Proper holomorphic mappings in the special class of Reinhardt domains

POLONICIMATHEMATICI

92.3(2007)

Proper holomorphi mappings in the spe ial

lass of Reinhardt domains

by

Łukasz Kosiński

(Kraków)

Abstra t. A omplete hara terization of proper holomorphi mappings between

domainsfromthe lassofallpseudo onvexReinhardtdomainsin

C ²

^withthelogarithmi imageequaltoastriporahalf-planeisgiven.

1.Statement of results. Weadoptthestandardnotationsof omplex

analysis. Given

γ = (γ ₁ , γ ₂ ) ∈ R ²

^and

z = (z ₁ , z ₂ ) ∈ C ²

^{we put}

|z ^γ | =

|z 1 | ^{γ 1} |z 2 | ^{γ 2}

^{whenever it makes sense. The unit dis in}

C

^{is denoted by}

D

andthesetofproperholomorphi mappingsbetweendomains

D, G ⊂ C ⁿ

^is

denotedby

Prop(D, G).

Inthispaperwe dealwiththosepseudo onvexReinhardt domainsin

C ²

whose logarithmi image is equal to a strip or a half-plane. Observe that

su h domainsarealwaysalgebrai ally equivalent to domainsof theform

D _α,r − ,r ⁺ := {z ∈ C ² : r ⁻ < |z ^α | < r ⁺ },

where

α = (α ₁ , α ₂ ) ∈ (R ² ) _∗ , 0 < r ⁺ < ∞, −∞ < r ⁻ < r ⁺ .

We say that

D _α,r − ,r ⁺

^{is of the irrational type if}

α ₁ /α ₂ ∈ R \ Q.

^{In the}

other ase itis ofthe rational type.

Re all that if

r ⁻ < 0 < r ⁺

^and

α ∈ (R ² ) _∗ ,

^then

D _α,r − ,r ⁺

^{are so- alled}

elementary Reinhardt domains.

Below we shall give a omplete des ription of all proper holomorphi

mappings from

D _α,r −

1 ,r ₁ ⁺

^to

D _β,r −

2 ,r ⁺ ₂

^{for arbitrary}

α, β ∈ (R ² ) _∗

^and

0 <

r _i ⁺ < ∞, −∞ < r ⁻ _i < r _i ⁺ , i = 1, 2.

^{Similar problems have been studied in}

theliterature.In[Shi1℄and[Shi2℄theproblemofholomorphi equivalen eof

elementary Reinhardt domainswas onsidered. Those results werepartially

extended by A. Edigarian and W. Zwonek [Edi-Zwo ℄ who gave a hara -

2000Mathemati sSubje tClassi ation:32H35,32A07.

Keywordsand phrases:properholomorphi mappings,Reinhardtdomains,elemen-

taryReinhardtdomains.

[285℄

^InstytutMatematy znyPAN,2007

terization of proper holomorphi mappings between elementary Reinhardt

domainsof therationaltype.

Set

A(̺ ⁻ , ̺ ⁺ ) := {z ∈ C : ̺ ⁻ < |z| < ̺ ⁺ }

^for

̺ ⁺ > 0, ̺ ⁻ < ̺ ⁺

^and

A _̺ := A(1/̺, ̺), ̺ > 1.

^{Moreover, put}

D _γ,r := {z ∈ C ² : 1/r < |z ₁ | |z 2 | ^γ < r}, γ ∈ R _∗ , r > 1, D γ := {z ∈ C ² : |z 1 | |z 2 | ^γ < 1}, γ ∈ R ∗ ,

D ^∗ _γ := {z ∈ C ² : 0 < |z ₁ | |z 2 | ^γ < 1}, γ ∈ R _∗ .

Notethatif

γ

^{is rational,i.e.}

γ = p/q

^{for some relatively prime}

p, q ∈ Z

^,

q > 0,

^then

D _γ,r

^is biholomorphi ally equivalent to

A _r q × C ∗

^and

D ^∗ _γ

^is

biholomorphi allyequivalent to

D _∗ × C.

^{Indeed, put}

ψ(z ₁ , z ₂ ) := (z ^q ₁ z ^p ₂ , z ₁ ^m z ₂ ⁿ )

^for

(z ₁ , z ₂ ) ∈ C ² ,

where

m, n ∈ Z

^{aresu hthat}

pm−qn = 1.

^{One an he kthatthemappings}

ψ| _{D γ,r} : D _γ,r → A _r ^q × C _∗

^and

ψ| _{D γ} ^∗ : D _γ ^∗ → D _∗ × C _∗

^arebiholomorphi .

Moreover, onemayeasily prove that

D _α,r − ,r ⁺

^isalgebrai allyequivalent to adomain of one ofthe following types:

(i) If

r ⁻ > 0

^:

(a)

A _̺ × C

^,

α ₁ α ₂ = 0,

(b)

A _̺ × C _∗

^,

α ₁ /α ₂ ∈ Q _∗ ,

( )

D γ,̺

^,

γ = α 2 /α 1 ∈ R \ Q

^.

(ii) If

r ⁻ = 0

^:

(a)

D _∗ × C

^,

α ₁ α ₂ = 0,

(b)

D _∗ × C ∗

^,

α 1 /α 2 ∈ Q ∗ ,

( )

D _γ ^∗

^,

γ = α ₂ /α ₁ ∈ R \ Q.

(iii) If

r ⁻ < 0

^:

(a)

D × C

^,

α 1 α 2 = 0,

(b)

D _γ

^,

γ = α ₂ /α ₁ 6= 0.

Ourmain resultisthefollowing:

Theorem 1.

(a)If

α ∈ R \ Q,

^{then thesetof proper} holomorphi mappings from

D _α,r

to

D _β,R

^{isnon-empty if andonly if}

(1)

log R

log r ∈ Z + βZ, α log R

log r ∈ Z + βZ.

(b) Let

α, β ∈ R\Q

^{and let}

r, R > 1

^{be su h that}

^{log R} _{log r} = k 1 + l 1 β

^and

α ^{log R} _{log r} = k ₂ + l ₂ β

^{for some integers}

k _i , l _i , i = 1, 2.

^{Then any proper}

holomorphi mapping

f : D α,r → D _β,R

^{is of one of the following}

(2)

( f (z) = (az ₁ ^{k 1} z ₂ ^{k 2} , bz ₁ ^{l 1} z ₂ ^{l 2} ),

f (z) = (az ₁ ^{−k 1} z ₂ ^{−k 2} , bz ₁ ^{−l 1} z ₂ ^{−l 2} ), z = (z ₁ , z ₂ ) ∈ D _α,r ,

where

a, b ∈ C

^satisfy

|a| |b| ^β = 1.

^{Moreover, any of the mappings}

given by (2)is proper.

Noti ethatinTheorem 1(a)we do notdemand

β

^tobe irrational.

Using Theorem 1 we will easily obtain analogous results for domains of

theforms (ii)and (iii)of the irrationaltype.

Theorem2. Let

α, β ∈ R \ Q.

^{The setof proper} holomorphi mappings from

D _α ^∗

^to

D ^∗ _β

^{is non-empty if and only if}

α = (k 2 + βl 2 )/(k 1 + βl 1 )

^for

some

k _i , l _i ∈ Z

^,

i = 1, 2.

^{Moreover, in that ase, if}

k ₁ + l ₁ β > 0,

^{then any}

proper holomorphi mapping

f : D _α ^∗ → D ^∗ _β

^{is of the form}

(3)

f (z ₁ , z ₂ ) = (az ₁ ^{k 1} z ^k ₂ ² , bz ₁ ^{l 1} z ₂ ^{l 2} ), (z ₁ , z ₂ ) ∈ D ^∗ _α ,

where

a, b ∈ C

^satisfy

|a| |b| ^β = 1.

Theorem3. Let

α, β ∈ R \ Q.

^{Then theset}

Prop(D _α , D _β )

^isnon-empty

if and only if

α = (k ₂ + βl ₂ )/(k ₁ + βl ₁ )

^{for some}

k _i , l _i ∈ Z ≥0 , i = 1, 2.

Moreover, in that ase any proper holomorphi mapping

f : D _α → D _β

^{is of}

the form

(4)

f (z ₁ , z ₂ ) = (az ₁ ^{k 1} z ^k ₂ ² , bz ₁ ^{l 1} z ₂ ^{l 2} ), (z ₁ , z ₂ ) ∈ D _α ,

where

a, b ∈ C

^{are su h that}

|a| |b| ^β = 1.

Nextweprove the following

Theorem 4. Let

α, β ∈ (R ² ) ∗ , r ⁺ _i > 0, r ⁻ _i < r ⁺ _i , i = 1, 2.

^Assume

that the sets

D _α,r −

1 ,r ₁ ⁺ , D _β,r −

2 ,r ⁺ ₂

^{are of the same type (either rational or}

irrational).If thereexistsaproper holomorphi mappingbetweenthem,then

either

r ⁻ ₁ r ₂ ⁻ > 0

^or

r ₁ ⁻ = r ⁻ ₂ = 0.

For domainsof dierent typeswe havethefollowing result:

Theorem 5. Let

α, β ∈ (R ² ) _∗ , r ⁺ _i > 0, r ⁻ _i < r _i ⁺ , i = 1, 2.

^{If the}

sets

D _α,r −

1 ,r ⁺ ₁

^and

D _β,r −

2 ,r ⁺ ₂

^{are of dierent types, then there is no proper}

holomorphi mapping betweenthem.

Finally, we dis uss the rational ase. As already mentioned, the set of

proper holomorphi mappings between elementary Reinhardt domains of

the rational type was des ribed in [Edi-Zwo ℄. Thus, in order to obtain the

desired hara terization, itsu esto prove the following threetheorems.

Theorem 6. Let

r, R > 1.

^If

R 6= r ^m

^{for any natural number}

m,

^then

Prop(A r × C, A R × C), Prop(A r × C ∗ , A R × C)

^and

Prop(A r × C ∗ , A R × C ∗ )

are empty. Moreover,for any

m ∈ N :

(a)

Prop(A r × C, A r ^m × C)

^{onsistsof the mappings of the form}

A _r × C ∋ (z, w) 7→ (e ^iθ z ^εm , a N (z)w ^N + · · · + a 0 (z)) ∈ A r ^m × C,

where

θ ∈ R, N ∈ N, ε = ±1

^and

a ₀ , . . . , a _N ∈ O(A r )

^{aresu h that}

|a 0 (z)| + · · · + |a _N (z)| > 0, z ∈ A _r .

(b)

Prop(A _r × C ∗ , A _r ^m × C)

^{onsistsof the mappings of theform}

A _r × C _∗ ∋ (z, w) 7→



e ^iθ z ^εm , a _N (z)w ^N + · · · + a 0 (z) w ^k



∈ A _r ^m × C,

where

θ ∈ R

^,

k, N ∈ N

^,

0 < k < N

^,

ε = ±1

^and

a _i ∈ O(A r )

^,

i = 1, . . . , N

^{, satisfy}

|a 0 (z)| + · · · + |a _k−1 (z)| > 0

^and

|a k+1 (z)| +

· · · + |a N (z)| > 0

^for

z ∈ A r .

( )

Prop(A _r × C ∗ , A _r ^m × C ∗ )

^{onsistsof the mappings of the form}

A _r × C ∗ ∋ (z, w) 7→ (e ^iθ z ^m , a(z)w ^k ) ∈ A _r ^m × C ∗ ,

where

ε = ±1, θ ∈ R, k ∈ N

^and

a ∈ O(A _r , C _∗ ).

Theorem 7. There are no proper holomorphi mappings from

A _r × C

to

A _R × C ∗

^{for any}

r, R > 1.

Theorem 8.

(a)

Prop(D _∗ × C, D ∗ × C)

^{onsistsof the mappings of the form}

D _∗ × C ∋ (z, w) 7→ (e ^iθ z ^m , a _N (z)w ^N + · · · + a ₀ (z)) ∈ D _∗ × C,

where

θ ∈ R

^,

N ∈ N

^,

m ∈ N

^and

a ₀ , . . . , a _N ∈ O(D ∗ )

^{are su h that}

|a 0 (z)| + · · · + |a N (z)| > 0, z ∈ D ∗ .

(b)

Prop(D _∗ × C ∗ , D _∗ × C)

^{onsistsof the mappings of the form}

D _∗ × C ∗ ∋ (z, w) 7→



e ^iθ z ^m , a _N (z)w ^N + · · · + a ₀ (z) w ^k



∈ D ∗ × C,

where

θ ∈ R

^,

m ∈ N

^,

k, N ∈ N

^,

0 < k < N

^and

a i ∈ O(D ∗ )

^,

i = 1, . . . , N

^satisfy

|a 0 (z)| + · · · + |a _k−1 (z)| > 0

^and

|a k+1 (z)| +

· · · + |a N (z)| > 0

^for

z ∈ D _∗ .

( )

Prop(D _∗ × C _∗ , D _∗ × C _∗ )

^{onsistsof the mappings of the form}

D _∗ × C ∗ ∋ (z, w) 7→ (e ^iθ z ^m , a(z)w ^k ) ∈ D ∗ × C ∗ ,

where

θ ∈ R, k ∈ N

^and

a ∈ O(D ∗ , C ∗ ).

(d)

Prop(D _∗ × C, D ∗ × C ∗ ) = ∅

^.

2. Proofs. The following result is probably known. However, we ould

not ndit inthe literature, so we present a proof.

Lemma 9. Let

D ⊂ C ⁿ

^{be a domain,}

α ∈ R \ Q

^{and let}

f, g : D → C

be holomorphi mappings with

|f (z)| = |g(z)| ^α , z ∈ D.

^{Then either}

f =

g = 0

^on

D

^{or there exists a} holomorphi bran h of the logarithm of

g

^{, i.e.}

a mapping

ψ ∈ O(D)

^{su h that}

e ^ψ = g

^on

D.

^In parti ular, there exists a

θ ∈ R

^{su h that}

f = e ^iθ+αψ

^on

D.

Proof. Comparing multipli ities of the roots of the fun tions

f

^and

g

omposed with ane mappings we may redu e our onsiderations to the

ase when

f, g : D → C _∗ .

^{Moreover, we may assume that}

g(x ^′ ) ∈ R _>0

^for

some

x ^′ ∈ D.

Obviously, there exists an

η ∈ R

^{su h that the set}

G η := {z ∈ D : e ^iη f (z) ∈ g(z) ^α }

^{is non-empty.} Considering, if ne essary, a mapping

e ^iη f

instead of

f

^{we mayassumethat}

η = 0.

Itiseasytoseethat

G ₀

^isanopen-and- losedsubsetof

D

^,andso

G ₀ = D.

Thus,thereexistsaholomorphi bran hof

g ^α

^{(alsodenotedby}

g ^α

^{)su hthat}

g ^α (x ^′ ) ∈ R _>0

^{.Itfollowsthatthereexist}

g ^t

^forany

t ∈ Q := {k+lα : k, l ∈ Z}.

Fixasequen e

(t m ) ^∞ _m=1 ⊂ Q

^{onverging to}

0

^{.InvirtueofMontel'stheorem,}

itis lear that

g ^{t m} → 1

^{lo ally uniformly.}

Put

ψ m := (g ^{t m} − 1)/t m .

^Then

lim m→∞ ψ m (x ^′ ) = log g(x ^′ )

^{and the}

sequen e

(ψ ^′ _m ) ^∞ _m=1 = (g ^{t m −1} g ^′ ) ^∞ _m=1

^{is lo ally uniformly onvergent with}

limit

(1/g)g ^′ .

^Thus

(ψ _m ) ^∞ _m=1

^{onverges lo ally uniformly on}

D

^{. Denote its}

limit by

ψ.

^{By the} Weierstrass theorem,

ψ

^is holomorphi on

D

^and

ψ ^′ = lim _m→∞ ψ _m ^′ = (1/g)g ^′ .

Let

D ⊂ D e

^{beanysimply onne ted}neighborhoodof

x ^′ .

^Let

ψ e

^beaholo-

morphi mapping on

D e

^{su h that}

g| _{D e} = e ^{ψ e}

^and

ψ(x e ^′ ) = log g(x ^′ ).

^{Itis easy}

to seethat

ψ = ψ e

^on

D, e

^{andso, bythe identity prin iple,}

g = e ^ψ

^on

D.

Lemma 10. Let

0 < r ⁺ _i , −∞ < r _i ⁻ < r ⁺ _i , i = 1, 2, α, β ∈ R.

^Let

(λ _n ) ^∞ _n=1 ⊂ A(r ⁻ ₁ , r ⁺ ₁ ).

^Let

φ ∈ Prop(D _(1,α),r −

1 ,r ⁺ ₁ , D _(1,β),r −

2 ,r ⁺ ₂ )

^{. Put}

v(λ) := |φ ₁ (λ, 1)| |φ ₂ (λ, 1)| ^β , λ ∈ A(r ⁻ ₁ , r ⁺ ₁ ).

If the sequen e

(λ _n ) ^∞ _n=1

^{has no} a umulation points in

A(r ⁻ ₁ , r ₁ ⁺ ),

^then

(v(λ _n )) ^∞ _n=1

^{has no} a umulation points in

A(r ₂ ⁻ , r ⁺ ₂ ).

Proof. Assume that

v(λ n ) → q.

^{It su esto show that}

q ∈ ∂A(r ⁻ ₂ , r ⁺ ₂ ).

Otherwise

q ∈ A(r ⁻ ₂ , r ⁺ ₂ ).

^{Notethat for any}

λ ∈ A(r ₁ ⁻ , r ₁ ⁺ )

^{thefun tion}

(5)

u _λ : C ∋ z 7→ |φ 1 (λe ^−αz , e ^z )| |φ 2 (λe ^−αz , e ^z )| ^β

isbounded andsubharmoni , so

u _λ

^{is onstant.}

Sin e

φ

^{is proper, the mapping}

C ∋ z 7→ φ ₂ (λ _n e ^−αz , e ^z ) ∈ C

^{is non-}

onstant for any

n ∈ N.

^{Pi ard's theorem implies that there is a sequen e}

(z _n ) ^∞ _n=0 ⊂ C

^{su h that}

|φ 2 (λ _n e ^{−αz n} , e ^{z n} )| ^β = 1.

^Obviously

u _λ (z) = u _λ (1) = v(λ)

^forall

z ∈ C

^and

v(λ _n ) → q,

^so

|φ 1 (λ _n e ^{−αz n} , e ^{z n} )| → q.

^Inparti ular,the set

{φ(λ n e ^{−αz n} , e ^{z n} ) : n ∈ N}

^{isrelatively ompa tin}

D _(1,β),r −

2 ,r ₂ ⁺

^;however,

((λ n e ^{−αz n} , e ^{z n} )) ^∞ _n=1

^{has no} a umulation points in

D _(1,α),r −

1 ,r ₁ ⁺

^{,a ontradi -}

Corollary11. Let

φ = (φ 1 , φ 2 ) : D α,r → D _β,R

^beaproperholomorphi mapping and let

α, β ∈ R _>0 , r, R > 1

^{. Put}

v(λ) := |φ ₁ (λ, 1)| |φ ₂ (λ, 1)| ^β , λ ∈ A _r .

^{Then either}

|λ|→1/r lim v(λ) = 1/R, lim

|λ|→r v(λ) = R or lim

|λ|→1/r v(λ) = R, lim

|λ|→r v(λ) = 1/R.

Lemma 12. Let

α ∈ R \ Q, β ∈ R, −∞ < r ⁻ _i < r _i ⁺ < ∞, 0 < r _i ⁺ , i = 1, 2,

^{and let}

φ : D _(1,α),r −

1 ,r ⁺ ₁ → D _(1,β),r ⁻

2 ,r ₂ ⁺

^{be a} holomorphi mapping. Then for any

λ ∈ A(r ₁ ⁻ , r ⁺ ₁ )

^,

φ({(z 1 , z 2 ) ∈ C ² : |z 1 | |z 2 | ^α = |λ|})

⊂ {(w 1 , w ₂ ) ∈ C ² : |w ₁ | |w 2 | ^β = |φ ₁ (λ, 1)| |φ ₂ (λ, 1)| ^β }.

Proof. Note thatfor any

λ ∈ A(r ⁻ ₁ , r ₁ ⁺ )

^{the fun tion}

u : C ∋ z 7→ |φ 1 (λe ^αz , e ^−z )| |φ 2 (λe ^αz , e ^−z )| ^β

issubharmoni and bounded. Hen e

u

^{is onstant.}

Invirtueof Krone ker's theorem, theset

{(|λ|e ^αz , e ^−z ) : z ∈ C}

^isdense

in

{(z 1 , z ₂ ) ∈ C ² : |z ₁ | |z 2 | ^α = |λ|}.

^{Thus, thereis}

t ∈ R

^{su h that}

φ({(z ₁ , z ₂ ) ∈ C ² : |z ₁ | |z 2 | ^α = |λ|}) ⊂ {(w ₁ , w ₂ ) ∈ C ² : |w ₁ | |w 2 | ^β = t}.

It iseasy to seethat

t = |φ ₁ (λ, 1)| |φ ₂ (λ, 1)| ^β .

Proof of Theorem 1(a). Let

φ : D _α,r → D β,R

^{be a proper} holomorphi mapping. Put

v(λ) := |φ ₁ (λ, 1)| |φ ₂ (λ, 1)| ^β , λ ∈ A _r .

^Obviously,

log v

^{is a}

harmoni fun tion.ApplyingCorollary11andHadamard'stheoremweinfer

that

v(λ) = |λ| ^{log R log r} , λ ∈ A _r

^or

v(λ) = |λ| ^{− log R log r} , λ ∈ A _r .

From thisand Lemma12 we easily on lude thatthereis

ε = ±1

^{su hthat}

(6)

|φ 1 (z)| |φ ₂ (z)| ^β = |z ₁ | ^{ε log R log r} |z 2 | ^{εα log R log r} , z ∈ D _α,r .

Let

z 2 = 1, z 1 = z ∈ A r , ψ i (z) := φ i (z, 1), i = 1, 2.

^Then

log(ψ 1 (z)ψ ₁ (z)) + β log(ψ 2 (z)ψ ₂ (z)) = ε log R

log r log(zz).

Dierentiatingwithrespe tto

z

^{we get}

(7)

ψ ₁ ^′ (z)

ψ 1 (z) + β ψ ₂ ^′ (z)

ψ 2 (z) = ε log R log r · 1

z , z ∈ A _r .

It follows that

Ind(ψ ₁ ◦ γ; 0) + β Ind(ψ 2 ◦ γ; 0) = ε log R log r ,

where

γ

^{is the unit ir le. Hen e}

^{log R} _{log r} ∈ Z + βZ.

^{The same argument with}

respe t to these ondvariableshows that

α ^{log R} _{log r} ∈ Z + βZ.

To provethe onverse, assumethat

log R

log r = k ₁ + l ₁ β, α log R

log r = k ₂ + l ₂ β,

where

k _i , l _i ∈ Z

^,

i = 1, 2.

^Dene

φ ₁ (z) := z ₁ ^{k 1} z ₂ ^{k 2} , φ ₂ (z) := z ₁ ^{l 1} z ₂ ^{l 2}

^for

z = (z ₁ , z ₂ ) ∈ C ²

^{, and}

φ := (φ ₁ , φ ₂ ).

^{Observe that}

φ| _{D α,r} ∈ Prop(D α,r , D _β,R ).

Indeed,it iseasy to he k that

(8)

|φ 1 (z)| |φ ₂ (z)| ^β = |z ₁ | log R/log r |z 2 | α log R/log r , (z ₁ , z ₂ ) ∈ D _α,r ,

so

φ| _{D α,r} ∈ O(D α,r , D _β,R ).

^{Sin e}

k ₁ l ₂ 6= k 2 l ₁ , φ

^{is a proper} holomorphi mappingfrom

(C _∗ ) ²

^{intoitself(see[Zwo ,Theorem2.1℄).Nowwe}immediately on lude from (8) that

φ| D α,r

^{is a proper} holomorphi mapping from

D α,r

to

D _β,R

^.

Lemma9and (6)leadto the following

Corollary 13. Let

α, β ∈ R \ Q

^and

φ ∈ Prop(D _α,r , D _β,R ).

^Assume

that

log R

log r = k ₁ + l ₁ β

^and

α ^{log R} _{log r} = k ₂ + l ₂ β

^forsome

k _i , l _i ∈ Z, i = 1, 2.

^Then

there are

θ ∈ R, ψ ∈ O(D _α,r )

^and

ε ∈ {1, −1}

^{su h that}

φ(z) = (z ₁ ^{εk 1} z ₂ ^{εk 2} e ^iθ e ^−βψ(z) , z ₁ ^{εl 1} z ₂ ^{εl 2} e ^ψ(z) ), z ∈ D α,r .

Remark

14

.

We may always assume that

ε

^{in Corollary 13 is equal}

to

1

^{(repla ing if ne essary}

φ

^by

φ ◦ h,

^where

h ∈ Aut(D _α,r )

^,

h(z ₁ , z ₂ ) :=

(z ₁ ⁻¹ , z ⁻¹ ₂ )

^).

To prove Theorem 1(b) we need the following notation. Put

X _α,r :=

{z ∈ C ² : − log r < Re z ₁ + α Re z ₂ < log r}

^and

Π(z ₁ , z ₂ ) := (e ^{z 1} , e ^{z 2} )

^for

(z ₁ , z ₂ ) ∈ C ² .

^{It is lear that}

(X _α,r , Π)

^{istheuniversal overing of}

D _α,r .

Lemma 15. Let

α, β ∈ R \ Q, r, R > 1

^{and assume that}

^{log R} _{log r} = k ₁ + l ₁ β, α ^{log R} _{log r} = k ₂ + l ₂ β,

^where

k _i , l _i ∈ Z, i = 1, 2.

^Let

f : D _α,r → D β,R

^{be a}

proper holomorphi mapping. Then every ontinuous lifting of the mapping

f ◦ Π : X α,r → D _β,R

^{is proper and} holomorphi .

Proof. By Corollary 13 and Remark 14 we may assume that

f (z) = (z ₁ ^{k 1} z ₂ ^{k 2} e ^{−βψ(z)+iθ} , z ^l ₁ ¹ z ^l ₂ ² e ^ψ(z) ) (z ∈ D _α,r ),

^where

θ ∈ R

^and

ψ ∈ O(D _α,r ).

Let

f e

^{be any ontinuous lifting of}

f ◦ Π : X _α,r → D β,R ,

^{that is,}

f : e X _α,r → X β,R

^and

f ◦ Π = Π ◦ e f .

^{Itis obviousthat}

f e

^is holomorphi .Then bytheidentityprin iple

(9)

( f e ₁ (z) = k ₁ z ₁ + k ₂ z ₂ − βψ(e ^{z 1} , e ^{z 2} ) + iθ + 2µ ₁ πi,

f e 2 (z) = l 1 z 1 + l 2 z 2 + ψ(e ^{z 1} , e ^{z 2} ) + 2µ 2 πi, z ∈ X α,r ,

for some

µ _i ∈ Z, i = 1, 2.

Suppose that

f e

^{is not proper, i.e. there is a sequen e}

(z ^m ) ^∞ _m=1 ⊂ X α,r

^,

z ^m = (z ₁ ^m , z ₂ ^m )

^,

m ∈ N

^{, without any} a umulation pointsin

X _α,r

^{su hthat}

( e f (z ^m )) ^∞ _m=1

^{is onvergent in}

X _β,R .

^Put

y ₀ := lim _m→∞ f (z e ^m ) ∈ X _β,R .

Obviously,

f (Π(z ₁ ^m , z ₂ ^m )) = Π( e f (z ^m ₁ , z ₂ ^m )) → Π(y 0 ).

^{Sin e}

f

^{is proper,}

theset

{Π(z ^m ) : m ≥ 1}

^{isrelatively ompa tin}

D _α,r .

^{Thuswemayassume}

that

(Π(z ^m )) ^∞ _m=1

^is onvergent, sayto

w ₀ ∈ D α,r .

^{From(9)wededu ethat}

(k 1 z ^m ₁ +k 2 z ₂ ^m ) ^∞ _m=1

^and

(l 1 z ₁ ^m +l 2 z ₂ ^m ) ^∞ _m=1

^{are onvergentin}

C ² .

^Thus

(z ^m ) ^∞ _m=1

isalso onvergent.

Put

z ₀ := lim _m→∞ z ^m .

^{Now it su es to observe that}

Π(z ₀ ) = w ₀ ∈ D _α,r ,

^so

z ₀ ∈ X α,r ;

^a ontradi tion.

Nowwe are able to give a des ription of theset of proper holomorphi

mappingsbetween the domains

D _α,r

^and

D _β,R

^{of theirrationaltype.}

Proof ofTheorem1(b). Let

f ∈ Prop(D _α,r , D _β,R )

^{.InvirtueofCorollary}

13 andRemark 14wemayassume that

f (z) = (z ₁ ^{k 1} z ₂ ^{k 2} e ^{−βψ(z)+iθ} , z ^l ₁ ¹ z ^l ₂ ² e ^ψ(z) ), z = (z 1 , z 2 ) ∈ D α,r ,

for some

θ ∈ R

^and

ψ ∈ O(D _α,r ).

^{Our aimis to showthat}

ψ

^{is onstant.}

To simplifynotation, for

γ ∈ R

^put

Λ _γ : C ² ∋ (z 1 , z ₂ ) 7→ (z ₁ + γz ₂ , z ₂ ) ∈ C ² .

It is lear that

Λ _γ (X _γ,̺ ) = S _̺ × C

^,

̺ > 1,

^where

S _̺ := {z ∈ C : − log r <

Re z < log r}.

^Moreover,

Λ _γ

^isbiholomorphi withinverse

Λ ⁻¹ _γ = Λ _−γ .

Notethatthe mapping

f : X e _α,r → X β,R

^{given by}

f (z) = (k e ₁ z ₁ + k ₂ z ₂ − βψ(e ^{z 1} , e ^{z 2} ) + iθ, l ₁ z ₁ + l ₂ z ₂ + ψ(e ^{z 1} , e ^{z 2} ))

isaliftingof

f ◦Π.

^{ThusLemma15impliesthat}

f e

^isproperandholomorphi . Put

H := (H 1 , H 2 ) := Λ _β ◦ e f ◦ Λ ⁻¹ _α : S r × C → S R × C.

^Obviously,

H

^is

properand holomorphi .

Applyingtherelations

log R

log r = k ₁ + l ₁ β

^,

α ^{log R} _{log r} = k ₂ + l ₂ β

^{we seethat}

(10) H(z) = (z ₁ (k ₁ + βl ₁ ) + iθ, l ₁ z ₁ + z ₂ (l ₂ − l 1 α) + ψ(e ^{z 1 −αz 2} , e ^{z 2} )),

z ∈ S _r × C.

Hen e for any

z ₁ ∈ S r

^{the mapping}

C ∋ z 7→ H 2 (z ₁ , z) ∈ C

^{is proper}

andholomorphi .Consequently,duetothe formofproperholomorphi self-

mappingsof

C,

^{there isa polynomial}

p = p _{z 1} ∈ P(C)

^{su h that}

H ₂ (z ₁ , z) = p(z).

^{Therefore, the polynomial}

q(z) := q _{z 1} (z) := p(z) − l ₁ z ₁ − z(l 2 − l 1 α)

satises theequation

(11)

ψ(e ^{z 1} e ^−αz , e ^z ) = q(z), z ∈ C.

Noti ethat

{(e ^{z 1} e ^−α2πim , e ^2πim ) : m ∈ N}

^{isa relatively ompa tsubset}

of

D α,r

^{and the sequen e}

{q(2πim)} ^∞ _m=1

^{isbounded.Thus the polynomial}

q

Put

c(z 1 ) := ψ(e ^{z 1 −αz 2} , e ^{z 2} ), z 1 ∈ S r .

^{Fix any}

1 < ̺ < R

^{and take}

a onstant

M = M (̺) > 0

^{su hthat}

|c(x)| < M

^forevery

x ∈ [− log ̺, log ̺].

Let

λ ∈ ̺D \ (1/̺)D

^{be arbitrary. Note that for any}

z ₂ ∈ C

^{we have}

|ψ(|λ|e ^{−αz 2} , e ^{z 2} )| = |c(log |λ|)| < M.

^ApplyingKrone ker's theoremweinfer thattheset

{(|λ|e ^−αz , e ^z ) : z ∈ C}

^isdensein

{(z 1 , z ₂ ) ∈ C ² : |z ₁ | |z 2 | ^α = |λ|}.

Consequently,

ψ| _{D α,̺}

^isbounded.

Nowitsu es to repeatthe proof of Lemma2.7.1 of [Jar-P1℄inorder

toshowthateveryboundedholomorphi mappingon

D _α,̺

⁽ⁱⁿparti ular,

ψ

⁾

is onstant.

On the other hand, we have already mentioned in the proof of Theo-

rem 1(a)thatanymapping given by(2)isproper.

Proof of Theorems 2and 3. We proveboththeoremsimultaneously.Let

f : D _α → D β

(respe tively,

f : D ^∗ _α → D _β ^∗

^)beaproperholomorphi fun tion.

We aim at redu ing the situationto that ofTheorem 1.Takeany

r > 1.

From Lemma 12 we seethat for any

t ∈ [0, 1)

^(resp.

t ∈ (0, 1)

^{) there is}

an

s(t) ∈ [0, 1)

^(resp.

s(t) ∈ (0, 1)

^{) su hthat}

f ({(z ₁ , z ₂ ) ∈ C ² : |z ₁ | |z 2 | ^α = t}) ⊂ {(w ₁ , w ₂ ) ∈ C ² : |w ₁ | |w 2 | ^β = s(t)}.

Note that

s(|λ|) = |f 1 (λ, 1)| |f 2 (λ, 1)| ^β

^{and the fun tion}

v

^{given by}

v : D ∋ λ 7→ s(|λ|) ∈ [0, 1]

^(resp.

v : D _∗ ∋ λ 7→ s(|λ|) ∈ [0, 1]

^{) is radial and}

subharmoni on

D

^{(inthe se ond asewemayremove the}singularityat

0

^).

Themaximumprin iple applied to

v

^impliesthat

s

^isin reasing.

In parti ular,

f | _D

(1,α),1/r2,1 : D _(1,α),1/r 2 ,1 → D _(1,β),1/R ² _,1

^{is proper for}

some

R > 1

^{. For}

̺ > 1

^put

Λ e ̺ : C ² ∋ (z 1 , z 2 ) 7→ (̺z 1 , z 2 ) ∈ C ²

^and

dene

ψ := e Λ _R ◦ f ◦ e Λ ⁻¹ _r | D α,r .

^{Note that}

ψ ∈ Prop(D _α,r , D _β,R )

^{. Applying}

Theorem 1 we ndthat

log R

log r = k ₁ + l ₁ β, α ^{log R} _{log r} = k ₂ + l ₂ β

^and

ψ(z ₁ , z ₂ ) = (az ^εk ₁ ¹ z ₂ ^{εk 2} , bz ₁ ^{εl 1} z ₂ ^{εl 2} )

^{for some}

k _i , l _i ∈ Z, i = 1, 2, ε = ±1

^and

a, b ∈ C

satisfying

|a| |b| ^β = 1.

^Oviously

α = (k 2 + l 2 β)/(k 1 + l 1 β)

^{andbytheidentity}

prin iple we obtain

(12) f (z 1 , z 2 ) = (ar ^{εl 1 β} z ^εk ₁ ¹ z ₂ ^{εk 2} , br ^{−εl 1} z ₁ ^{εl 1} z ₂ ^{εl 2} ),

(z ₁ , z ₂ ) ∈ D _α (

^resp.

(z ₁ , z ₂ ) ∈ D _α ^∗ ).

If

f : D _α ^∗ → D _β ^∗ ,

^{then it su es to noti e that}

|f 1 (z)| |f ₂ (z)| ^β = (|z ₁ | |z 2 | ^α ) ^{εk 1 +εl 1 β} , z = (z ₁ , z ₂ ) ∈ D ^∗ _α ,

^{hen e}

ε(k ₁ + l ₁ β) > 0.

When

f : D α → D _β ,

^{we on lude that}

εk i , εl i ≥ 0, i = 1, 2.

Hen ewe easily gettherequired formulas.

On the other hand, one an he k that any of the mappings given in

Theorem 3is proper (sin e

α

^isirrational,

k ₁ l ₂ − k 2 l ₁ 6= 0

^).

Lemma 16. Let

r ⁺ > 0

^,

r ⁻ < r ⁺

^,

t ∈ R.

^{Suppose that the fun tion}

v : A(r ⁻ , r ⁺ ) → [−∞, t)

^is subharmoni , radial (i.e.

v(|λ|) = v(λ)

^,

λ ∈

A (r ⁻ , r ⁺ )

^{) and harmoni on the set}

{z ∈ A(r ⁻ , r ⁺ ) : v(z) 6= −∞}.

^Then

there exist

a, b ∈ R

^{su h that}

v(λ) = a log |λ| + b, λ ∈ A(r ⁻ , r ⁺ ).

Proof. Itsu estoobserve thatsin e

v

^isradial,

A(r ⁻ , r ⁺ ) \ {0} ⊂ {z ∈ A(r ⁻ , r ⁺ ) : v(z) 6= −∞}

^{(and next one maypro eedina standard way,i.e.}

solve aneasy dierential equation).

Proofof Theorem4. First, onsiderthe asewhen

D _α,r −

1 ,r ⁺ ₁

^and

D _β,r − 2 ,r ⁺ ₂

are of the irrational type. Then we may assume that

α = (1, α ₁ )

^{for some}

α 1 ∈ R \ Q.

^Let

v : A(r ⁻ ₁ , r ₁ ⁺ ) ∋ λ 7→ log |ψ ₁ (λ, 1)| ^{β 1} |ψ 2 (λ, 1)| ^{β 2} ∈ R.

By Lemma12 we seethat

ψ({(z ₁ , z ₂ ) ∈ C ² : |z ₁ | |z ₂ | ^α = |λ|} ⊂ {(w ₁ , w ₂ ) ∈ C ² : |w 1 | ^{β 1} |w 2 | ^{β 2} = e ^v(λ) }.

^Therefore,

v

^{isradial.Observemoreoverthat}

v

^is

subharmoni on

A(r ⁻ ₁ , r ₁ ⁺ )

^{and harmoni on}

{λ ∈ A(r ₁ ⁻ , r ⁺ ₁ ) : v(λ) > −∞}.

Sin e

ψ

^issurje tive, we on lude that

(13)

v(A(r ₁ ⁻ , r ₁ ⁺ )) =

 (log r ₂ ⁻ , log r ⁺ ₂ )

^if

r ⁻ ₂ ≥ 0, [−∞, log r ⁺ ₂ )

^if

r ⁻ ₂ < 0

(we put

log 0 := −∞

^{). However, by Lemma 16,}

v(λ) = a log |λ| + b, λ ∈ A(r ⁻ ₁ , r ₁ ⁺ )

^{,for some}

a, b ∈ R,

^{whi h easilynishes theproof inthis ase.}

Nowsupposethat

D _α,r −

1 ,r ⁺ ₁

^and

D _β,r −

2 ,r ⁺ ₂

^{areoftherationaltype;wemay}

assume that

β = (p, q) ∈ Z ²

^and

α = (1, α 1 )

^{for some}

α 1 ∈ Q.

^Applying

Lemma10 one ansee thatthemapping

A(r ₁ ⁻ , r ₁ ⁺ ) ∋ λ 7→ ψ ₁ (λ, 1) ^p ψ ₂ (λ, 1) ^q ∈ A(r ₂ ⁻ , r ⁺ ₂ )

isproper.Hen ethis asefollows dire tlyfromtheform ofthesetof proper

holomorphi mappingsfrom

A(r ₁ ⁻ , r ₁ ⁺ )

^to

A(r ₂ ⁻ , r ₂ ⁺ ).

Proof of Theorem 5. Assume that

D _α,r −

1 ,r ⁺ ₁

^{is of the rational type and}

D _β,r −

2 ,r ⁺ ₂

^{is of theirrationaltype;without loss of generality}

α = (1, p/q)

^for

some

p, q ∈ Z

^and

β = (1, β 2 )

^{for some}

β 2 ∈ R \ Q

^.

Suppose that

ψ ∈ Prop(D _α,r −

1 ,r ⁺ ₁ , D _β,r −

2 ,r ⁺ ₂ )

^{. Note that for any}

λ ∈ A (r ⁻ ₁ , r ₁ ⁺ )

^{the mapping}

(14)

u _λ : C ∗ ∋ z 7→ |ψ 1 (λz ^p , z ^−q )| |ψ 2 (λz ^p , z ^−q )| ^{β 2}

is onstant.Fix

λ ₀

^and

c 6= 0

^{su hthat}

u _{λ 0} ≡ c.

^{One an seethat}

C _∗ ∋ z 7→

ψ _i (λ ₀ z ^p , z ^−q ) ∈ C _∗

^{is a proper} holomorphi self-mapping of

C _∗ , i = 1, 2.

Therefore,thereare

a _i ∈ C ∗

^and

µ _i ∈ Z ∗ , i = 1, 2,

^{su hthat}

ψ _i (λ ₀ z ^p , z ^−q ) = a i z ^{µ i}

^for

z ∈ C ∗

^,

i = 1, 2.

^{Applying (14)it is lear that}

|a 1 | |a 2 | ^{β 2} |z| ^{µ 1 +µ 2 β 2}

= c

^for

z ∈ C _∗ .

^Inparti ular,

β ₂ ∈ Q

^,a ontradi tion.

Now, suppose that there exists

ψ ∈ Prop(D _β,r −

2 ,r ⁺ ₂ , D _α,r −

1 ,r ⁺ ₁ ).

^Put

u(λ) := |ψ 1 (λ, 1)| |ψ 2 (λ, 1)| ^{β 2}

^for

λ ∈ A(r ₂ ⁻ , r ⁺ ₂ ).

Applying Lemmas 10 and 12 we nd that

u

^{satises the} assumptions of Lemma 16. Thus, there are

a, b ∈ R

^{su h that}

log u(λ) = a log |λ| + b

for

λ ∈ A(r ₂ ⁻ , r ⁺ ₂ ).

^In parti ular,

u

^{is either stri tly in reasing or stri tly}

de reasing. Takeany

̺ ⁻ ₂

^,

̺ ⁺ ₂

^{su hthat}

̺ ⁻ ₂ > max{0, r ⁻ ₂ }, ̺ ⁺ ₂ < r ₂ ⁺ , ̺ ⁻ ₂ < ̺ ⁺ ₂ .

Put

̺ ⁻ ₁ := min{u(̺ ⁻ ₂ ), u(̺ ⁺ ₂ )}, ̺ ⁺ ₁ := max{u(̺ ⁻ ₂ ), u(̺ ⁺ ₂ )}.

^Then

ψ| D

β,̺− 2 ,̺ + 2

: D _β,̺ −

2 ,̺ ⁺ ₂ → D _(1,α),̺ − 1 ,̺ ⁺ ₁

isobviouslyaproperholomorphi mapping.InvirtueofTheorem1(a)there

are

k _i , l _i ∈ Z

^,

i = 1, 2,

^{su h that}

β = (k ₁ + l ₁ α)/(k ₂ + l ₂ α).

^In parti ular,

β ∈ Q,

^a ontradi tion.

Lemma 17. Let

A, B ⊂ C ⁿ

^{be domains andassume that}

B

^{is bounded.}

(a) Amapping

f : A×C _∗ → B ×C

^isproperandholomorphi ifandonly ifthere are

m ∈ Prop(A, B), k ∈ N, 0 < k < N, N ∈ N, a _i ∈ O(A)

^,

i = 1, . . . , N,

^with

|a ₀ (z)| + · · · + |a _k−1 (z)| > 0

^and

|a _k+1 (z)| + · · · +

|a N (z)| > 0

^for

z ∈ A

^,satisfying

f (z, w) =



m(z), a N (z)w ^N + · · · + a 0 (z) w ^k



, (z, w) ∈ A × C ∗ .

(b) A mapping

f : A × C → B × C

^isproperandholomorphi ifandonly ifthereare

a ₀ , . . . , a _N ∈ O(A)

^,

N ∈ N

^,with

|a 0 (z)|+· · ·+|a _N (z)| > 0

for

z ∈ A

^{, and there is a proper} holomorphi mapping

m : A → B

su h that

f (z, w) = (m(z), a _N (z)w ^N + · · · + a ₀ (z)), (z, w) ∈ A × C.

( ) A mapping

f : A × C _∗ → B × C

^{is proper and} holomorphi if and only ifthere are

m ∈ Prop(A, B), a ∈ O(A, C _∗ )

^and

k ∈ N

^{su hthat}

f (z, w) = (m(z), a(z)w ^k ), (z, w) ∈ A × C _∗ .

(d) There isno proper holomorphi mapping from

A × C

^to

B × C _∗

^.

Proof. First of all,noti e that for any

z ∈ A

^themapping

w 7→ f ₁ (z, w)

∈ C ⁿ

^{isbounded on}

C

^(or

C _∗

^{),so itis onstant.}

(a)Observe that

C _∗ ∋ w 7→ f 2 (z, w) ∈ C

^{isproperfor any}

z ∈ A.

^Thus,

for any

z ∈ A

^{there is a polynomial}

p(z, ·), p(z, 0) 6= 0,

^{and a natural}

k(z)

su h that

(15)

φ 2 (z, w) = p(z, w)

w ^k(z) , (z, w) ∈ A × C ∗ .

One an see that there is a

k

^{su h that}

k = k(z)

^for

z ∈ A

^{(use Rou he's}

theorem).Consequently,

p ∈ O(A × C _∗ )

^.

Fixanydomain

A ^′ ⊂⊂ A

^{and put}

A µ :=



z ∈ A ^′ : ∂ ^µ p

∂w ^µ (z, w) = 0

^forany

w ∈ C

 .

Theabove onsiderations implythat

S _∞

µ=1 A _µ = A ^′ .

^{Applying Baire'stheo-}

rem we nd that there exists

N ∈ N

^{su h that}

A _N

^{has non-empty interior.}

By theidentity prin iple,

A N = A.

Thus, there are holomorphi mappings

a ₀ , . . . , a _N : A → C

^{su h that}

p(z, w) = a _N (z)w ^N + · · · + a ₁ (z)w + a ₀ (z)

^for

(z, w) ∈ A × C,

^i.e.

(16)

f ₂ (z, w) = a _N (z)w ^N + · · · + a ₁ (z)w + a ₀ (z)

w ^k , (z, w) ∈ A × C.

By properness of

f ₂ (z, ·)

^{we on lude that}

0 < k < N

^{, and}

|a N (z)| + . . . +

|a _k+1 (z)| > 0

^and

|a _k−1 (z)| + · · · + |a ₀ (z)| > 0

^{for any}

z ∈ A.

Put

m(z) := f ₁ (z, 1), z ∈ A.

^{We laim that}

m

^isproper.

Indeed, take any sequen e

(z _n ) ^∞ _n=1

^{and assume that it has no a umu-}

lation points in

A.

^{We may assume that}

a 0 (z n ) 6= 0

^{for any}

n ∈ N

^{(if ne -}

essary repla e

a ₀

^with

a ₁

^{et .).Then there exists a sequen e}

(w _n ) ^∞ _n=1 ⊂ C ∗

su h that

a _N (z _n )w _n ^N + · · · + a ₁ (z _n )w _n + a ₀ (z _n ) = 0

^{for any}

n ∈ N.

^{Sin e}

f (z _n , w _n ) = (m(z _n ), 0),

^{it is obvious that}

(m(z _n )) ^∞ _n=1

^{has no} a umulation pointsin

B.

Conversely, one an he k that every mapping

f

^{dened in this way is}

proper.

(b)It iseasy to seethat

C ∋ w 7→ f 2 (z, w) ∈ C

^{is a proper} holomorphi mappingforany

z ∈ A.

^{Fromtheformofsu hmappingswe on ludethatfor}

every

z ∈ A

^{the mapping}

f ₂ (z, ·)

^{is a omplex} polynomial.Nowwe pro eed exa tly as intheproofof (a).

( )We pro eedsimilarly to theproofs of (a)and (b).

(d)Supposethat

f : A × C → B × C _∗

^{isa proper}holomorphi fun tion.

Fix

z ∈ A.

^Then

C ∋ w 7→ f 2 (z, w) ∈ C ∗

^isproper.

Take

ψ ∈ O(C)

^{su h that}

f ₂ (1, ·) = exp ◦ψ.

^{Observe that}

ψ

^{is a proper}

holomorphi self-mapping of the omplex plane, hen e

ψ

^{is a} polynomial.

From thisweeasily geta ontradi tion.

Proof of Theorems 6, 7 and 8. These are dire t onsequen es of Lem-

ma17.

Finally, Itake this opportunityto express mydeep gratitudeto Profes-

sor Wªodzimierz Zwonek for introdu ing me to the subje t and numerous

Referen es

[Edi-Zwo℄ A.EdigarianandW.Zwonek,Properholomorphi mappingsinsome lassof

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InstytutMatematyki

UniwersytetJagiello«ski

Reymonta4

30-059Kraków,Poland

E-mail:lukasz.kosinskiim.uj.edu.pl

Re eived27.7.2007

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