On h -extendible domains and associated models C.R. Acad. Sci. Paris, Ser.I

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Complex analysis

On h-extendible domains and associated models ^✩

Sur les domaines h-extensibles et les modèles associés

Feng Rong, Ben Zhang

DepartmentofMathematics,SchoolofMathematicalSciences,ShanghaiJiaoTongUniversity,800DongChuanRoad,Shanghai,200240, PR China

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received5April2016

Acceptedafterrevision18July2016 Availableonline29July2016 PresentedbyJean-PierreDemailly

A boundarypointof asmoothpseudoconvexdomain inC^{n issaid tobe}h-extendibleif its Catlin’s multi-type coincides with its D’Angelo’s multi-type. There is a local model defined by Catlin’s multi-weight. Inthis paper, weshow that a domain in C^{n with a} noncompactautomorphismgroupisbiholomorphicallyequivalenttoitsassociatedmodel ifthereexistsasequenceofautomorphismsofthedomainthathasanorbitconvergingto anh-extendibleboundarypointnon-tangentiallyinaconeregion.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

r é s um é

Un point frontière d’un domaine pseudo-convexe lisse de C^{n est dit} h-extensible si son multi-type de Catlin coïncide avec son multi-type de D’Angelo. Le multi-poids de Catlindéfinitunmodèlelocal.Nousmontrons iciqu’undomainedeC^{n avecungroupe} d’automorphismes non compact est bi-holomorphiquement équivalent à son modèle associé s’ilexisteunesuited’automorphismesdudomaineayantuneorbiteconvergeant nontangentiellementdansuncône,versunpointfrontièreh-extensible.

©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

1. Introduction

Oneofthemainproblemsinthestudyofweaklypseudoconvexdomainsistounderstandthepropertiesofdomainsof finitetype.Thereareseveralclassesofdomainsoffinitetypethathavebeenrelativelybetterunderstood:domainsoffinite typeinC^{2,convexdomainsoffinitetype,anddecoupleddomainsoffinitetype.Allthesedomainsarecontainedinaclass} ofdomainscalledh-extendibledomains[14] orpseudoconvexdomainsofsemiregulartype[4].Inthispaper,westudythe h-extendibledomainswithanoncompactautomorphismgroup.

Let

⊂

C^{n beadomain,anddenotebyAut}

()

thegroupofholomorphicautomorphismsequippedwiththecompact- open topology. Ifthe automorphism group isnot compact, then by a theorem of Cartan(see, e.g., [9]), there are points

✩ TheauthorsarepartiallysupportedbytheNationalNaturalScienceFoundationofChina(GrantNo.11371246).

E-mailaddresses:[email protected](F. Rong),[email protected](B. Zhang).

http://dx.doi.org/10.1016/j.crma.2016.07.005

1631-073X/©^{2016Académiedessciences.PublishedbyElsevierMassonSAS.Allrightsreserved.}

p

∈ ∂

,q

∈

,andautomorphisms fj

∈

^Aut

()

suchthat fj

(

q

)

convergesto pas jgoestoinfinity.Thepoint piscalleda boundaryorbitaccumulationpoint.If

isboundedandstronglypseudoconvexnearp,then

isbiholomorphictotheunit ball inC^{n [12,13].Whenthe boundaryof}

isrealanalyticandconvexnear p,Kim gaveacompletedescriptionofsuch domains[8].In[1],BedfordandPinchukobtainedthefollowingresult:

Theorem1.1.[1,Theorem2]AnyconvexsmoothlyboundeddomainoffinitetypeinC^n+1,havingnon-compactautomorphismgroup, isbiholomorphicallyequivalenttoadomainoftheform

{ (

w

,

z

) ∈

C

×

Cⁿ

:

^Rew

+

^P

(

z

) <

0

}

^,whereP

(

z

)

isaweightedhomogeneous polynomial.

If

issmooth,convex,andoffinitetype2mnearp,thenGaussierprovedthefollowingresult:

Theorem1.2.[5,Theorem1]Let

beadomaininC^n+1andp

∈ ∂

.Assumethatp isanaccumulatingpointforasequenceof automorphismsof

.If

∂

issmooth,convex,andoffinitetype2m nearp,then

isbiholomorphicallyequivalenttoarigidpolynomial domain

D

= {(

w

,

z

) ∈ C × C

ⁿ

:

Rew

+

P

(

z

) <

0

},

whereP

(

z

)

isarealnondegenerateconvexpolynomialofdegreelessthanorequalto2m.

Intheabove statement,“finitetype”meansthatthereisnonon-trivialanalytic settangenttoarbitrarilyhighorderto theboundaryof

atp.Accordingly,thenondegeneracyofP

(

z

)

isgivenbytheconditionthat

{

^z

∈

Cⁿ

:

^P

(

z

) =

⁰

}

^contains nonontrivialanalyticset.

Inthispaperweextendtheaboveresultstotheh-extendibledomainsunderaconeconvergencecondition.Forq

∈

, p

∈ ∂

and f_j

∈

^Aut

()

,wesaythat f_j

(

q

)

convergestopnon-tangentiallyinaconeregionif

f_j

(

q

) ∈

_α

(

p

) := {

^z

∈ : |

^z

−

^q

| < α δ

_p

(

z

) }

for all j large enough forsome

α >

1.Here

δ

p

(

z

) =

^min

{

^dist

(

z

, ∂),

dist

(

z

,

Tp

∂) }

^{. If}

is convexnear p,then

δ

p

(

z

) =

dist

(

z

, ∂)

.By[10,Lemma3],theconeregioncanbedescribedas

_α

(

p

) ⊂ {

^z

∈ :

⁰

<

zpq

<

arccos

(

1

/ α ) } .

(1.1)

Wewrite

for

α

(

p

)

fromnowonbecausethevalueof

α

isnotimportantinourproof.Thefollowingisourmainresult.

Theorem1.3.Let

⊂

C^{n+1beasmoothly}pseudoconvexdomain,p

∈ ∂

ish-extendiblewithCatlin’smulti-type

(

1

,

m1

, · · · ,

mn

)

. Ifthereisapointq

∈

and fj

∈

^Aut

()

suchthatfj

(

q

)

convergestop

∈ ∂

non-tangentiallyinaconeregionasj goestoinfinity, then

isbiholomorphicallyequivalenttoadomainoftheform:

{ (

w

,

z

) ∈ C × C

ⁿ

:

^Rew

+

^P

(

z

) <

0

} .

HereP

(

z

)

isa

(

1

/

m1

, · · ·,

1

/

mn

)

-homogeneouspolynomialwithnopluriharmonicterms.

Insection2,werecallsomebasicdefinitionsandpreparatoryresults.Insection3,wegivetheproofofourmainresult.

2. Preliminaries

Let

beadomaininC^{n andp}

∈

.WefirstrecallthedefinitionofCatlin’smulti-type(seee.g.[3]).

LetLn denotethesetofalln-tuples

μ = ( μ

1

, · · · , μ

n

)

with1

≤ μ

i

≤ ∞

^suchthat (i) 0

< μ

1

≤ μ

2

≤ · · · ≤ μ

n

≤ ∞

^;

(ii) Foreachk,either

μ

k

= ∞

^{orthereisasetof}nonnegativeintegersa1

, · · · ,

akwithak

>

0 suchthatk

j=1aj

/ μ

j

=

^1.

An element ofLn will be referred to asa list. The set of listscan be ordered lexicographically, i.e.if

μ

₌ ( μ

₁

, · · · , μ

_n

)

and

μ

= ( μ

1

, · · ·, μ

n

)

,then

μ

< μ

ifforsome k,

μ

_j

= μ

j forall j

<

k,but

μ

_k

< μ

k.A list

μ ∈

^Ln withrational componentsiscalleddistinguishedifthereexistholomorphiccoordinates

(

z1

, · · · ,

zn

)

centeredatpsuchthat

If

n

i=1

α

_i

₊ β

_i

μ

i

<

1

,

then D^αD^βr

(

p

) =

⁰

.

Here Dα andD^βdenotethepartialdifferentialoperators

∂

^|α|

∂

z^α₁¹

· · ·∂

z^α_nⁿ and

∂

^|β|

∂

z^β₁¹

· · · ∂

z_n^βn

.

Definition2.1. [2]Themulti-type M(∂,p

) = (

m1

, · · · ,

mn

)

isdefinedtobe theleastlistMinLn such thatM

≥ μ

for everydistinguishedlist

μ

.

Themulti-typeisanuppersemicontinuousholomorphicinvariantwithrationalcomponents.Ifthemulti-typeM

(∂,

p

)

of pisfinite,i.e.mn

< ∞

^{,thenitwasshownin[2]that} M(∂,p

) = μ

forsome distinguishedelement

μ

ofLn.Since p isasmoothpoint,itiseasy toseethatthefirstentrym₁

=

^{1 in}M^.Notethatif

ispseudoconvexnear p,theneachm_k, 2

≤

^k

≤

^{n,isanevennumber.Wecall}

= (λ

1

, · · · , λ

n

) = (

1

/

m1

, · · · ,

1

/

mn

)

themulti-weightof p.

Definition2.2.Let f

(

z

)

beafunctiononCⁿ and

= (λ

1

, · · · , λ

n

)

isamulti-weight.Foranyrealnumbert

≥

0,set

π

t

(

z

) = (

t^λ1z1

, · · · ,

t^λnzn

) ∀

^z

∈ C

ⁿ

.

We saythat f is

-homogeneouswithweight

α

if f

(

πt

(

z

)) =

^tαf

(

z

)

foreveryt

>

0 and z

∈

Cⁿ

/ {

⁰

}

^.If

α ₌

1,then f is simplycalled

-homogeneous.

Let

beadomaininCⁿ⁺¹

,

n

≥

^{1.Suppose p}

∈ ∂

isoffinitemulti-type

(

1

,

m₁

, · · ·,

^mn

)

.Theninsuitablelocalcoordi- nates,thedefiningfunctionof

nearphastheform:

r

(

w

,

z

) =

^Rew

+

^P

(

z

) +

^R

(

w

,

z

),

where w

∈

C^{, z}

∈

C^{n, P isa}

(

1

/

m₁

, · · · ,

1

/

m_n

)

-homogeneous plurisubharmonicpolynomialthat contains nopluriharmonic terms,andR issmoothandsatisfies

|

R

(

w

,

z

)| ≤

C

(|

w

| +

n

i=1

|

zi

|

^mi

)

^γ

,

forsomeconstant

γ >

1 andC

>

0.WecallD

= { (

w

,

z

) ∈

C

×

Cⁿ

:

^Rew

+

^P

(

z

) <

0

}

^{anassociatedmodelfor}

atp.

Nowwegivethedefinitionofh-extendibledomains.

Definition2.3.Let

∈

C^{n+1beadomainandsupposethattheCatlin’smulti-typeofaboundarypoint pis}

(

1

,

m1

, · · · ,

mn

)

withmn

< ∞

^.Set

= (

1

/

m1

, · · · ,

1

/

mn

)

.If D

= { (

w

,

z

) ∈

C

×

Cⁿ

:

^Rew

+

^P

(

z

) <

0

}

^{isanassociatedmodelof}

nearp, then Discalledh-extendibleatpifthereisaC^{1 functiona}

(

z

)

onCⁿ

/{

⁰

}

^{satisfyingthefollowing}conditions:

(i) a

(

z

) >

0 wheneverz

=

^0;

(ii)a

(

z

)

is

-homogeneous;

(iii) P

(

z

) −

a

(

z

)

isstrictlyplurisubharmoniconCⁿ

\ {

⁰

}

^when0

< ≤

^1.

Wesaythat

ish-extendibleatpifitsassociatedmodelDish-extendibleatp.And

iscalledh-extendibleifeachone ofitsboundarypointsish-extendible.

Wecalla

(

z

)

abumpingfunctionforP

(

z

)

.Theseconditionsstatethatthemodeldomainfor

atpcanbeapproximated fromtheoutsidebythepseudoconvex domains

{ (

w

,

z

) ∈

C

×

Cⁿ

:

^Re

(

w

) +

^P

(

z

) −

a

(

z

) <

0

}

^{havingthesamehomogene-} ity as

.The key geometric property to theapplications of h-extendible domains is the following relationship between h-extendibledomainsandh-extendiblemodels.

Theorem2.4.[15,Theorem4.7]Let

beasmoothdomaininCⁿandp anh-extendibleboundarypointof

∂

.Thentherearelocal holomorphiccoordinates

(

z

,

w

)

andanh-extendiblemodelQ (withthesamemulti-weightasp)suchthatinthesecoordinatesp

=

⁰ and

\ {

^p

} ⊂

^{Q nearp.}

Foranyintegern

≥

^1,let

= (λ

1

, · · · , λ

n

)

beafixedn-tupleofpositivenumbersand

μ >

0.DenotebyO(

μ , )

theset ofsmoothfunctions f definedneartheoriginofC^{n suchthat}

D^αD^βf

(

0

) =

0

,

whenever

n

i=1

( α

i

+ β

i

)λ

i

≤ μ .

Ifn

=

^{1 and}

= (

1

)

,thenweuseO(

μ )

todenotethefunctionsvanishingtoorderatleast

μ

attheorigin.Then wehave thefollowinglemma.

Lemma2.5.[15,Lemma4.11]Let

beadomaininC^{n+1 andp}

∈ ∂

h-extendible.SupposethattheCatlin’smulti-typeof p is

(

1

,

m₁

, · · ·,

^mn

)

withm_n

< ∞

^andlet

= (

1

/

m₁

, · · · ,

1

/

m_n

)

.Thentherearelocalholomorphiccoordinates

(

^z

,

^w

)

,suchthatinthese coordinatesp

=

^0and

canbedescribednearp asfollows:

= { (

w

,

z

) ∈ C × C

ⁿ

:

^Re

w

+

P

(

z

) +

R₁

(

z

) +

R₂

(

Im

w

) + (

Im

w

)

R

(

z

) <

0

} .

HereP

(

z

)

isa

-homogeneousplurisubharmonicreal-valuedpolynomialcontainingnopluriharmonicterms,R1

∈

O

(

1

, )

,R

∈

O(1

/

2

, )

andR2

∈

O(2

)

.

Remark2.6.Assumethat

isgivenby

{ (

w

,

z

) ∈

C

×

Cⁿ

:

^r

(

w

,

z

) <

0

}

^nearp

= (

0

,

0

)

and

⊂

isaconewithvertexatp.

Convexityisnotabiholomorphicinvariant,butfromtheproofof[15,MainTheorem] and[15, Lemma4.10,Lemma4.11], oneeasilyseesthatconesarepreservedinbothTheorem 2.4andLemma 2.5.

3. ProofoftheMainTheorem

Wefirstprovethemaintheoreminthespecialcasewhere

isdescribedasinLemma 2.5.Theprocesscanbedivided intothreestepsbythestandardscalingmethod(see,e.g.,[7]).Firstweshowthatanycompactsetin

canbemappedinto someneighborhoodU ofp.Thenwemove f_j

(

q

)

totheoriginbycomplexlinearmapsandstretchingthecoordinatesatthe origin. WeprovethattheimagesofU

∩

underthestretchingmaphaveanontriviallimit.The limitisbiholomorphic to theassociatedmodel.Composingtheautomorphismsof

withthestretchingmaps,wehaveasequencefromanycompact set of

to its associatedmodel, andwe call thissequence thescaling sequence. Finally,we show that the limit ofthe scaling sequencegivesabiholomorphic mappingbetween

anditsassociatedmodel.Forthegeneralcase,we construct biholomorphic mapsofcoordinatesthat keep theconeconvergence.Intheabove process,we haveusedseveraldifferent localcoordinates,whichwillresultindifferentmodeldomains.However, in[11],Nikolovshowed, usingascalingmethod, thatallmodeldomainsindifferentcoordinatesarebiholomorphicallyequivalent.Therefore,wecanreducethegeneralcase tothespecialcase.

Firstweneedalocalizationlemma.

Lemma3.1.[7,Proposition9.2.8]ForanyneighborhoodU ofp inC^{n,letK beanycompactsubsetof}

,thenthereisaN

>

0,such that fj

(

K

) ⊂ ∩

^{U forallj}

≥

^{N .}

Nowwedefine

t

(

w

,

z

) = (

t w

,

t^1/m1z1

, · · ·,

t^1/mnzn

) = (

t w

, π

t

(

z

)),

t

∈ R.

Let

beasinLemma 2.5andU aneighborhoodof0

∈ ∂

.Forany

ξ = (ξ

0

,

ξ ) ∈ ∩

^U,where

ξ ∈

C^{n and}

ξ

0

∈

C^,set t

= |

^r

(ξ )|

^and

ξ ˆ = (ˆ ξ

0

,

ξ ) ˆ :=

_t−1

(ξ )

.As

ξ ∈

,it iseasy to seethat

|ξ| |

^Re

ξ

0

| ≈

^t

= |

^r

(ξ )|

^.Here

|

^A

| |

^B

|

^meansthat thereisaconstantC

>

0 whichonlydependson

,

U and

suchthat

|

^A

| ≤

^C

|

^B

|

^,and

|

^A

| ≈ |

^B

|

^if

|

^A

| |

^B

|

^and

|

^B

| |

^A

|

^. Soif

ξ ∈ ∩

and

| ξ |

^{issmall,thenwehave}

|

ξ ˆ | = |π

1/t

(

ξ )|

n

i=1

t^−λi

|ξ

i

|

n

i=1

t^−λit

→

^{0 (3.1)}

since0

< λ

i

<

1 forall1

≤

ⁱ

≤

^n,and

|ˆ ξ

0

|

^t−1

|ξ

0

|

^t−1

|ξ|

¹

.

(3.2)

Thuswegetthat

|ˆ ξ |

^1.

Now,forany

ξ ∈ ∩

^U

∩

,define L_ξ

(

w

,

z

) = (

w

,

z

) − ξ.

ItisaholomorphicautomorphismofC^nthatmoves

ξ

totheoriginandL⁻_ξ¹

=

^L−ξ.Thenconsidertheholomorphicmappings D_ξ

:=

_t−1

◦

^Lξ

( ∩

^U

)

andU_ξ

:=

_t−1

◦

^Lξ

(

U

)

.Set

rξ

(

w

,

z

) =

t⁻¹r

(

L₋ξ

◦

t

(

w

,

z

))

and r0

(

w

,

z

) =

Rew

−

1

+

P

(

z

).

(3.3) Thenlocally Dξ isdefinedby

D_ξ

= { (

w

,

z

) ∈ C × C

ⁿ

:

^r

(

L_−ξ

◦

_t

(

w

,

z

)) <

0

} ∩

^Uξ

= {(

^w

,

z

) ∈ C × C

ⁿ

:

^t−1r

(

L_−ξ

◦

t

(

w

,

z

)) <

0

} ∩

^Uξ

= { (

w

,

z

) ∈ C × C

ⁿ

:

^rξ

(

w

,

z

) <

0

} ∩

^Uξ

.

Then bytheestimates(3.1)and(3.2) itiseasytoseethatU_ξ convergesnormallytoC^{n+1 as}

ξ

tends tozero.Onereadily checksthat

∩^U∩limξ→⁰r_ξ

(

w

,

z

) =

^r0

(

w

,

z

)

(3.4) wheretheconvergenceisuniformoncompactsubsetsofC^{n+1,whichmeansthatD}ξ convergesnormallytoD0

:= { (

w

,

z

) ∈

Cⁿ⁺¹

:

^r0

(

w

,

z

) <

0

}

^{.(Forthedefinitionofthenormal}convergence,see,e.g.,[7,Definition9.2.2].)

Ontheotherhand,byTheorem 2.4,thereisanh-extendiblemodel

Q

= {(

^w

,

z

) ∈ C × C

ⁿ

: ρ (

w

,

z

) =

^Rew

+

^Q

(

z

) <

0

},

^(3.5)

whereQ

(

z

)

isa

-homogeneousfunctiononCⁿ

\ {

⁰

}

^{,suchthatifU isasmall}neighborhoodofpthen

∩

^U

\ {

⁰

} ⊂

^Q

∩

^U. Furthermore, shrinking U if necessary, there is a constant C, only depending on

, U and

, such that, for all small

ξ ∈ ∩

^U

∩

,wehave

D_ξ

(

Q

∩

^U

) ⊂

^Q0

:= { (

w

,

z

) ∈ C × C

ⁿ

:

^Rew

+

^Q

(

z

) −

^C

<

0

} .

(3.6) Nowweproveasimplifiedversionofthemaintheorem.

Proposition 3.2. Let

be a smooth pseudoconvex domain in C^{n+1. Assume that p}

∈ ∂

is h-extendible with multi-type

(

1

,

m1

, · · · ,

mn

)

,mn

< ∞

^andlet

= (

1

/

m1

, · · · ,

1

/

mn

)

.Supposethat,nearp

= (

0

,

0

)

,

hasadefiningfunctionr

(

z

,

w

)

ofthe form:

r

(

w

,

z

) =

Rew

+

P

(

z

) +

R1

(

z

) +

R2

(

Imw

) + (

Imw

)

R

(

z

) <

0

.

HereP

(

z

)

isa

-homogeneousplurisubharmonicreal-valuedpolynomialcontainingnopluriharmonicterms,andR1

∈

O(1

, )

,R

∈

O(¹

/

2

, )

andR₂

∈

O(²

)

.Ifthereisapointq

∈

andf_j

∈

^Aut

()

suchthatf_j

(

q

)

convergestop non-tangentiallyinaconeregion

,then

isbiholomorphictoadomainoftheform:

D

:= { (

w

,

z

) ∈ C × C

ⁿ

:

^Rew

+

^P

(

z

) <

0

} .

(3.7)

Proof. LetU be aneighborhoodofp andK beanarbitrarycompactsubsetof

withq

∈

^K.Bythelocalizationprinciple inLemma 3.1, f_j

(

K

) ⊂ ∩

^{U forall jlargeenough.Set}

j

(

w

,

z

) =

_|fj(q)|−1

(

w

,

z

),

L_j

(

w

,

z

) =

^Lf_j(q)

(

w

,

z

),

r_j

(

w

,

z

) =

^rf_j(q)

(

w

,

z

),

D_j

=

^Df_j(q)

.

Bydefinition, Lj

(

fj

(

q

)) = (

0

,

0

)

and

j

◦

Lj

(

fj

(

q

)) = (

0

,

0

)

.Then forall j largeenough, wedefine thefollowing biholo- morphicmappings:

σ

j

:=

j

◦

^Lj

◦

^fj

.

Inordertoshowthat

isbiholomorphicto D₀

= {(

^w

,

z

) ⊂

C

×

Cⁿ

:

^r0

(

w

,

z

) <

0

}

^{,firstweshowthatthereisalimitof}

σ

j,say

σ

,whichdefinesaholomorphicmapfrom

toD0andisbiholomorphicnearq.Second,weconstructaholomorphic mapfromD0 to

.Finallyweshowthattheyareholomorphicinversestoeachother.

Firstweshowthat alimitof

σ

j definesa holomorphicmap from

toD₀.Notethat

σ

j

|

K isamapfromK to D_j and Dj convergesnormallyto D0 (by(3.4)),and Dj

⊂

^Q0 (by(3.6))forall j largeenough.As Q0 isa tautdomain,

{ σ

j

}

^form a normalfamily andlet

σ

be one of the limits. Since D_j convergesnormally to D₀,we have

σ |

K

:

^K

→

^D0. Since K is arbitrary,

σ

isdefinedonthewhole

.Thenbythetautnessof D0,either

σ () ⊂ ∂

D0 or

σ () ⊂

^D0.Ontheother hand,

σ

j

(

q

) =

j

◦

^Lj

◦

^fj

(

q

) = (

0

,

0

)

.But

(

0

,

0

) ∈

^D0andD0isopen,so

σ () ⊂

^D0.

Nextweshowthat

σ

isbiholomorphic nearq

∈

.Since D₀ isopen,thereisaconstant

δ >

0 suchthatforall jlarge, we havetheballcentered at

(

0

,

0

)

andwithradius

δ

such that B

((

0

,

0

), δ) ⊂

^Dj

=

j

◦

^Lj

( ∩

^U

)

.Since

∂

is offinite type at p, itis offinitetype at points p in

∂

near p,since finitetype is anopen condition (see, e.g., [3]). As p isan accumulatingpoint,by[6,Example3.1.2],

istaut, hencehyperbolic. Sothereisaneighborhood W ofq andaconstant

δ

1

>

0 sothat

|

^F

(ζ,

X

) | ≥ δ

1

|

^X

|

^forany

ζ ∈

^{W andX}

∈

^Tζ

.Thenwehave

δ

₁

|

^X

| ≤

^F

(

q

,

X

) =

^Ffj()

(

f_j

(

q

),

df_j

(

q

)

X

)

(3.8)

≤

^F∩U

(

f_j

(

q

),

df_j

(

q

)

X

)

(3.9)

=

^Fj◦^Lj(∩^U)

((

0

,

0

),

d

_j

◦

^dLj

◦

^dfj

(

q

)

X

)

(3.10)

≤

^FB((0,0),δ)

((

0

,

0

),

d

σ

j

(

q

)

X

)

(3.11)

≤ δ

2

|

^d

σ

j

(

q

) ||

^X

| ,

(3.12)

forall0

=

^X

∈

^Tq

.Equations(3.8)and(3.10)holdbecausetheKobayashimetricisabiholomorphicinvariantandinequal- ities(3.9)and(3.11)holdbecauseKobayashimetricdecreasesunderholomorphicmappings.Inequality(3.12) holdsbythe explicitKobayashimetricoftheballwithradius

δ

attheorigin.Soweget

|

^d

σ

j

(

q

)| ≥ δ

1

δ

2

.

Theconstants

δ

1 and

δ

2donotdependon j.Soletting jgotoinfinity,andsetting

δ

3

= δ

1

/δ

2,weget

|

^d

σ (

q

)| ≥ δ

3

>

0

.

Thisshowsthat

σ

isbiholomorphicnearq.

NowweconstructaholomorphicmapfromD₀ to

.Set

ω

j

=

f⁻_j¹

◦

⁻_j¹

◦

L⁻_j¹

.

ForanycompactsubsetK

⊂

^D0,as Dj convergesnormallyto D0,sofor j largeenough,K

⊂

^Dj

=

j

◦

^Lj

( ∩

^U

)

,thatis L⁻_j¹

◦

⁻_j¹

(

K

) ⊂ ∩

^{U.Thusthesequence}

ω

j

|

K

:

K

→

iswelldefined.As

istaut,

{ ω

j

}

^{formanormallyfamilyandlet}

ω

beoneofthelimits.SinceK isarbitrary,

ω

isdefinedonD0 and

ω (

D0

) ⊂

.But

ω

j

(

0

,

0

) =

^f−j¹

◦

⁻_j¹

◦

^L−j¹

(

0

,

0

) =

^q, so

ω (

D₀

) ⊂

.

Now we show that

σ

is a biholomorphism. As

σ

j convergesto

σ

uniformly on compactsets of

and

σ

is a local biholomorphism nearq, there exists a constant

δ

4

>

0 and a compact set N

⊂

with q

∈

^{N, such that B}

((

0

,

0

), δ

4

) ⊂ σ

j

(

N

) ⊂

^D0 foralllarge j.Thisimpliesthat

ω

j

(

B

((

0

,

0

), δ

4

)) ⊂

^f−_j¹

(

f_j

(

N

)) =

^{N.Thusforlarge j,themapping}

σ

j

◦ ω

j

|

B((0,0),δ4)

=

id_B((0,0),δ4)

iswelldefined. Let j gotoinfinity,wehave

σ ◦ ω _|

_B((0,0),δ4)

=

^idB((0,0),δ4),andhence

σ ◦ ω ₌

idD0.Ontheotherhand,on anycompactsetK

⊂

withq

∈

^K,wehave

ω

_j

_◦ σ

_j

_|

K

=

^idK

.

Let jgotoinfinity,weseethat

ω ◦ σ |

K

=

^idK,thus

ω ◦ σ =

^id.Hence

σ

isabiholomorphicmappingbetweenD0 and

. BythedefiningfunctionsofD0 in(3.3) andD in(3.7),itiseasy toseethat

anditsassociatedmodelD arebiholo- morphicallyequivalent. 2

Nowwegivetheproofofthemaintheorem(Theorem 1.3):

ProofofTheorem 1.3. Supposethat

isgivenby

{ (

w

,

z

) ⊂

C

×

Cⁿ

:

^r

(

w

,

z

) <

0

}

^inaneighborhoodU ofp.Byassumption, there exist holomorphic automorphisms f_j

∈

^Aut

()

andq

∈

such that f_j

(

q

)

converge to theboundary point p non- tangentially in a cone

as j goes toinfinity. Let

(

^w

,

^z

) =

^Hp

(

w

,

z

)

be thelocal coordinatechangegiven inLemma 2.5.

Then, byRemark 2.6,theimageofthecone

underHp,definedby

:=

^Hp

()

,isalsoaconewithvertexp

=

^0.Soone gets Hp

(

fj

(

q

)) ∈

.ByProposition 3.2,

isbiholomorphictothemodel D

= { (

w

,

z

) :

^Rew

+

P

(

z

) <

0

}

^{.By[11],allmodels} indifferentcoordinatesarebiholomorphicallyequivalent.Thiscompletestheproof. 2

Acknowledgements

We sincerelythankthereferee, whoread thepaperverycarefullyandgavemanyexcellentsuggestions,whichgreatly improvedthepresentationofthispaper.

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