ChiaradeFabritiis Complexgeometryofgeneralizedannuli

(de Gruyter 2003

Complex geometry of generalized annuli

Chiara de Fabritiis

(Communicated by G. Gentili)

Abstract. We study the complex geometry of a class of domains inCⁿ which generalize the annuli inC, i.e., which are quotients of the unit ballBⁿofCⁿfor the action of a group gen- erated by a hyperbolic element of AutBⁿ. In particular, we prove that the degree of holomor- phic maps between two such domains is bounded by a constant which depends on the ‘‘radii’’

of the domains only and we give some results on the existence of complex geodesics for the Kobayashi distance in these domains.

Key words.Holomorphic maps, annuli, degree of a map, complex geodesics.

2000 Mathematics Subject Classification. Primary 32A10, 32A40; secondary 32H15, 32A30

1 Introduction

The aim of this paper is to study the complex geometry of a class of domains in Cⁿ which are quotients of the unit ball BⁿHCⁿ for the action of a group gen- erated by a hyperbolic element of AutBⁿ (see Section 2 for definitions). Since the annuli in C are obtained as quotients of the unit disk D¼ fxAC:jxj<1g for the action of a group generated by a hyperbolic element of AutD, the domains we study can be seen as a generalization of annuli to several complex variables.

In Section 2 we give some definitions we need in the sequel of the paper and we recall some statements concerning this class of domains, which were introduced in [3]. In Section 3 we generalize to several complex variables a result which is due to Schi¤er in the one-dimensional case: the degree of a holomorphic map between two annuli is bounded by a constant which depends on the moduli of the annuli only.

In Section 4 we study the geometry of extremal mappings and complex geodesics for the Kobayashi distance in this class of domains. In particular we prove that there always exists an extremal mapping through two given points of a ‘‘generalized annu- lus’’ and we give several results on existence and non-existence of complex geodesics according to the ‘‘radius’’ of the domain and to other parameters which classify these domains.

2 Preliminaries and statements

We denote the unit ball for the Euclidean metric inCⁿbyBⁿ; the following result is well known.

Theorem 2.1.Any holomorphic automorphismgofBⁿcan be extended holomorphically to a neighborhood of the closure ofBⁿ;ifghas no fixed points inBⁿ,then its extension has either one or two fixed points inqBⁿ.

From now on we shall denote by the same symbol a holomorphic automorphism of Bⁿand its extension to the closure ofBⁿ.

Definition 2.2.LetgAAutBⁿ: ifghas at least one fixed point inBⁿ, thengis said to beelliptic; ifghas no fixed points inBⁿand has one fixed point inqBⁿ, it is said to be parabolic; ifghas no fixed points inBⁿ and has two fixed points inqBⁿ, it is said to behyperbolic.

To generalize the construction of annuli to several complex variables, we will focus our attention to the action of hyperbolic elements on Bⁿ. First of all, we recall a result which is due to de Fabritiis and Gentili (see [5]).

Proposition 2.3.Let gbe a hyperbolic element inAutBⁿ;then there exist TAR and y₂;. . .;y_nARsuch thatgis conjugate to

g₀:z7! z₁coshTþsinhT

z1sinhTþcoshT; e^iy2z₂

z1sinhTþcoshT;. . .; e^iynz_n z1sinhTþcoshT

: ð2:1Þ

In the sequel it will be useful to consider the problem on the Siegel half-spaceHⁿ ¼ fwACⁿjIw₁>jw2j²þ þ jwnj²g which is biholomorphic to Bⁿ via the Cayley transformC(see e.g. Rudin [7] or Abate [1]), so we also give the form of hyperbolic elements in AutHⁿ.

Corollary 2.4. Let mAAutHⁿ be hyperbolic; then there exist lAR^þnf1g and y2;. . .;ynARsuch thatmis conjugate to

Cg₀C¹ ¼m₀:w7! ðl²w₁;le^iy2z₂;. . .;le^iynz_nÞ; ð2:2Þ wherel¼e^T.

This result enables us to consider the quotients ofBⁿ (resp.Hⁿ) for the action of the groupGðMÞgenerated by a hyperbolic elementgAAutBⁿ ðmAAutHⁿÞ. Since the quotients Hⁿ=M1 andHⁿ=M2 are biholomorphic i¤ M1 andM2 are conjugate in AutHⁿ, then it is enough to consider the case of a group generated by an element of the form (2.2). As we are interested in the groupM generated bym₀, rather than in the element m₀ itself, we can always suppose that l>1, that is T>0. Let

ln:H¹!R ð0;pÞbe a branch of the logarithm, set b¼1=lnl¼1=T and con- sider the holomorphic map pr:Hⁿ!Cⁿgiven by

prðwÞ ¼ ðe^ibplnw1;e^{bðlnlþiy2Þlnw1=2}w₂;. . .;e^{bðlnlþiynÞlnw1=2}w_nÞ: ð2:3Þ In [3] and [4] the following result is proved:

Theorem 2.5.Let m₀ be given by(2.2)and M be the group generated bym₀.Then the mappr:Hⁿ!Cⁿ given by(2.3)factors onHⁿ=M giving a biholomorphism between

Hⁿ=M and the bounded domain

Wðr;y₂;. . .;y_nÞ ¼

xACⁿ

r<jx1j<1;Xⁿ

j¼2

jxjj²jx1j^yj=p<sin plnjx1j lnr

;

where r¼e^p2=lnl¼e^p2=T Að0;1Þ, y₂;. . .;y_nAR. In particular Hⁿ=M is a Stein manifold which is biholomorphic to a bounded domain inCⁿ.

As a consequence of the previous theorem, the domainsWðr;y₂;. . .;y_nÞcan be seen as a generalization to several complex variables of the annuli, which are the quotients of the unit diskD¼B¹ for the action of the groups generated by hyperbolic auto- morphisms ofD. In factWðrÞ ¼ fxACjr<jxj<1gis an annulus inCand for this reason we shall often call these domains ‘‘generalized annuli’’.

To simplify notation, when no confusion can arise we only write W instead of Wðr;y2;. . .;ynÞ. Of course, via the Cayley transform we can also study the problem onBⁿ: in this case the covering will be given byðBⁿ!^w WÞwherew¼prC.

Remark 2.6. Notice that the domainWðr;y2;. . .;ynÞretracts by deformation on the annulusWðrÞ, in particular it is doubly connected.

From now on, we shall denote byW₁the domainWðr1;y₂;. . .;y_n1ÞHCⁿ¹ and byW₂ the domainWðr2;Q₂;. . .;Q_n2ÞHCⁿ²; in order to simplify the notation, the symbolH will stand for HolðW1;W₂Þ ¼ ff :W₁!W₂jf holomorphicg. For all f AH, we will denote the degree of f bydðfÞ.

Definition 2.7.Let f;gAH. We say that f andgarehomotopicif there exists a con- tinuous mapF :½0;1 W₁!W₂ such that

(i) Fð0;Þ ¼ f,Fð1;Þ ¼g;

(ii) Fðt;ÞAHfor alltA½0;1.

Of course, the fact of being homotopic is an equivalence relation on H; we shall denote the homotopy class of f AHby½f.

At last, ifDis a domain inCⁿ, we shall denote bykDð;ÞtheKobayashi distance on D and bykDð;Þthe Kobayashi metric on D(for a comprehensive reference on this topic see [1] or [6]).

3 Holomorphic maps between generalized annuli

In this section we generalize the estimate of the degree of a holomorphic map between two annuli which is due to Schi¤er (see [8]) in the one-dimensional case.

Theorem 3.1.If f AHolðWðr1Þ;Wðr2ÞÞ,thenjdðfÞjc½lnr₂=lnr₁.Moreover,if equal- ity holds there existsyARsuch that for allxAWðr1Þ

fðxÞ ¼ e^iyx^dðfÞ if dðfÞ>0;

r2e^iyx^dðfÞ if dðfÞ<0:

(

LetW1¼Wðr1;y2;. . .;yn1ÞHCⁿ¹ andW2¼Wðr1;Q2;. . .;Qn2ÞHCⁿ² be two general- ized annuli; since the fundamental group of bothW1 andW2 is isomorphic toZ, we can define the degree dðfÞAZof a holomorphic map f AH by choosing genera- tors aa~j of p1ðWjÞ represented by ajðtÞ ¼ ðpffiffiffiffirj

e^2pit;0;. . .;0Þ for j¼1;2 and setting fðaa~₁Þ ¼dðfÞaa~₂.

Theorem 3.2.If f AH,thenjdðfÞjc½lnr₂=lnr₁.

Proof.LetS_n ¼ fwACⁿjIw₁Að0;pÞ;Ie^w1>jw2j²þ þ jwnj²g; it is easily verified that the mapE:S_nCw! ðe^w1;w₂;. . .;w_nÞAHⁿis a biholomorphism.

Then we can consider the coveringsðSnj !^qj WjÞfor j¼1;2, where the mapsq1and q₂are given by

q₁ðwÞ ¼ ðe^ib1pw1;e^{b1ðlnl1þiy2Þw1=2}w₂;. . .;e^{b1ðlnl1þiyn1Þw1=2}w_n1Þ; q2ðwÞ ¼ ðe^ib2pw1;e^{b2ðlnl2þiQ2Þw1=2}w2;. . .;e^{b2ðlnl2þiQn2Þw1=2}wn2Þ;

withbj ¼ lnrj=p² and lnlj¼1=bj. The group of deck-transformations of the cov- eringsðSnj !^qj WjÞis generated bynj ¼E¹m_jE:Snj !Snj given by

n₁ðwÞ ¼ ðw1þ2 lnl₁;l₁e^iy2w₂;. . .;l₁e^iyn1w_n1Þ;

n2ðwÞ ¼ ðw1þ2 lnl2;l2e^iQ2w2;. . .;l2e^iQn2wn2Þ;

respectively. Since the domainSn1is simply connected, there exists a continuous map ff~:Sn1 !Sn2 such that q2 ff~¼ f q1; the mapsq1 andq2 being local biholomor- phisms, we immediately obtain that ff~is holomorphic. Interpreting the degree of f via the isomorphism between the fundamental groups ofW_j and the groups of deck- transformations of the coverings, we obtain the following equality

ff~n1 ¼n₂^dðfÞ ff~; ð3:1Þ the comparison between (3.1) and the contracting property of the Kobayashi dis- tance will yield the conclusion. Let us consider the points w⁰¼ ðip=2;0;. . .;0Þand

w¹¼n₁ðw⁰Þ ¼ ð2 lnl₁þip=2;0;. . .;0ÞAS_n1; if k_D denotes the Kobayashi distance onD, we have the following chain of inequalities

k_Sn

2ðff~ðw⁰Þ;ff~ðw¹ÞÞck_Sn

1ðw⁰;w¹Þ ¼kHⁿ¹ðEðw⁰Þ;Eðw¹ÞÞ

¼k_H¹ði;l₁²iÞ ¼lnl1¼ p²=lnr1;

where the second equality is due to the fact thatEðw⁰Þ;Eðw¹ÞAH¹ f0gwhich is a holomorphic retract ofHⁿ. Since

k_Sn

2ðff~ðw⁰Þ;ff~ðw¹ÞÞ ¼k_Sn

2ðff~ðw⁰Þ;ff~ðn1ðw⁰ÞÞÞ ¼k_Sn

2ðff~ðw⁰Þ;n₂^dðfÞðff~ðw⁰ÞÞÞ;

considering the projection on the first component and denoting f1ðw⁰Þ by c, the above inequality and the contracting property of the Kobayashi distance yield

k_S1ðc;cþ2dðfÞlnl₂Þck_Sn

2ðff~ðw⁰Þ;n₂^dðfÞðff~ðw¹ÞÞÞcp²=lnr₁:

Via the biholomorphismE:S1!H¹, we can evaluatekS1ðc;cþ2dðfÞlnl2Þobtain- ing

k_S1ðc;cþ2dðfÞlnl₂Þ ¼k_H¹ðe^c;l₂^2dðfÞe^cÞdp²jdðfÞj=lnr₂;

asr₂<1, this impliesjdðfÞjclnr₂=lnr₁. r This means that the ratio of the logarithms of ‘‘inner radii’’ (that is, the generaliza- tion of the ratio of the moduli) bounds the degree of holomorphic maps between gen- eralized annuli. The following proposition gives an even deeper interest to the above theorem, since it tells us that the degree is a complete homotopy invariant.

Proposition 3.3.Let f;gAH.Then dðfÞ ¼dðgÞif and only if½f ¼ ½g,that is if and only if f and g are homotopic.

Proof.The ‘‘if ’’ part is obvious. In order to prove the converse implication, we write f ¼ ðf1;. . .;fn2Þandg¼ ðg1;. . .;gn2Þ; using the retraction by deformation ofW2onto Wðr2Þ f0ggiven byrðt;xÞ ¼ ðx₁;tx₂;. . .;tx_nÞ, it is easily seen that we can limit our- selves to the casen2¼1.

Settingd¼dðfÞ ¼dðgÞ, we then have 1

2pi ð

ffiffiffir1

p S¹

qf

qx₁ðx1;0;. . .;0Þ=fðx1;0;. . .;0Þdx₁

¼ 1 2pi

ð ffiffiffir1

p S¹

qg

qx₁ðx₁;0;. . .;0Þ=gðx₁;0;. . .;0Þdx₁ ¼d:

Letj;c:W₁!Cbe given by jðxÞ ¼x^d₁ fðxÞandcðxÞ ¼x^d₁ gðxÞ; it is easily veri- fied that

1 2pi

ð ffiffiffir1

p S¹

qj

qx₁ðx₁;0;. . .;0Þ=jðx₁;0;. . .;0Þdx₁

¼ 1 2pi

ð ffiffiffir1

p S¹

qc

qx₁ðx₁;0;. . .;0Þ=cðx₁;0;. . .;0Þdx₁¼0:

Consider the maps j;c:p1ðW1Þ !p1ðCÞ; the above equality implies that both these maps are trivial and therefore j;c can be lifted to continuous maps j~

j;cc~:W1!Csuch that expjj~¼jand expcc~¼c. Since exp:C!C is a local biholomorphism and both j and c are holomorphic, the two maps jj~ and cc~ are holomorphic. Then we have found holomorphic maps jj;~cc~:W1!C such that

fðxÞ ¼x₁^de^~jjðxÞ andgðxÞ ¼x₁^de^ccðxÞ~ for allxAW1. Now set Hðt;xÞ ¼x₁^de^{jjðxÞþtð~ ccðxÞ~~ jjðxÞÞ};

obviouslyH is continuous, and the mapHðt;Þis holomorphic for alltA½0;1; more- overHð0;Þ ¼ f andHð1;Þ ¼g, so we are left to verify thatHð½0;1 W1ÞHWðr2Þ.

FixxAW1and consider the map

b:½0;1Ct7! jHðt;xÞj ¼ jx₁^dje^<~jjðxÞþt<ðccðxÞ~~ jjðxÞÞAR;

the mapb is monotonic and we haver₂<jfðxÞj ¼bð0Þ<1,r₂ <jgðxÞj ¼bð1Þ<1.

Thus r₂<jHðt;xÞj ¼bðtÞ<1 for all tA½0;1; so Hð½0;1 W₁ÞHWðr2Þ and this concludes the proof of the assertion, since H is the required homotopy between the

maps f andg. r

As a consequence of the proof of Proposition 3.3 we obtain the following

Corollary 3.4. Let f AH, then there exists a holomorphic map u:W1!Csuch that f1ðxÞ ¼x₁^dðfÞe^uðxÞfor allxAW1.

Moreover, gathering Theorem 3.2 and Proposition 3.3, we obtain that, if the ‘‘hole’’

inW₁is smaller than the ‘‘hole’’ inW₂, any holomorphic map fromW₁toW₂is homo- topic to a constant map.

Corollary 3.5.Let f AH;if r₁<r₂ then f is homotopic to a constant.

Now we turn to the study of the homotopy classes Hd ¼ ff AHjdðfÞ ¼dg. If jdj<lnr2=lnr1 the following remark and proposition ensure that the family Hd is very ‘‘ample’’.

Remark 3.6.For all f AH0, there exists a holomorphic map ff~:W1!Hⁿ²such that pr₂ ff~¼ f. Vice versa, for all holomorphic map ff~:W1!Hⁿ², the map pr₂ ff~ belongs toH0.

Proof. Since Hⁿ¹ is simply connected, the existence of a continuous ff~:W1 !Hⁿ²

such that pr₂ ff~¼ f is equivalent to the triviality of the map f(that is todðfÞ ¼0).

As pr₂ is a local biholomorphism, the assertion follows. r This simple remark classifies all elements of H0; moreover the boundedness of W₁ ensures that there exists a huge family of holomorphic maps fromW₁toHⁿ²; in this sense we can say thatH0is ‘‘big’’.

Now let us consider the case when 0<jdj<lnr2=lnr1. First of all, ifn2d2 we set L¼maxfjQjj=pjj¼2;. . .;n2g. If 0<d <lnr2=lnr1, choose CAR such that r2 <

Cr₁^d <r₁^d <C; if 0<d <lnr2=lnr1, choose CAR such that r2<Cr2 <r2r₁^d <

Cr2r₁^d <1 and set

d¼ minfð1CÞ=2;ðCr₁^dr₂Þ=2g if d >0;

minfðCr2r₂Þ=2;ð1Cr₁^dr₂Þ=2g if d <0;

K¼

min sin ^plnðð1þCÞ=2Þ lnr2

;sin ^plnððCr

d 1þr₂Þ=2Þ lnr2

n o

if d >0;

min sin ^plnððCr_ln²_r^þr2Þ=2Þ

2

;sin ^plnðð1þCr_lnr^1dr2Þ=2Þ

2

n o

if d <0:

8>

<

>:

Then we can describe an ample set of the elements belonging toHd:

Proposition 3.7.Let dAZbe such that0<jdj<lnr2=lnr1and C;d;L;K be as above.

For anyyAR,any holomorphic functions:W1!Csuch thatkskycdand any holo- morphic map h:W1!Cⁿ²¹such thatPn21

j¼1 jhjðxÞj²<r₂^LK for allxAW1,the map f :W₁Cx7! ðCe^iyx₁^dþsðxÞ;hðxÞÞAW2; if d>0;

ðCr2e^iyx₁^dþsðxÞ;hðxÞÞAW2; if d<0;

(

belongs toHd.

Proof.We shall perform the proof in the cased >0, the cased <0 can be obtained from this by minor changes. First of all, let us prove that the map f belongs toH.

Since it is obvious that f is a holomorphic map, we are left to prove that it mapsW₁ intoW₂. LetxAW₁; the choice ofCanddimplies that the first component of f sat- isfies the following inequality

r₂<Cr₁^dþr₂

2 <jf₁ðxÞj<Cþ1

2 <1; ð3:2Þ

which yields

0<plnððCþ1Þ=2Þ

lnr₂ <plnjf1ðxÞj

lnr₂ <plnððCr₁^dþr2Þ=2Þ lnr₂ <p:

The map sin:½0;p !Ris concave and therefore the choice ofKentails

Kcsin plnjf1ðxÞj lnr₂

ð3:3Þ

for allxAW₁. By (3.2) and (3.3) we then have Xⁿ²

j¼2

jfjðxÞj²jf1ðxÞj^Qj=pcr^L₂ ⁿX²¹

j¼1

jhjðxÞj²cK<sin plnjf₁ðxÞj lnr2

and therefore f mapsW1toW2. At last, we prove thatdðfÞ ¼d, that is f AHd. Let H :½0;1 W1!Cⁿ² be given byHðt;xÞ ¼ ðCe^iyx₁^dþtsðxÞ;thðxÞÞ. The mapHis of course continuous and by the above reasoning we haveHð½0;1 W1ÞHW2; more- overHð1;Þ ¼ f andHð0;xÞ ¼ ðCe^iyx₁^d;0;. . .;0Þ. Since the degree ofHð0;Þis equal

tod, we are done. r

Now we consider the case when jdðfÞj ¼lnr2=lnr1 which of course can occur i¤

lnr2=lnr1AN. The following results describe the maps f AHd whenjdj ¼lnr2=lnr1: the restriction of f to Wðr1Þ f0g has to follow a prescribed pattern, while outside this annulus the behaviour of f can be quite ‘‘free’’.

Theorem 3.8. Let f AHd where jdj ¼lnr₂=lnr₁. Then there exist yAR and pAHolðW1;CÞsuch that pj_Wðr1Þf0g10,_qx^qp

j

Wðr1Þf0g10for all j¼2;. . .;n₁and f1ðxÞ ¼ e^iyx₁^de^pðxÞ if d>0;

e^iyr2x₁^de^pðxÞ if d<0:

(

ð3:4Þ

Proof.Since we are interested in the behaviour of f₁only andW₂retracts by deforma- tion onto Wðr2Þ f0g, it is enough to consider the case n₂¼1. Moreover we shall perform the proof only in the cased >0, the cased <0 being obtained from this by a few minor changes.

By Corollary 3.4 we can find a holomorphic mapu:W1!Csuch that

fðxÞ ¼x₁^de^uðxÞ: ð3:5Þ Let us consider the map t:Wðr1ÞCx₁7! fðx₁;0;. . .;0ÞAWðr2Þ; since Wðr1Þ is a deformation retract ofW1 the degree oftis equal tod. AsS1¼R ð0;pÞis simply connected andq2:S1!Wðr2Þis a local biholomorphism, there exists a holomorphic lifting ~ttof tq1, that is a holomorphic map ~tt:S1!S1 such that q2~tt¼tq1. Interpreting the degree oftvia the action of the group of deck-transformations yields

~ttn₁ ¼n₂^d~tt, that is

~ttðzþ2 lnl₁Þ ¼~ttðzÞ þ2dlnl₂

for allzAS1. Takingz¼ip=2 and using the equalitydlnl2 ¼lnl1 we obtain

~tt ip

2 þ2 lnl1

¼~tt ip

2 þ2 lnl1: ð3:6Þ

Transferring the problem onH¹, we obtain as a consequence of the Schwarz lemma that~ttis an automorphism ofS₁of the form~tt:z7!zþxfor a suitablexAR.

In fact, let s¼Ett~E¹:H¹!H¹ and set ~ttðip=2Þ ¼xþiy. Since sðiÞ ¼ expðxþiyÞandsðil²₁Þ ¼l₁²expðxþiyÞ, by the contracting property of the Kobaya- shi distance we obtain

k_H¹ðsðiÞ;sðil₁²ÞÞ ¼tanh¹ l²₁1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l₁⁴þ12l₁²cos 2y q

0 B@

1

CAck_H¹ði;il₁²Þ ¼lnl1:

Now the fact that tanh is increasing yield l₁²1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi l⁴₁þ12l₁²cos 2y

q cl²₁1 l²₁þ1

and hence cos 2y¼ 1, that is y¼p=2. Moreover the equality at one point of the Kobayashi distance yields, by the Schwarz lemma, that s is an automorphism of H¹; sincesðiÞ ¼ie^x andsðil₁²Þ ¼ie^xl₁², we havesðwÞ ¼e^xwfor all wAH¹, that is

~ttðzÞ ¼zþxfor a suitablexAR. Then there existsyARsuch thattðxÞ ¼e^iyx^d for allxAWðr1Þand we have proved that fðx1;0;. . .;0Þ ¼e^iyx₁^d for allx₁ AWðr1Þ. Com- paring this equality with (3.5) we infer that u is a holomorphic map which takes values iniyþ2piZonWðr1Þ f0gand hence, setting pðxÞ ¼uðxÞ uðpffiffiffiffir₁

;0;. . .;0Þ, we obtain that pðx1;0;. . .;0Þ ¼0 for allx₁AWðr1Þ.

In order to prove that _qx^qp

j

Wðr1Þf0g10 for all j¼2;. . .;n₁, it is enough to con- sider the map Wðr1;yjÞCðx1;x_jÞ 7! fðx1;0;. . .;0;x_j;0;. . .;0ÞAWðr2Þ and hence we can limit ourselves to the casen1¼2.

Since for anyx₁AWðr1Þand anyR<r^jy₁^2j=2pðsinðplnjx₁j=lnr1ÞÞ¹⁼²the setfx₁g DR

is contained inW1, we can find holomorphic maps a1:Wðr1Þ !Canda2 :W1!C such that pðxÞ ¼a1ðx₁Þx₂þa2ðxÞx₂². Our last assumption is therefore equivalent to a110. As f maps W1 in Wðr2Þ, using the form of f we have that for all xAW1 the following inequality holds:

lnr2dlnjx₁j<<pðxÞ<dlnjx₁j: ð3:7Þ Fix x₁⁰AWðr1Þ and for any 0<e<sinðplnjx⁰₁j=lnr₁Þ denote by R the number r^jy₁^2j=2pðsinðplnjx⁰₁j=lnr₁Þ eÞÞ¹⁼². Then for allx₂AD_R the pointx¼ ðx₁⁰;x₂Þbelongs toW₁. Now let us setr¼R=2; then we have

rja1ðx₁⁰Þjcr max

jx2jcrja1ðx₁⁰Þ þa₂ðxÞx2j ¼r max

jx2j¼rja1ðx₁⁰Þ þa₂ðxÞx2j ¼max

jx2j¼rjpðxÞj: The Borel–Carathe´odory theorem and (3.7) imply that

jxmax₂j¼rjpðxÞjc R Rr max

jx₂j¼r<pðxÞ ¼2 max

jx₂j¼r<pðxÞc2dlnjx₁⁰j;

and therefore ja1ðx⁰₁Þjc4r^jy₁ ^2j=2pdlnjx₁⁰jðsinðplnjx₁⁰j=lnr₁Þ eÞ¹⁼²; the arbitrari- ness ofeyields that for allx⁰₁ AWðr1Þthe following inequality holds

ja1ðx₁⁰Þjc 4r^jy₁ ^2j=2pdlnjx₁⁰j ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sinðplnjx₁⁰j=lnr₁Þ

q : ð3:8Þ

It is easily seen that the right hand term of (3.8) goes to 0 whenjx⁰₁j !1, yielding lim_jx⁰

1j!1ja1ðx₁⁰Þj ¼0. The same reasoning can be performed for jx₁⁰j !r^þ₁; also in this case we obtain that lim_jx⁰

1j!r^þ₁ja1ðx₁⁰Þj ¼0. Then we can invoke the maximum

modulus principle and we are done. r

Now we consider the behaviour of the last n21 components of f on the annulus Wðr1Þ f0g.

Proposition 3.9. Let f AHd where jdj ¼lnr₂=lnr₁; then f_jðx₁;0;. . .;0Þ ¼0 for all x₁AWðr1Þand all j¼2;. . .;n₂.

Proof.First of all note that it is enough to prove the assertion forn1¼1 andn2¼2;

in fact for any jAf2;. . .;n2gconsider the map

g:Wðr1ÞCx₁ 7! ðf1ðx₁;0;. . .;0Þ;fjðx₁;0;. . .;0ÞÞAWðr2;QjÞ;

it is obvious that f andghave the same degree, thus we can limit ourselves to study the casen1¼1,n2¼2. Moreover, as in the previous theorem, we perform the proof ford >0 only, leaving the proof of the cased <0 to the reader.

Theorem 3.8 tells us thatjf1ðxÞj ¼ jxj^d; since f mapsWðr1ÞintoWðr2;QjÞwe have that for allxAWðr1Þthe following inequality holds

jf₂ðxÞj²jxj^dQj=p¼ jf₂ðxÞj²jf₁ðxÞj^Qj=p<sinðplnjf₁ðxÞj=lnr₂Þ ¼sinðplnjxj=lnr₁Þ; that is

jf2ðxÞj²<jxj^dQj=psinðplnjxj=lnr1Þ ð3:9Þ for all xAWðr1Þ. Taking the limit of the right hand side both for jxj !1 and jxj !r^þ₁ we obtain that

lim

jxj!r^þ₁jf2ðxÞj²¼ lim

jxj!1jf2ðxÞj²¼0;

the maximum modulus principle yields the conclusion. r The big di¤erence inHd between the one-dimensional and the multi-dimensional case arises just whenjdj ¼lnr2=lnr1. In the one-dimensional caseHdis a one-dimensional topological space which is isomorphic to S¹, while in the multidimensional case it contains an open set in an infinite-dimensional Fre´chet space. In fact, provided n₁þn₂d3, the following proposition gives a ‘‘large’’ family of maps belonging to

Hd whered ¼lnr₂=lnr₁ (and of course the same can be done, mutatis mutandis, for d ¼ lnr₂=lnr₁).

Since the domainWðr;y₂;. . .;y_n1Þis biholomorphic toWðr;y₂þ2pl;. . .;y_n1þ2plÞ for alllAZ(see [4] for a proof ), in order to simplify computations we can suppose thaty_jc0 for all j¼2;. . .;n₁andQ_jd0 for all j¼2;. . .;n₂. Now set

K¼r2lnr₂

p and K1¼lnð1þK=ðn11ÞÞ

d if n1d2:

For any yAR, sjk AHolðW1;CÞ for j;k¼2;. . .;n1 and hjk AHolðW1;CÞ for j¼ 2;. . .;n1,k¼2;. . .;n2, settðxÞ ¼Pn1

j;k¼2x_jx_ksjkðxÞ.

Theorem 3.10.If Xⁿ¹

j;k¼2

jsjkðxÞjcK1 and ⁿX^1;n2

j;k¼2

jhjkðxÞj²c1þpðn11ÞK1

lnr₁ p²K₁²

2 ln²r₁; ð3:10Þ then the map

f :W₁Cx7!

e^iyx₁^de^dtðxÞ;Xⁿ¹

j¼2

x_jh_j2ðxÞ;. . .;Xⁿ¹

j¼2

x_jh_jn2ðxÞ

AW₂

belongs toHd where d ¼lnr2=lnr1.

Proof.By definition the map f is obviously holomorphic. Now we prove that f maps W₁intoW₂; after that a simple remark will show that the degree of f is equal todand therefore f belongs toHd. First of all, we give two estimates oftwhich will be useful in the sequel. By the definition oftwe havejtðxÞjcPn1

j;k¼2jxjx_ks_jkðxÞj; sincey_kc0 for k¼2;. . .;n1, then we obtain jx_kj<1 for all xAW1 and for k¼2;. . .;n1, and therefore

jtðxÞjcXⁿ¹

j;k¼2

jsjkðxÞjcK1 ð3:11Þ

for allxAW₁. Moreover for allxAW₁the following bound ontalso holds jtðxÞjc Xⁿ¹

j;k¼2

jxjj jxkj Xⁿ¹

j;k¼2

jsjkðxÞj

cK₁Xⁿ¹

j¼2

jxjj 2

cðn11ÞK1

Xⁿ¹

j¼2

jxjj²cðn11ÞK1

Xⁿ¹

j¼2

jxjj²jx1j^yj=p

cðn11ÞK1sin plnjx1j lnr1

: ð3:12Þ

Let us notice that f₁ðxÞcan be written ase^iyx₁^dþe^iyx₁^dqðxÞ, where

qðxÞ ¼e^dtðxÞ1¼X

l>0

ðdtðxÞÞ^l l! : Now we estimateqðxÞ: by (3.11) and (3.12) we obtain

jqðxÞjcX

l>0

jdtðxÞj^l

l! cdjtðxÞj X

l>0

ðdK1Þ^l1 l!

!

cdðn11ÞK1

X

l>0

ðdK1Þ^l1 l!

!

sin plnjx₁j lnr1

cðn11Þðe^dK11Þsin plnjx1j lnr₁

¼Ksin plnjx1j lnr₁

:

As for anyxAW1 we havejx₁jc1, the following inequalities yield jx1j^dKsin plnjx1j

lnr₁

cjf₁ðxÞjcjx1j^dþKsin plnjx1j lnr₁

: ð3:13Þ

Now consider the functionsF;C:½r1;1 !Rgiven by FðtÞ ¼t^dKsin plnt

lnr1

; CðtÞ ¼t^dþKsin plnt lnr1

;

it is easily seen thatFðr1Þ ¼Cðr1Þ ¼r₂, thatFð1Þ ¼Cð1Þ ¼1 and that both of them are increasing (in fact their derivatives on½r1;1are always positive due to the choice ofK). Then (3.13) implies thatr₂ <jf₁ðxÞj<1 for allxAW₁.

In order to prove that f maps W₁ to W₂, we have to check the second condition, namely thatPn2

j¼2jfjðxÞj²jf1ðxÞj^Qj=p<sin ^plnj_ln^f_r^1ðxÞj

2

for allxAW1. The definition of f1

and the relationdlnr1¼lnr2entail sin plnjf1ðxÞj

lnr2

¼sin plnjx₁^de^dtðxÞj lnr2

!

¼sin plnjx1j lnr1

cos p<tðxÞ lnr1

þcos plnjx1j lnr1

sin p<tðxÞ lnr1

dsin plnjx₁j lnr₁

1p²jtðxÞj² 2 ln²r₁

!

þpjtðxÞj lnr₁ ; by (3.11) and (3.12) we obtain that

sin plnjf₁ðxÞj lnr2

dsin plnjx1j lnr1

1 p²K₁²

2 ln²r₁þpðn11ÞK1

lnr1

: ð3:14Þ

SinceQ_k is non-negative fork¼2;. . .;n₂, then for allxAW₁we havejf₁ðxÞj^Qk=pc1 fork¼2;. . .;n₂, therefore the second inequality in (3.10) yields that for allxAW₁ the following chain of inequalities holds

Xⁿ²

k¼2

jfkðxÞj²jf1ðxÞj^Qk=pcXⁿ²

k¼2

jfkðxÞj² ¼Xⁿ²

k¼2

Xⁿ¹

j¼2

x_jhjkðxÞ

2

cXⁿ²

k¼2

Xⁿ¹

j¼2

jxjj²Xⁿ¹

j¼2

jhjkðxÞj²

cXⁿ¹

j¼2

jx_jj²jx₁j^yj=p X^n1;n2

j;k¼2

jhjkðxÞj²

<sin plnjx1j lnr1

1þpðn11ÞK1

lnr1

p²K₁² 2 ln²r₁

; ð3:15Þ

together with (3.14) this ensures that fðW1Þis contained inW2. Now consider the map

H:½0;1 W₁Cx7!

e^iyx₁^de^dttðxÞ;tXⁿ¹

j¼2

x_jh_j2ðxÞ;. . .;tXⁿ¹

j¼2

x_jh_jn2ðxÞ

ACⁿ²;

since for anytA½0;1the inequalities contained in (3.10) are both satisfied, we obtain that H maps ½0;1 W₁ intoW₂. MoreoverHðt;Þis holomorphic for any tA½0;1, and Hð1;Þ ¼ f, while it is easily seen that Hð0;Þ has degree d and therefore

degðfÞ ¼d, which concludes the proof. r

Even if in the general case the bounds given by (3.10) can be non-optimal, there is at least one case in which they are optimal. IfW₁¼Wðr1;0ÞandW₂¼Wðr2;0Þthen the following corollary holds.

Corollary 3.11.For anyyARand hAHolðW1;DÞthe map f :W1Cx7! ðe^iyx₁^d;x₂hðxÞÞAW2

belongs to Hd where d¼lnr2=lnr1. Vice versa, for any map f in Hd of the form fðxÞ ¼ ðf1ðx₁Þ;f2ðxÞÞ there exist yAR and hAHolðW1;DÞ such that the equality fðxÞ ¼ ðe^iyx₁^d;x₂hðxÞÞholds for allxAW₁.

Proof.Since the su‰ciency of the condition can be obtained by direct computation as in the proof of the previous theorem, we are left to prove its necessity. By Theorem 3.8 there existsyARsuch that f1ðx₁;0Þ ¼e^iyx₁^d; as f1does not depend onx₂we have that f1ðxÞ ¼e^iyx₁^d for any xAW1. By Proposition 3.9 there exists a holomorphic

functionh:W₁ !Csuch that f₂ðxÞ ¼x₂hðxÞ. Since fðW1ÞHW₂ we obtain that for allxAW₁

jx₂hðxÞj²<sin plnje^iyx₁^dj lnr2

!

¼sin plnjx₁j lnr1

:

Ifx⁰¼ ðx₁⁰;x⁰₂ÞAW1choosee>0 such thatjx⁰₂j²csinðplnjx₁⁰j=lnr1Þ eand setR¼ ðsinðplnjx⁰₁j=lnr₁Þ eÞ¹⁼²; then the following chain of inequalities holds

jhðx⁰Þj²c max

jx2jcRjhðx₁⁰;x₂Þj²¼max

jx2j¼Rjhðx⁰₁;x₂Þj²¼R²max

jx2j¼Rjf₂ðx₁⁰;x₂Þj²

¼maxjx₂j¼Rjf2ðx₁⁰;x₂Þj²

sinðplnjx₁⁰j=lnr1Þ ec sinðplnjx₁⁰j=lnr1Þ sinðplnjx₁⁰j=lnr1Þ e:

Lettingego to 0 we obtain thatjhðx⁰Þjc1 and then we are done. r 4 Complex geodesics for generalized annuli

In this section we prove some results on complex geodesics in generalized annuli.

Definition 4.1. Given x;zAD an extremal map j through x and z is a holomor- phic map j:D!D for which there exist t;sAD such that jðtÞ ¼x, jðsÞ ¼zand kDðx;zÞ ¼kDðt;sÞ. Acomplex geodesicfor the domainDis a holomorphic isometry j:D!Dwith respect to the Kobayashi distance ofDandD(that is, a holomorphic map which is extremal through any point of its image).

Analogous definitions can be given replacing the Kobayashi distance with the Kobayashi metric: in this case we speak of aninfinitesimal extremal mapand of an infinitesimal complex geodesic. Recall that for anyx;zAWand for anyz₀Aw¹ðxÞ, w₀Aw¹ðzÞwe have

k_Wðx;zÞ ¼inffkBⁿðz0;wwÞ~ : ~wwAw¹ðzÞg ¼inffkBⁿðz0;g₀^jðw0ÞÞ: jAZg; ð4:1Þ this equality yields both the existence of extremal maps through any couple of points in generalized annuli and a characterization of complex geodesics which will be use- ful in order to solve some problems concerning existence of complex geodesics inW.

Remark 4.2.For anyx;zAWthere exists an extremal map throughxandz.

Proof. Consider the covering ðBⁿ!^w WÞ: since Bⁿ is complete hyperbolic we can choose z;wABⁿ such that wðzÞ ¼x, wðwÞ ¼z and k_Wðx;zÞ ¼k_Bⁿðz;wÞ. Let j~

j:D!Bⁿ be a complex geodesic throughzandw(jj~does exist sinceBⁿis a strictly convex bounded domain inCⁿ) and setj¼wjj. Setting~ t¼jj~¹ðzÞands¼jj~¹ðwÞ, it is easily seen thatjis an extremal map throughxandz. r

Proposition 4.3. Let jj~:D!Bⁿ be a complex geodesic in Bⁿ; then the holomorphic mapj¼wjj~:D!Wis a complex geodesic inWi¤

kBⁿðjjðtÞ;~ jjðsÞÞ ¼~ inffkBⁿðjjðtÞ;~ g₀^jðjjðsÞÞÞ~ : jAZg ð4:2Þ for any t;sAD.Vice versa,ifjis a complex geodesic inWthen any liftingjj~ofjtoBⁿ is a complex geodesic inBⁿ for which(4.2)holds for any t;sAD.

Proof.Ifjj~:D!Bⁿis a complex geodesic for which (4.2) holds for anyt;sAD, then (4.1) and (4.2) imply thatkDðt;sÞ ¼kWðwjjðtÞ;~ wjjðsÞÞ~ for allt;sADand therefore j¼wjj~is a complex geodesic inW.

Vice versa, ifj:D!Wis a complex geodesic, thenkDðt;sÞ ¼kWðjðtÞ;jðsÞÞholds for anyt;sAD. Fixs0AD, choosea0ABⁿsuch thatwða0Þ ¼jðs0Þand letjj~:D!Bⁿ be the lifting ofj through a0, i.e. the unique holomorphic map fromDto Bⁿ such that j¼wjj~andjjðs~ 0Þ ¼a₀. Since the Kobayashi distance is contracted by holo- morphic maps, we then have

k_Dðt;sÞ ¼k_WðjðtÞ;jðsÞÞ ¼k_WðwjjðtÞ;~ wjjðsÞÞ~ ck_BⁿðjjðtÞ;~ jjðsÞÞ~ ck_Dðt;sÞ and therefore equality holds at each t;sAD. Equation (4.1) implies that (4.2) holds

for anyt;sADand this concludes the proof. r

It is well known that there exist no complex geodesics in the annuli WðrÞ: this state- ment can be generalized to any couple of points belonging toWðrÞ f0gHW.

Proposition 4.4.For anyx₁;z₁AWðrÞwithx₁0z₁there are no complex geodesics inW throughx¼ ðx₁;0;. . .;0Þandz¼ ðz₁;0;. . .;0Þ.For anyx₁AWðrÞand v1ACthere are no infinitesimal complex geodesics in Wthroughx¼ ðx₁;0;. . .;0Þwith tangent vector v¼ ðv1;0;. . .;0Þ.

Proof. We perform the proof in the case of complex geodesics, the case of infinites- imal complex geodesics is analogous and is left to the reader.

Suppose j is a complex geodesic through x and zand let jj~be a lifting of j to Bⁿ; by Proposition 4.3 the mapjj~is a complex geodesic inBⁿ. The form ofwimplies that w¹ðWðrÞ f0gÞ ¼D f0g, and hence jjðDÞ~ intersects D f0g in two distinct pointsz¼ ðz1;0;. . .;0ÞAw¹ðxÞandw¼ ðw1;0;. . .;0ÞAw¹ðzÞ. Since the image of a complex geodesic inBⁿ is an a‰ne subset of Bⁿ, i.e. the intersection of Bⁿ with an a‰ne line, we havejjðDÞ ¼~ D f0g, and thereforejðDÞ ¼wð~jjðDÞÞ ¼WðrÞ f0g. As WðrÞ f0gis a holomorphic retract ofW, we obtain that the map

j^

j:DCt7!j₁ðtÞAWðrÞ

is a complex geodesic inWðrÞand this is a contradiction. r As we already noticed, complex geodesics in W are projections on W of complex

geodesics in Bⁿ for which (4.2) holds for any t;sAD. To simplify computations, which are very long in general, we will focus our attention on complex geodesics in Wpassing through the pointP₀¼ ðpffiffir

;0;. . .;0Þ, i.e. complex geodesics inBⁿ passing through the origin. Up to holomorphic automorphisms of Dwe can therefore sup- pose thatjð0Þ ¼0; then, since complex geodesics inBⁿpassing through the origin of Bⁿ are given by maps of the formDCt7!tpfor any pAqBⁿ, we are led to investi- gate the following question, which is equivalent to the existence of a complex geode- sic inWpassing through the pointP0and with tangent vectordw₀ðpÞatP0:

Does the equality

k_Dðs;tÞ ¼inffkBⁿðg₀^jðspÞ;tpÞ: jAZg ð4:3Þ hold for anyt;sAD?

Denote by h;i the standard Hermitian product in Cⁿ and for any aABⁿnf0g definePa;Qa:Cⁿ!CⁿandsaARby

P_aðzÞ ¼hz;ai

ha;aia; Q_aðzÞ ¼zP_aðzÞ; s_a¼ ð1 kak²Þ¹⁼² and considerg_a:Bⁿ!Cⁿgiven by

g_aðzÞ ¼aPaðzÞ saQaðzÞ 1hz;ai :

Theng_ais an involution in AutBⁿ which mapsato the origin and

1 kg_aðzÞk²¼ ð1 kak²Þð1 kzk²Þj1hz;aij² ð4:4Þ holds for anyzABⁿ (for a proof see [1] p. 152–153).

Let g_tp be the involution defined above which mapstp to the origin; since tanh is increasing, by developing computations and by (4.4) we obtain that (4.3) is equiva- lent to

1 jsj²

j1tsj² d 1 kg₀^jðspÞk²

j1htp;g₀^jðspÞij²¼ 1 kg₀^jðspÞk² j1hg₀^jðspÞ;tpij² for anyt;sADand any jAZ.

Setting cj¼coshðjTÞ, sj¼sinhðjTÞ, p⁰¼ ðp2;. . .;pnÞ, W ¼diag½e^iy2;. . .;e^iyn and developing computations, we obtain the following question which is again equi- valent to the existence of a complex geodesic inWpassing through the pointP₀and with tangent vectordw₀ðpÞatP₀:

Does

j1tsj²cjcjþsjsp1hðcjsp1þsj;W^jðsp⁰ÞÞ;tpij² ð4:5Þ hold for anyt;sADand any jAZ?

A first, very simple algebraic remark again stresses the fact thatp⁰cannot be equal to zero. In fact, if p⁰¼0, takings¼ p1AqDandt¼ s, by continuity (4.5) implies 2c2ðcjs_jÞfor all jAZ, which is impossible sinceT >0.

Remark 4.5.Inequality (4.5) holds for anyt;sADand any jAZif and only if it holds for anyt;sAqDand any jAZ.

Proof. If j¼0, it is obvious that (4.5) is an equality for any t;sAD and there is nothing to prove. Now suppose that (4.5) holds for anyt;sAqDand any jAZ. First of all we prove thatcjþsjsp1hðcjsp1þsj;W^jðsp⁰ÞÞ;tpi 00 for allt;sADand for all j00. In fact, if cjþsjsp1hðcjsp1þsj;W^jðsp⁰ÞÞ;tpi¼0 for some t;sADand some jAZnf0g, then we obtain thathg₀^jðspÞ;tpi¼1, and thereforetp¼g₀^jðspÞand jtj ¼ kg₀^jðspÞk ¼1 which implies t;sAqD. Then the fact that (4.5) holds for any t;sAqDgives 1ts¼0, that is t¼s. Since j00, the unique fixed points of g₀^j are Ge₁, and hence sp¼Ge₁, that is p⁰¼0, which is a contradiction to the previous

remark. So, for any jAZnf0g, the holomorphic maps hj:DDCðt;sÞ 7! 1ts

cjþsjsp1hðcjsp1þsj;W^jðsp⁰ÞÞ;tpi AC

extend continuously to the boundary; ifjhjðt;sÞjc1 on the Shıˇlov boundary of the bidisk, thenjhjðt;sÞjc1 for anyt;sADand any jAZnf0g; this implies (4.5) for any t;sADand any jAZ. The other implication is trivial by continuity. r Then we are led to investigate on the following question: for which pAqBⁿ does

j1tsj²cjcjþsjsp1hðcjsp1þsj;W^jðsp⁰ÞÞ;tpij² ð4:6Þ hold for anyt;sAqDand any jAZ?

To simplify notation, we denote byq_j the quantityhp⁰;W^jp⁰iand obtain j1tsj²cjcjtsðcjjp1j²þq_jÞ þs_jðsp1tp₁Þj²;

settingz¼ts(which belongs toqDif bothtandsdo) we get j1zj²cjcjþs_jsp₁zðcjjp1j²þq_jþs_jsp₁Þj²

for all z;sAqD and jAZ. A simple computation proves that the above inequality holds for anyz;sAqDand jAZif and only if

2þ2jðcjjp1j²þqjþsjsp1Þðcjþsjsp1Þ 1jcjcjþsjsp1j²þ jcjjp1j²þqjþsjsp1j² for allsAqDand jAZ. SettingGj¼cjjp1j²þqj we get

2þ2jcjG_j1þs_jðcjþG_jÞsp1þs²_jðsp1Þ²jcjcjþs_jsp₁j²þ jGjþs_jsp₁j²