1)−kz˜k2 > R}.
For v ∈Cn and z ∈Hn we let kvkHn,z be the Kobayashihyperbolic length of v.
Let c 0. We define the class (H ,c) to be the set of all infinitesimal generators H on H
n n
≥ K
satisfying the inequality
c
kH(z)kHn,z ≤ uH (z)2
n
for all z H . Then we have
n
∈
(B ,c)= C′(C−1) (H C−1) H (H ,c) .
n n
K { · ◦ | ∈K }
From now on we will stay in the upper half-space H , where most of the computations we need
n
take a simpler form.
3 Slices
3.1 Normalized geodesics and slices
For any H Inf(H ) one can consider one-dimensional slices by using the so called Lempert pro-
n
∈
jection devices; see [BS14], Section 3.
If w H , then there exists a unique complex passing through w and . Let us choose a
n
param∈etrizationϕ:H H ofthisgeodesic. Thereexistsauniqueholomorph∞icmapP :H H
n n n
with P2 =P and P ϕ→=ϕ. Define P˜ =ϕ−1 P. Then →
◦ ◦
h :H C, h (ζ)=dP˜(ϕ(ζ)) H(ϕ(ζ)),
ϕ ϕ
→ ·
is an infinitesimal generator on H; see [BS14], p. 6.
We will need special parametrizations of these geodesics: In [BP05], p. 516, it is shown that for
any complex geodesic ϕ:H H with ϕ( )= , there exists a >0 such that
n ϕ
→ ∞ ∞
uHn(ϕ(ζ))=aϕ·uH(ζ)
for all ζ H. Call a geodesic ϕ:H H normalized if ϕ( )= and a =1.
n ϕ
∈ → ∞ ∞
Lemma 3.1. Let a C and γ Cn−1 such that (a,γ) H . Then the map
n
∈ ∈ ∈
ϕ :H H , ϕ (ζ):=(ζ+i γ 2,γ),
γ n γ
→ k k
is a normalized geodesic through (a,γ). Furthermore, if H = (H ,H˜) Inf(H ), then the slice
1 n
∈
h :=h of H with respect to ϕ is given by
γ ϕγ γ
h (ζ)=H (ϕ (ζ)) 2iγT H˜(ϕ (ζ)). (3.1)
γ 1 γ γ
− ·
Proof. Let ψ :D B be a complex geodesic with ψ(1)=e . As a parametrizationfor ψ one can
n 1
choose(see Sectio→n 3 in[BS14]) ψ(ζ)=(α2(ζ 1)+1,α(ζ 1)β), whereα>0 andβ Cn−1 such
− − ∈
that β 2 =1 α2. Then C−1(ψ(ζ))=(i2+α2(ζ−1),iβ/α) and
k k − α2(1−ζ)
ζ+i ζ 1 α2 ζ β 2
ζ C−1(ψ(C (ζ)))=( i+ ,iβ/α)=( +i − ,iβ/α)=( +i ,iβ/α)
7→ 1 − α2 α2 α2 α2 α
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13)
5 (cid:13) (cid:13)
is a complex geodesic from H to H . A reparametrization [ζ/α2 to ζ] and setting γ = iβ/α gives
n
the geodesic
ϕ (ζ)=(ζ +i γ 2,γ). (3.2)
γ
k k
This complex geodesic is normalized because it satisfies ϕ ( )= and
γ
∞ ∞
uHn(ϕγ(ζ))=Im(ζ+ikγk2)−kγk2 =Im(ζ)=uH(ζ).
The projection onto ϕ (H) is given by
γ
P(z ,z˜)=(z 2iγT z˜+2i γ 2,γ). (3.3)
1 1
− · k k
Clearly, P is holomorphic and maps H onto ϕ (H) because
n γ
Im(z 2iγT z˜+2i γ 2) = Im(z ) 2Im(iγT z˜)+2 γ 2
1 1
− · k k − · k k
z˜ 2 2 γ z˜ + γ 2+ γ 2 =( γ z˜ )2+ γ 2 γ 2.
≥ k k − k kk k k k k k k k−k k k k ≥k k
Furthermore,
(P P)(z ,z˜)=(z 2iγTz˜+2i γ 2 2iγTγ+2i γ 2,γ)=(z 2iγTz˜+2i γ 2,γ)=P(z ,z˜).
1 1 1 1
◦ − k k − k k − k k
Thus, the inverse P˜ :H H,P˜ =ϕ−1 P, is given by P˜(z ,z˜)=(z 2iγTz˜+i γ 2).
2 → γ ◦ 1 1− k k
If H(z)=(H (z),H˜(z)) is a generator on H , we get the slice reduction
1 n
h (ζ)=dP˜(ϕ (ζ)) H(ϕ (ζ))=H (ϕ (ζ)) 2iγT H˜(ϕ (ζ)).
ϕγ γ · γ 1 γ − · γ
3.2 Some explicit formulas
Later on we will need explicit formulas of the Kobayashinorms of dP(z)H(z) and H(z) dP(z)
− ·
H(z). The following lemma is proven in the Appendix.
Lemma 3.2. Let a C,p,v Cn−1 and z =(z ,z˜) H . Then the following formulas hold:
1 n
∈ ∈ ∈
a a
= | | , (3.4)
(cid:13)(cid:18)0(cid:19)(cid:13)Hn,z |uHn(z)|
(cid:13) (cid:13)
(cid:13) (cid:13)
(cid:13) (cid:13) T
2ipTv v 2 uH (z) + (p z˜) v 2
=2 k k | n | | − | , (3.5)
(cid:13)(cid:18) v (cid:19)(cid:13)Hn,z q |uHn(z)|
(cid:13) (cid:13)
T(cid:13) (cid:13)T 2 T 2 T 2
a 2iz˜ v(cid:13) 2iz˜(cid:13)v a 2iz˜ v 2iz˜ v
− + = − + . (3.6)
(cid:13) 0 v (cid:13) (cid:13) 0 (cid:13) (cid:13) v (cid:13)
(cid:13)(cid:18) (cid:19) (cid:18) (cid:19)(cid:13)Hn,z (cid:13)(cid:18) (cid:19)(cid:13)Hn,z (cid:13)(cid:18) (cid:19)(cid:13)Hn,z
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
By using Lem(cid:13)ma 3.2 we obtain the foll(cid:13)owing ex(cid:13)plicit expressi(cid:13)ons. (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13) (cid:13) (cid:13)
Lemma 3.3. Let H = (H ,H˜) Inf(H ) and fix z H . Denote by P the projection onto the
1 n n
∈ ∈
complex geodesic through z and . Then the following formulas hold:
∞
dP(z) H(z)=(H (z) 2iz˜TH˜(z),0), H(z) dP(z) H(z)=(2iz˜TH˜(z),H˜(z)). (3.7)
1
· − − ·
Furthermore,
H(z) 2 = dP(z) H(z) 2 + H(z) dP(z) H(z) 2 , (3.8)
k kHn,z k · kHn,z k − · kHn,z
H (z) 2iz˜TH˜(z)
1
kdP(z)H(z)kHn,z = | u−H (z) |, (3.9)
| n |
H˜(z)
kH(z)−dP(z)·H(z)kHn,z =2 kuH (zk) . (3.10)
| n |
Proof. The formulas for dP(z)H(z) and H(z) dP(z)H(z)pfollow from the explicit form (3.3).
Equation (3.8) follows from (3.6) with a=H (−z) and v =H˜(z).
1
Furthermore, equation (3.9) follows directly from (3.4) with a = H (z) 2iz˜TH˜(z) and equation
1
(3.10) from (3.5) by setting p=z˜and v =H˜. −
6
3.3 Slices of generators in (Hn,c) and examples
K
Proposition 3.4. Let c 0 and H (H ,c). Then every normalized slice h of H belongs to
n γ
(H,c). ≥ ∈ K
K
Proof. Fix γ Cn−1 and ζ H and let z =ϕ (ζ).
γ
Furthermore,∈let P be the p∈rojection onto ϕ (H). Now we write H(z) as
γ
H(z)=dP(z) H(z)+(H(z) dP(z)H(z)).
· −
As H (H ,c), equation (3.8) implies
n
∈K
c2
H(z) 2 = dP(z) H(z) 2 + H(z) dP(z)H(z) 2 .
k kHn,z k · kHn,z k − kHn,z ≤ uH (z)4
n
In particular,
c
kdP(z)·H(z)kHn,z ≤ uH (z)2. (3.11)
n
By the definition of the slice h we have
γ
dP(ϕ (ζ)) H(ϕ (ζ))=(dϕ )(ζ) h (ζ)
γ γ γ γ
· ·
and consequently
kdP(ϕγ(ζ))·H(ϕγ(z))kHn,ϕγ(ζ) =k(dϕγ)(ζ)·hγ(ζ)kHn,ϕγ(ζ) =|hγ(ζ)|H,ζ.
The last equality holds as ϕ is a complex geodesic. Equation (3.11) implies
γ
c c
|hγ(ζ)|H,ζ ≤ uHn(ϕγ(ζ))2 = uH(ζ)2,
where the last equality holds as ϕ is normalized. Hence, h (H,c).
γ γ
∈K
Remark3.5. IftwoholomorphicfunctionsH ,H :H Cnhavethesameslices,i.e. dP(z)H (z)=
1 2 n 1
dP(z)H (z) for all z H , then H =H ; see the proof→of Theorem 3.2 in [Cas10].
2 n 1 2
∈
Example 3.6. The family Φt(z) = (z1,e−it/z1z2) t≥0 is a semigroup on H2. Its generator H is
{ }
given by
z
2
H(z ,z )=(0, i ).
1 2
− z
1
Thus, for γ C the slice h has the form
γ
∈
γ 2γ 2
h (z)= 2iγ i = − | | .
γ − ·− z+iγ 2 z+iγ 2
| | | |
Consequently,the limit lim y h(iy) =2γ 2 exists, but does not havean upper bound that is
y→∞
independent of γ. Proposition 3.4·i|mplies| that| f|or any c 0, H (H ,c). ⋆
2
≥ 6∈K
Example 3.7. Let
−1
H :H2 →C2, H(z1,z2)= zz12 .
(cid:18)2z2(cid:19)
1
For γ C the slice h is given by
γ
∈
1 γ ζ 2iγ 2 ( ζ 2iγ 2)(ζ2 2iγ 2ζ γ 4)
h (ζ)= − 2iγ = − − | | = − − | | − | | −| | .
γ ζ+iγ 2 − · 2(ζ+iγ 2)2 (ζ+iγ 2)2 ζ+iγ 2 4
| | | | | | | | | |
Let us write ζ =x+iy; x R,y (0, ). Then a small calculation gives
∈ ∈ ∞
y(x2+y2)+4y2 γ 2+5y γ 4+2γ 6
Im(h (ζ))= | | | | | | >0.
γ ζ+iγ 2 4
| | | |
Furthermore,
limsupy h (iy) =1.
γ
| |
y→∞
7
Hence, h (H,1). So each slice is an infinitesimal generator in H and by [BS14], Proposition
γ
3.8, the fun∈ctKion H is an infinitesimal generator in H .
2
Now let (z ,z ) H and write z = x+iy, x,y R. Then we get (an explicit formula of the
1 2 2 1
∈ ∈
Kobayashimetric is given in the appendix)
x2+y2+3z 2y x2+y2+3y2 x2+4y2
uH2(z)4·kH(z)k2H2,z =(y−|z2|2)2· (x2+y2|)22| y≥≤|z2|2 y2· (x2+y2)2 ≤ x2+y2 ≤4.
Consequently, H (H ,2). ⋆
2
∈K
Question 3.8. Let H : H Cn be an infinitesimal generator. Assume there exists c 0 such
n
that h (H,c) for every γ→ Cn−1. Does this imply that H (H ,C) for some C c≥?
γ n
∈K ∈ ∈K ≥
4 Univalent functions with hydrodynamic normalization
Motivatedby Remark1.6 we define the followinggeneralizationofthe class P,where idstands for
the identity mapping on H .
n
Definition 4.1.
P := f :H H f is univalent and f id (H ,c) for some c 0 .
n n n n
{ → | − ∈K ≥ }
Remark 4.2. It is important to note that if f :H H is a holomorphic self-mapping, then the
n n
→
map f id is automatically an infinitesimal generator; see [RS05], p. 207.
−
4.1 Basic properties of P
n
The following proposition summarizes some basic properties of P .
n
Proposition 4.3.
a) P contains no automorphism of H except the identity.
n n
b) Letα:H H beanautomorphismofH withα( )= .Iff P ,thenα−1 f α P .
n n n n n
→ ∞ ∞ ∈ ◦ ◦ ∈
c) Let f ∈Pn. Then f(EHn(∞,R))⊂EHn(∞,R) for every R>0.
d) Let f P and write f(z)=z+H(z) with H =(H ,H˜) (H ,c). Then
n 1 n
∈ ∈K
H˜(z) 2 H (z) 2iz˜TH˜ for all z =(z ,z˜) H . (4.1)
1 1 n
k k ≤| − | ∈
e) Let f ∈Pn. Then there exists R>0 such that EHn(∞,R)⊂f(Hn).
Proof. The statements a) and b) can easily be shownby using the explicit form of automorphisms
of H ; see Proposition 2.2.4 in [Aba89].
n
The statement c) is just Julia’s lemma: Write f(z)=z+H(z) and let us pass to the unit ball and
define f˜:B B ,f˜=C f C−1. Then
n n
→ ◦ ◦
1 (1 z )H (C−1(z))
f˜= − 1 1 +2iz .
2i+H (C−1(z)) z H (C−1(z)) 2(1 z )H˜(C−1(z))
1 − 1 1 (cid:20)(cid:18) − 1 (cid:19) (cid:21)
By taking the sequence z =(1 1/n,0) it is easy to see that
n
−
1 f˜(z )
lim f˜(z )=e and lim −k n k =1,
n 1
n→∞ n→∞ 1 zn
−k k
i.e. e isaboundaryregularfixedpointoff˜withboundarydilatationcoefficient 1.Julia’slemma
1
I(nseeequTahlietoyredm)f2o.l2lo.w21sidnir[eActblay8f9r]o)mimcp):lieLsetthza=t f˜((zE1,Bz˜n)(e∈1,HRn).)A⊂nEotBhne(re1f,oRrm)ufolartiaonny≤oRfc>)i0s.−uHn(z+
H(z)) uH (z), or more explicitly
≥− n
Im(z )+Im(H (z)) z˜+H˜(z) 2 Im(z ) z˜ 2
1 1 1
−k k ≥ −k k
Im(H (z)) z˜+H˜(z) 2 z˜ 2 =2Re(z˜TH˜(z))+ H˜(z) 2
1
⇐⇒ ≥k k −k k k k
Im(H (z) 2iz˜TH˜(z)) H˜(z) 2.
1
⇐⇒ − ≥k k
From this inequality it follows that H˜(z) 2 H (z) 2iz˜TH˜ for all z H .
1 n
k k ≤| − | ∈
8
Finally we prove e):
Let f P and write f(z) = z+H(z) with H (H ,c). Because of c) f maps the horosphere
n n
∈ ∈ K
EH ( ,1) into itself. Hence the statement is proven if we can show that uH is bounded on
n ∞ n
f(∂EH ( ,1)).
Let z ∈nH∞n with z ∈∂EHn(∞,1), i.e. |uHn(z)|=1. Furthermore, we choose ζ ∈H and γ ∈C such
that ϕγ(ζ)=z. Note that this implies uH(ζ) =Im(ζ)=1.
Let P be the projection onto ϕ (H). | |
γ
Then we have uH (f(z)) = uH (z + H(z)) = uH (z+dP(z)H(z)+H(z) dP(z)H(z)). As
| n | | n | | n − |
=:w =:v
dP(z) dP(z)=dP(z), we have dP(z) v =0. A small calculation (see also Lemma 3.1 in [Cas10])
TgiwCv∂esEvH·n∈(∞T,z|Cu∂HEnH(wn()∞|−,11).).AFsuErtHhne(r∞m,o|ru·eH,na(lwso)|−w1)∈=ϕ{γz(H∈|)Hann|d{|uzdHPn((zz))}|=>||duPH(nw(w))a|{}nzdiswcoengve}etxvthi∈s
implies
|uHn(w+v)|≤|uHn(w)|=|uHn(z+dP(z)H(z))|Lemm=a3.3|uHn(z+(hγ(ζ),0))|
= Im(z ) z˜ 2+Im(h (ζ)) Im(z ) z˜ 2+ h (ζ)
1 γ 1 γ
−k k ≤ −k k | |
c
= |uHn(z)|+|hγ(ζ)|=1+|hγ(ζ)|≤1+ Im(ζ) =1+c.
Consequently, f(Hn)⊃f(EHn(∞,1))⊃EHn(∞,1+c).
Theorem 4.4. P is a semigroup: If f,g P , then f g P .
n n n
∈ ◦ ∈
Proof. Let f,g P with F =(F ,F˜):=f id,G=(G ,G˜):=g id and
n 1 1
∈ − −
c d
kF(z)kHn,z ≤ uH (z)2, kG(z)kHn,z ≤ uH (z)2
n n
for all z H . Let z =(z ,z˜) H and p=(p ,p˜):=z+G(z).
n 1 n 1
FromRe∈mark4.2weknowthat∈f g idisaninfinitesimalgeneratoronH .Itremainstoestimate
n
◦ −
the hyperbolic metric of this generator. We have
k(f ◦g)(z)−zkHn,z =kG(z)+F(z+G(z))kHn,z
d
≤ kG(z)kHn,z +kF(z+G(z))kHn,z ≤ uH (z)2 +kF(p)kHn,z
n
d
≤ uH (z)2 +k(F1(p)−2ip˜TF˜(p),0)kHn,z +k(2ip˜TF˜(p),F˜(p))kHn,z.
n
Note that F (p) 2ip˜TF˜(p) corresponds to the slice of F with respect to the geodesic through p
1
−
and infinity. Because of Proposition 3.4 we know that
c c
F (p) 2ip˜TF˜(p) ,
1
| − |≤ uH (p) ≤ uH (z)
| n | | n |
where the second inequality follows from Proposition 4.3 c). Together with equation (3.4), this
implies
(F (p) 2ip˜TF˜(p) c
k(F1(p)−2ip˜TF˜(p),0)kHn,z = | 1 u−H (z) | ≤ uH (z)2. (4.2)
| n | n
It remains to show that there exists a constant C >0 such that
C
k(2ip˜TF˜(p),F˜(p))kHn,z ≤ uH (z)2.
n
First, equation (3.5) gives
k(2ip˜TF˜(p),F˜(p))kHn,z =2qkF˜(p)k2|uHn(uzH)|(+z)|(p˜−z˜)TF˜(p)|2
| n | (4.3)
F˜(p) 2 uH (z) + (p˜ z˜) 2 F˜(p) 2 F˜(p)
2 k k | n | k − k ·k k =2 k k uH (z) + G˜(z) 2.
≤ q uH (z) uH (z) | n | k k
| n | | n |q
9
Now we differentiate between two cases. Case 1: uH (z) 1.
| n |≥
The equations (3.8) and (3.10) imply √2k|uF˜H(np()pk)| ≤kF˜(p)kHn,p ≤ uHnc(p)2, thus
c c
F˜(p) . (4.4)
k k≤ 2uH (p)3/2 ≤ 2uH (z)3/2
| n | | n |
In the same way we get
d
G˜(z) . (4.5)
k k≤ 2uH (z)3/2
| n |
Combining (4.4) with (4.3) gives
c
k(2ip˜TF˜(p),F˜(p))kHn,z ≤ uH (z) uH (z)3/2 |uHn(z)|+kG˜(z)k2
| n || n | q
c G˜(z) 2 c d2 c 1+ d42
= 1+ k k 1+ .
|uHn(z)|2s |uHn(z)| (4≤.5) |uHn(z)|2s 4|uHn(z)|4 ≤ |uqHn(z)|2
Case 2: uH (z) 1.
| n |≤
From equation (4.2) we know that F (p) 2ip˜TF˜(p) c and equation (4.1) implies
| 1 − |≤ |uHn(z)|
√c
F˜(p) .
k k≤ uH (z)
| n |
Similarly we get p
√d
G˜(z) .
k k≤ uH (z)
| n |
Hence, we obtain with (4.3): p
√c
k(2ip˜TF˜(p),F˜(p))kHn,z ≤2 uH (z)3/2 |uHn(z)|+kG˜(z)k2
| n | q
√c d √c
2 uH (z) + =2 uH (z)2+d
≤ uH (z)3/2s| n | uH (z) uH (z)2 n
| n | | n | | n |
p
√c
2 √1+d.
≤ uH (z)2
| n |
4.2 Semigroups with generators in (Hn,c)
K
Theorem 4.5. Let Φ be a semigroup on H with generator H (H ,c). Then Φ P
t t≥0 n n t n
{ } ∈ K ∈
for every t 0.
≥
Proof. Firstly, for every t≥0 and R>0, the map Φt maps the horosphere EHn(∞,R) into itself,
i.e.
|uHn(Φt(z))|≥|uHn(z)| (4.6)
for every z H . This can be seen as follows:
n
∈
Let G be the corresponding generator in the unit ball, i.e. G = C′(C−1) (H C−1). Then G
· ◦
satisfies the inequality
c c1 z 4
1
kG(z)k≤kG(z)kBn,z ≤ uB (z)2 = (1| −z 2|)2.
n −k k
Putting z =r e gives
1
· c(1 r)4 c(1 r)2
G(re ) − = − .
k 1 k≤ (1 r2)2 (1+r)2
−
From this it follows immediately that
G (re )
1 1
lim G(re )=0 and lim =0.
1
(0,1)∋r→1 (0,1)∋r→1 r 1
−
10