An introduction to Teichmüller spaces

(elr)

(1.12)

9a=9 m8

m8

comparing with (1.10), we conclude that an isothermal coordinate w for ds 2 exists if the partial differential equation esp roJ /'1 al€urprooc l€uraqlo$

'

t

z

o

uorlenba prluereJrp lerlred eql JI s?s1xe ue leql apnlcuoc erra'(0I'I) qlrrn Sur.reduroc

t

, l z ni +

r

l

z P l z l z m=l"dl n n l d Since an isothermal coordinate

for ds 2 satisfies

sagsr??sesp roJ nl eleurProoc leruraqlosl u3 a?uIS W

sernlrnrls I"ruroluoC pu€ sarnlf,ulg xalduro3 'g'1

21

1.5. Complex Structures and Conformal Structures

tz

22 22

1. l. Teichmiiller Teichmriller Space Space of Genus 9g

z

\

a

y

x Fig. F i g . 1.12. 1.12. = (a (o+ g ) ccos o sB, 0, 6 cos c o s
z=bsin
1

b

rl,@)== [' ---!4o, t/J,1,== t/J(
2d'(t/J)(dB ( { ) ( d 0 22++ddt/J2). rl}). ds 2 to = B 0+ fry' is an isothermal Thus w on M, isothermal coordinate coordinate for for ds ds2 M, which defines defines aa + it/J complex structure on M. M. Hence Hence R is aa torus. A little little more more calculation shows ft is shows generated by 1 and that R is / r, where is biholomorphic to C C/f, where r l- is aa lattice group generated ib/\/i,a2 --F.b2 . ib/V

1.5.2. of L.5.2. Reconstruction Reconstruction of Teichmiiller Teichmiiller Spaces by by Riemannian Riemannian Metrics Metrics genus gg ~ ) 1. Fix Fix aa closed closed Riemann surface surface R r? of of genus L. Take any local coordinate coordin ate zz on

R. R. 2 For an arbitrary on R, arbitrary Riemannian metric ds ds2 rR, from the uniqueness uniqueness of the (1.10), we p on R, expression we obtain aa globally defined coefficient J.I expression in (1.10), defined Beltrami Beltrami coefficient p (-1,1) and is called being < 1. Such a J.I is called the being aa differential differential form of type (-1,1) and IIJ.1l1oo 1. a Such llpllBeltrami melric. Bellrarni coefficient coefficientinduced inducedby by a Riemannian Riemannian metric. Let us coefficient of an us observe observe the relationship between between the Beltrami Beltrami coefficient orientation-preserving orientation-preserving diffeomorphism diffeomorphism and the one one induced induced by aa Riemannian metric. metric.

'@)+llvo/(a)w = uw '@)'llto l(a)w = @)t M g ~ M(R)/Diff+(R).

T(R) ~ M(R)/Diffo(R),

Theorem 1.8. The mappings which send an element [S,f) in T( R) to the equivalence and strong equivalence class, respectively, of a metric corresponding to f give the following identifications: :suotToc{zyuapt, 6urmo11ol ayy aatf

acuapatnba6uo.t7spuD nuelD { o7 0utpuodser,ror)uleru o to'fr1aa4cadsa.r,'sso1c 'g'I uraroaqJ - a m b aa q 7o t ( A ) J u ! l t ' S l l u e u e p u o p u e sq n q m s f u r d d v u e q J 'uorlress€

on each coordinate neighborhood (U, z), we obtain a Riemann surface R' whose complex structure is induced by the isothermal coordinate (U, w). Then the identity mapping fo: R -+ R' is an orientation-preserving diffeomorphism and its Beltrami coefficient coincides with J-I. Conversely, let an orientation-preserving diffeomorphism f: R -+ S be given. We can take a Riemannian metric dsr on S so that the complex structure induced by dsr is equivalent to the original one of S. In fact, the universal covering surface of S is biholomorphic to the Riemann sphere, the complex plane, or the upper half-plane (Theorem 2.2 of Chapter 2). We take the metric on S which is induced by the spherical, Euclidean, or Poincare metric, respectively (cf. §1.3 of Chapter 3). Then the pull back f*(dsi) of dsr under f gives the same Beltrami coefficient as that of f. Note that this assertion does not depend on the choice of a metric dsr inducing the complex structure of S. Such a metric f*(dsi) is said to be a Riemannian metric on R corresponding to f. In this way, we see that the set B(Rh of Beltrami coefficients of orientationpreserving diffeomorphisms on R is equal to the set of Beltrami coefficients induced by Riemannian metrics on M. Now, using the set M(R) of Riemannian metrics on R, we reconstruct the Teichmiiller space T(R) of R. Two elements ds 2 and dsr in M(R) are defined to be equivalent if there exists an element w in Dif f+(R) such that w: (R, ds 2 ) -+ (R, dsi) is conformal. Further, ds 2 and dsr are defined to be strongly equivalent if this w belongs to Dif fo(R). Denote by M(R)/ Dif f+(R) and M(R)/ Dif fo(R) the set of all equivalence and all strong equivalence classes of M( R), respectively. This observation and the Corollary to Theorem 1.6 lead to the following assertion. 3ur,rao11o; eql of p€al g'I ruaroeql o1 ,t.re1oro3 eql pu€ uorle^Jasqo srql

',,t1e,rr1cedser'(A)W p sass'elc acuaprr,rnba 3uor1sIIe pue ecuele.,lrnbe IIeJo las eq? pue (a)+IlpO/(A)W rq eloue6l '(A)"lIlO o1s3uo1aqo srql @)"ttpo/(A)W y Tualoamba fi16uo.t7s eq o1 paugepa.re|sp pue usp 'reqlrnJ 'l€ruroJuocq (I*p'U) - (zsp'A),o ?tsqtqrns (A)+ttlO ur r,l luerueleue slsrxe areqlJr Tualoaznba aq 01 paugap erc (g)4r ur |sp pue 6s'pslueuale o rI 'g Jo (U)J eceds rallnurqcraa aql llnrlsuocer a^r '9, uo scrrlaur u€ruueurelg Jo (U)hf les eql Sursn 'no51 'w uo s)rrleru ueruu€ruaru ,{q parnput sluarrlg:aof,ttu€rllag Jo las aql ol lenbe sr Ur uo srusrqdrouroagrp Surarasard -uorleluarroJosluer)Ueo, rurerllagJo I(U)5. ?eseql leql eesa,r.'.{e.nsqt uI 't o7 0utpuodsatloc uo culeur uvruuoulery A e eq ol pr€s sr (|rp)-t crrlaru e qcns'S'Jo ernlcn.rlsxalduroc eq1 Surcnpur[sp crrleru € Jo ecroq) eql uo puadep lou saop uotlress€ slql l€r{l eloN '/ Jo }€r{t se '(g luel)cgeor rur€rlleg erueseql se,rr3/ .repun |sp l" (trtp) -l >1ceq11ndeq1 uaql 'crrleru er€rurod .ro 'ueaprlcng 'pcrraqds eql ,(q raldeq3 Jo t'I$ 'gc) ,t1a,rr1cedse.r pe)npul sr qcrq/( S' uo crrleu eql e{€t alA '(U .ra1deq3Jo 6'Z ueroeqa) eueld-;pq raddn aq1 ro 'aue1d xalduoc eql 'araqds uu€ruerg eqt ol crqdroruoloqlq sl S Jo ac"JrnsSurra.roclesrelrun aq1 '1ce; uI 'SJo auo leur8rro aql ol lual€^rnbe sr Ltp fq pa?npur arnl?nrls xelduroo eql leql os S' uo |sp or.rlaurueruu€ruerll " a{€l uer eM 'ue,rr3eq * :/ ursrqdrouoegrp Surrr.raserd-uorleluerro ue 1e1'd1asra,ruo3 .g U 'rl qtl* saprcurocluercgeoc rureJllag sll pue ursrqdrouroagrpSurrr,.rasard-uorleluarJo u€ sl ,U * g. : { Eurddeur ,t1r1uepr aq? ueqJ '(^'n) eleurproof, leurreqlosl aql ^q pernpur sr arnlcnrls xaldtuoc asoq^r /U a)eJrns uuer.uerge urclqo all.'(z 'p) pooqroqq3rau eleurpJoof,q)Ba uo

zo --:- fl m8

zo n8

= ::-

uorlenbe rurcrlleg aql Surr'1os'I'g$ ul uaes ueeq s€q sV 'A to zsp rrrleur ueruueruerll e ,tq pecnpur luerrlgeor rtuerllag eql s! r/ 1eq1 esoddns '1srrg

First, suppose that J-I is the Beltrami coefficient induced by a Riemannian metric ds 2 on R. As has been seen in §5.1, solving the Beltrami equation 23

8,2

sarnlf,nrls l"urroJuoC pu" sarnl)nrlg xaldurop 'g'1

1.5. Complex Structures and Conformal Structures

1. 1. Teichmriller Teichmiiller Space Space of Genus Genus g

24 24

Notes Notes function theory originated with with Riemann's 1851 Gottingen dis1851 Gottingen The geometric function [181] and his 1857 1857 paper [182]. [182]. In connection connection with with multi-valued sertation [181] such as as algebraic algebraic functions, he introduced the concept concept of of the analytic functions such Riemann as the Riemann sphere. the sphere. He surface as a branched covering surface over Riemann Riemann also recognized recognized clearly the intimate intimate relationship between between holomorphic functions also plane. In [181], he proved and conformal mappings on a domain in the complex plane. [181], he Riemann's mapping mapping theorem theorem which asserts asserts that that any simply domain simply connected domain Riemann's plane with with mor'e mor~ than one one boundary point point is biholomorphic to in the complex pla.ne unit disk. In [182], he obtained the Riemann-Roch theorem. theorem. By By using this the unit [182], he determined the degree degree of of freedom of of finite finite branched coverings coverings over theorem, he determined Riemann sphere which which represent closed closed Riemann Riemann surfaces of genus genus g, and surfaces of the Riemann obtained the complex complex dimension mo m g of of the moduli moduli space of closed closed Riemann Riemann space of he obtained l, and g that is, m g = 0, 1, 3g -- 3 for g 9 = 0, 0, 1, 9 > ~ 2, 2, surfaces genus g, that is, ms 1, and 39 surfaces of genus of Riemann's work, work, we refer to Ahlfors Ahlfors exposition of respectively. For more complete exposition [4] [A-53]. [a] and Klein [A-53]. The standard surface, that that is, a one-dimensional standard definition definition of a Riemann surface, complex manifold was classic [A-1l1] was introduced for the first time in Weyl's classic [A-111] "Die Idee "Die 1913. Fl6che" in 1913. Idee der Riemannschen Riemannschen Flache" celebrated books The material of classical.Some of this chapter is classical. Some of the many celebrated on Riemann surfaces are Ahlfors and Sario Bers [A-6], Bers [A-13], are surfaces Sario Riema.nn [A-6], [A-13], Cohn [A-22], [A-22], Farkas Forster [A-32], Griffiths and Harris [A-39], Forster Griffths Kra [A-28], Farkas and Kra [A-39], Gunning [A-32], [A-28], [A-40], Siegel [A-98], Schlichenmaier [A-95], Jones and Singerman Singerman [A-48], [A-95], Siegel [A-98], and [A-48], Schlichenmaier [A-40], Jones are Springer [A-99]. For details of topology on surfaces, there are further books by details surfaces, Springer [A-99]. Moise Birman 6, Moise [A-75], Birman [A-18], [A-75], Stillwell [A-I0l], [A-101], [A-4lj, Chapters 1 and 6, [A-18], Harvey [A-41], curves, we we refer refer to and Zieschang, algebraic curves, Ziescha,ng,Vogt and Coldewey Coldewey [A-114]. [A-114]. For algebraic the books by Arbarello, Cornalba, Griffiths and Harris [A-9] , Griffiths [A-38], Grffiths books [A-9], [A-38], Mumford [A-78], Namba [A-82], and Shafarevich [A-97]. space The moduli space of Shafarevich Mumford [A-78], [A-97]. [A-82], tori considered in §2 is studied in the context of elliptic curves, elliptic integrals, curves, integrals, isstudied considered $2 see Clemens and theta functions. Clemens functions. For an an interesting exposition on this subject, see [A-21], Jones and Singerman Singerman [A-48]. [A-48]. [A-21], and Jones For textbooks on Teichmiiller spaces, spaces,there there are are Abikoff Abikoff [A-I], [A-1], Ahlfors [A-2], [A-2], Nag [A-80]. Gardiner [A-34], [A-80]. [A-68], and Nag [A-41], Krushkal' [A-60], [A-60], Lehto [A-68], [A-34], Harvey [A-41], For and [11], papers on this subject, consult articles articles by Ahlfors [8] For expository papers [11], [8] and spaces as in and Bers [22], §4 approaches to Teichmller Teichmller spaces as $4 and §5 $5 [40]. The approaches [22], [29], [29], and [40]. respectively. are tomba [71], Eells [62], Fischer and Tromba are found in Earle and Eells [7L], respectively. [62], [63], [63], and Fischer

=

=

' H J o ' J ' 3 s e c o t t n su u D u r e l y st acoltns-uuouaty peloauuo? 'tuaroaqtr, uoltBznuJoJrun

Uniformization Theorem. (Klein, Poincare and Koebe) Every simply connected Riemann surface is biholomorphically equivalent to one of the three Riemann surfaces C, C, or H. a?rtn e1t lo auo o7 TualoatnbaQparydloruoptfq fi1du.tzsfueag (aqeox pue ?recurod '.r!"IX)

'(w6d '[y-y] sroylqy eas) {$p }run aq} ot crqdrouroloqrq sr lurod f.repunoq euo u"q? arotu qlra C ur ul€urop pelceuuot fldurts dra,ra 1€rll slresse qcrq/rr uraJoall ,utdrlou,t s.uuDur?ta ol Pa)nPer sI rueroaq? sHl'c aueld xelduroc eqt ur sur€urop ro; 'r(gercadsg 'sac€Jrns uu€tuerlf roJ sploq uaro -eql uollezlruloJlun eql Pallec sl q?lq,$ ?oeJalqe{r€IueJ e }sq} u^lou{ 11ams! 1I

It is well known that a remarkable fact which is called the uniformization theorem holds for Riemann surfaces. Especially, for domains in the complex plane C, this theorem is reduced to Riemann's mapping theorem which asserts that every simply connected domain in C with more than one boundary point is biholomorphic to the unit disk (see Ahlfors [A-4], p.247).

rrreroaql uorlBzlruroJlun'I'z

2.1. Uniformization Theorem '6 snuaS;o tg aceds rellnurq)Iel eql ql!,u PagluePl q ll }"qt alord pue'g-ngll uI lasqns e se td aceds alcr.rg aql eu$ep aal '(6 l)f snueS3o sereJJnsuu€r.uarlr pasolc Surluasardar suelsfs Ierruorr€cEursn'g uorltas ur 'fgeurg sdno.rEuersrl?ndJos.ro1e.raua3;o 'ralel pasn ele qtlq^\ 'sdno.r3 uersqcr\{ yo satl.redord freluaurala euros a,rord airrr'7 pue t suolltes uI '; dno.r3 u€rsrlf,nde Aq H Jo J/H aceds luatlonb e fq paluasardar q (Z ?)n snua3 ;o ereJrns uuetuerg pasolc fre^e leql epnl?uot a,rl 'dem srql uI 'suolleuroJsueJl snlqgl 'dnor3 'relnarlred uersqrnd e uI tr J II€r ell'' 11 = U uerl^r 'Il ro'C '? ;o Surlsrsuocdnor3 e sB g uo rtlsrionurluo?srpfpadord slce J pu" o1 lualelrnba fllecrqd.rouioloqlq sr g, 'ruaroaql uorleznuJoJrun eql fg '6 uorlaai ur palrnrlsuoc s-rJ dnor3 uorleur.r5gsue.rlSurra,roc sll 'U Jo U eceJrns Eutreloc tg, Iesra^run eql JePrsuo?arn aae;rns uueuaru frelrqre ue ol ueroeql uollezfluroJ -run eql {1dde o1 repro uI 'g aueld-g1eq.reddn eq} ro '9 aueld xalduroc eql 'e a.raqdsuu€rueq aql :sae"Jrns uueruenr eerql eql Jo euo o1 crqdrouroloqlq $ eceJ -rns uusruarg palreuuoc fldurrs .,(Jea leqt slrass'e qcrqilr 'aqaox pue 'atecuto4 'ura1y o1 enp rueroeql uorlsznuroJlun aql urc1dxa er,r '1 uotlcag u1 'sdno.r3 u€rsqrnJ pue 'suorleurroJsueJl snlqgl [ 'sece3:tnsSurralroc lesrellun uo s]c€J f,Issq 'asodtnd 'sace3:rns sn{l roJ uu€ruerg Jo ruaroaql uort"z-turoJlun eql apuro.rda,u 'd snua8 (e-rsg ur }asqns e se pezrper sr (6 l)f snua3 ;o eceds e{llq aq} pag"c fl rlcrq^{ ;o aceds rellnurq)ral eql l"qt ^{oqs ol u .raldeqc luesard eql Jo esodrnd aqa

The purpose of the present chapter is to show that the Teichmiiller space of genus g(~ 2) is realized as a subset in R 6g-6, which is called the Fricke space of genus g. For this purpose, we provide the uniformization theorem of Riemann surfaces, basic facts on universal covering surfaces, Mobius transformations, and Fuchsian groups. In Section 1, we explain the uniformization theorem due to Klein, Poincare, and Koebe, which asserts that every simply connected Riemann surface is biholomorphic to one of the three Riemann surfaces: the Riemann sphere C, the complex plane C, or the upper half-plane H. In order to apply the uniformization theorem to an arbitary Riemann surface R, we consider the universal covering surface R of R. Its covering transformation group r is constructed in Section 2. By the uniformization theorem, R is biholomorphically equivalent to C, C, or H, and r acts properly discontinuously on R as a group consisting of Mobius transformations. In particular, when R = H, we call r a Fuchsian group. In this way, we conclude that every closed Riemann surface of genus g(~ 2) is represented by a quotient space H / r of H by a Fuchsian group r. In Sections 3 and 4, we prove some elementary properties of Fuchsian groups, which are used later. Finally, in Section 5, using canonical systems of generators of Fuchsian groups representing closed Riemann surfaces of genus g(~ 2), we define the Fricke space Fg as a subset in R 6 g- 6 , and prove that it is identified with the Teichmiiller space Tg of genus g.

Fricke Space

ar€ds a{rlqjt u raldBrlc

Chapter 2

26 26

2. 2. Fricke Fricke Space Space

Remark. These Riemann Riemann surfaces surfaces 0, C, C, C, and and /y' H are are not not mutually mutually biholomorphibiholomorphiRemark. These The Mobius transformation transformation tr w = (z --i)lQ i)j(z *+ f)i) maps biholocally equivalent. The morphically 11 H onto onto the unit unit disk 4, Li, and hence hence we often use use the unit unit disk 4Li morphically of the the upper half-plane IfH.. instead of

=

Corollary. A A closed closed Riemann Riemann surface surface of of genus genusO biholomorphic equiualent equivalent to 0 is biholomorphic Corollary. C.. Thus modu./i space space Ms M o of of closed closed Riemann surfaces surfaces the Riemann spheree Thus the moduli the Riemann sphere of genus genus 0 consists consists of of one one poinl. point. of

closed Riemanil Riemann surface surface R R of of genus genus 0 is simply simply connected, connected, the Proof. Since Since a closed uniformization theorem implies that that .R R is biholomorphic to one of of the three uniformization C, C, C, and 11. H. Since Since .R R is compact, itit should be biholomorphibiholomorphisurfaces e^, Riemann surfaces equivalent to C. C. !0 cally equivalent For proofs of of the uniformization uniformization theorem, we we refer to to books on Riemann also Ahlfors notes of See also [A-3]. For historical surfaces listed 1. in the notes of Chapter See surfaces [A-3]. and expository accounts, accounts, see Abikoff [2], Bers [29] [36]. and Bers and see Abikoff [29] [36]. [2], Among standard proofs proofs for the uniformization uniformization theorem, theorem, there is a method in which a mapping function is is constructed by using Green Green functions. functions. Let us us example from elucidate the concept concept of Green Green functions by using using an intuitive intuitive example elucidate electromagnetism. We rega.rd regard a Green function on a Riemann surface R as as the surface .R Green function electromagnetism. given at a point p and whose whose electric potential potential on R where where a positive charge charge is given boundary is earthed. Mathematically, when z is a local coordinate around p is is earthed. p as minimal on R, we define the Green function g(.,p) on R with pole at as the minimal ,R we define Green are harmonic function in the family of positive superharmonic superharmonic functions which are -loglz p. existenceof a in R The existence z(p)lI at singularity -log Iz - z(p) and have have the singularity .R - {p} {p} and "capacity" of the boundary of R is indepent of Green rt and is depends on "capacity" Green function depends with disk Li 4 with the choice example, the Green Green function on the unit disk choice of aa point p. For example, there are no the other hand, pole 1(1 - Zo z)j(z - zo)l. given by log z")1. On the other hand, there are no pole at Zo logl(l-z;z)/(z zo is is given C or C. Green functions on C Green = g(-, g = g(,p)p) on R. Then we we Now, assume exists aa Green Green function 9 assume that there exists obtain aa biholomorphic mapping ff:: RR'-'+ -- Li; A;

exp(-g(q) (-s(q)+ ig*(q)), ic.k)), I(q) f k) == exp

(2.1) (2.1)

g* is is aa where on R -- {p g* is where g* is the conjugate conjugate harmonic function of 9g on {p }. }. Note that g* 1r (n E multi-valued function whose periods are becauseof the singularity of whoseperiods are2n 2ntr e Z) because is aa single-valued single-valuedholomorphic holomorphic 9g and Hence, I/ itself is R. Hence, and simple simple connectedness connectednessof R. i.e., aa is univalent, univalent, i.e., function on we see see that I/ is argument principle we on R. .R. By the the argument biholomorphic mapping of R (see Fig. 2.1). 4 (see Fig. 2.1). -Ronto Li on R. We functions on R. We Next, we where there there exist exist no no Green Green functions casewhere we deal deal with the case .R' is such connected subdomains E take of simply connected subdomains of R such that Rn is aa take aa sequence sequence{{ Rn n' }~=1 }Lr for at coversR .Rexcept except for n, that U~=l Rn covers relatively eachn, Uf=rR" subsetof Rn+l E +r for each compact subset relatively compact g,. with with the the common common most the Green function gn point, and every Rn has the Green function and that that every ftr has one point, most one biholomorphic pole we can can construct construct aa biholomorphic way as as before, before, we pole p. p. Then, the same same way Then, in in the - Li constant, by aa suitable suitable constant, eachIn mapping for every every n. n. By By multiplying multiplying each Alor mapping In Rn n -/' by fn:: R - {{ w r' }} SO so tr E C Ilwl mappings F F'n :: Rn R- -we < get aa sequence biholomorphic mappings we get sequenceof of biholomorphic eC I lr.rll< Tn

sBurra,rog lesrel-ru1'Z'Z

LZ

27

2.2. Universal Coverings

+--I I I

lrnf/1 So1 (*)"6;o qder3

((4t)"0 - (r)6 uort?unJuaarg erlt Jo qder3

graph of the Green function 9(Z) = 9o(f(Z»

graph of 9o(W) = log

l/lwl

f

[€ruJoJuot -~

conformal l r > l m l l :v

fldurrs e

a simply connected Riemann surface

L1={lw!
Fig. 2.1.

'r'z'8tJ

'Ur*n ;o Eurddeu erqdrotuoloqrq parrsape sef,npur

uo uorl?unJ fr-uliaqf leql

that the limit function on R onto C or C.

U~=lRn

induces a desired biholomorphic mapping of

'c ro oluo c u

stur.ra,ro3 lesJa,rrun'e'Z

2.2. Universal Coverings 'e?sJrns uueruarg f.rerlrqre ue Jo aceJrns Surrarroc pelceuuoc fldurs € lcnrlsuoc aaa'uraroeql uorlszrruJoJruneql ,f1ddeo1 rapro uJ

In order to apply the uniformization theorem, we construct a simply connected covering surface of an arbitrary Riemann surface. sdno.rg rrorlerrrroJsue4l, Eurre^oC puB saceJrns Eurraaog Jo suor+Iugo1, 'TZ'Z

2.2.1. Definitions of Covering Surfaces and Covering Transformation Groups 'U I€sJelrun y Jo acottns |ut.taaoclDsrearun" U pue'g 1o 6uuaao) IDsrearune (A'o'A) II€) e^\ 'palcauuoe flduns sr U uaqm '.raillrnd 'U otuo g 1o uo4cato"tdeq| pa11ec6qe sr z deur Surra,rocaql 'Ujo acn!.rns|uuaaoc e g iue 'g 1o |uttaaoc e (A'v'U) IIec aaa'r1ooq srql uI 'crqdlouroloqlq q /? * ]1 :t deur palcr.rlsar aql '11 p (A)r.-" a3eun asra.rlura{l;o 1 luauodruoc pe}cauuor qcee roJ leql qcns pooq.roqqSreue s€q d fra,re y dout, Fuueaoc e eq ol pres sr lurod , A Jo 'sac€JJnsuueuaru aq pue Ur Ar ]erl A - U : ! Surddeur crqdroruo-Joqanrlcalrns y

Let Rand R be Riemann surfaces. A surjective hol'omorphic mapping 11": R ---+ R is said to be a covering map if every point p of R has a neighborhood U such that for each connected component V of the inverse image 11"-1 (U) of U, the restricted map 11": V ---+ U is biholomorphic. In this book, we call (R, 11", R) a covering of R, and R a covering surface of R. The covering map 11" is also called the projection of Ronto R. Further, when Ris simply connected, we call (R, 11", R) a universal covering of R, and R a universal covering surface of R. A universal

2. 2. Fricke Fricke SPace Space

28 28

covering of of R R mea,ns means that that itit is is the the "highest" "highest" covering covering surface surface of of all all coverings coverings of of covering R (cf. Theorems 2.2 and 2.4, and the remark remark in in $2.2). §2.2). Theorems2.2 and,2.4, r? Example I.1. We give a few simple simple examples of of covering surfaces. surfaces. Example

'11": C C -- C C -- {0} {O} be given given by r(z) 'II"(z) == e'. eZ • Then C C is a universal universal covering covering (i) Let r: ssurface u r f a c of e oC f C-- {{O}. 0}. niversal i v e n bby - .d e22".iz. ((ii) i i ) LLet y r'II"(z) ( z ) -= e ' i " . TThen h e n IH/ iiss a uuniversal bbe e ggiven : H + A --{ 0 {O} } e t r'11": covering surface surface of of A .d - {{ 0 }. }. covering } ositive *- C -- {{O} i v e n bby h e r e n iiss a ppositive ( z ) = zzn, n, w where ((iii) i i i ) LLet e ggiven y r'II"(z) : C --{ 0{O} e t r'11": 0 } bbe universal but it is not a integer. Then Then C -- {{O} covering surface of but it not of itself, a covering surface is 0} covering surface. surface. covering nd A For given 1),) , sset = exp and A== {{w E C Il tr < < lIwl < i v e n ^A(> e x p(-2'11"2jlog ( - 2 r 2 / l o g , \A) ) a ((iv) iv) F (> 1 et r = or a g .l < w € w h e r e l o g z d e n otes l}. Define'll": exp(2'11"ilogzjlogA), (z) = e x p ( 2 t r i l o g z / l o g . \ ) , where logz denotes A bby y r'II"(z) e f i n e r ; H - --- + A 1 ). D its principal principal branch. Then Then 11 H becomes becomes a universal covering surface of of the its annulus ,4. A. e /, a n d llet e t r'II"b be (v) bee aal alattice byy 1 aand and n d aa ppoint o i n t r €E IH, t t i c e ggroup r o u p ggenerated e n e r a t e db ( v ) tLet et 4T b C is a universal projection of of C onto onto the quotient quotient space Then space C fj rfr.T • Then the projection covering surface of the torus C/ C j lr.T • covering surface of

=

r

r

=

'1, R - R wittr '1ro^l is-called Any biholomorphic mapping "I: 1"0"1 = z'II" is called a coaering covering fr, E with Any (R,r, denote by (R,r,R). given covering covering(ft, transformation '11", R). For a given '11", R), denote covering(R, transformalion of a covering mappings, rl' the set covering transformations. By the composition of mappings, set of all its covering group (,R.'r,,R). forms a group, which is called the covering transformation group of (R, 11", R). transfonnation coaering called lgroup of (R, (-R,'r,.R) In particular, particular, we 11", R) lransformation group uniaersal covering coaering transformation we call I the universal if R is a universal covering surface of R. r?. if ,R

r

r

Example 2. We give covering transformation groups (i)', ... , (v), of coverings (i), ... , (v) in Example 1. The notation ("11,' .. ,"In) expresses the group generated by "11, ... ,"In' - *2tri. with "I1(Z) (i)' 2'11"i. (i)' r f = ("11) (rr) with 71(z) zz + (ii)' (ii)' r f ==( r("11) r ) w with i t h 7"I1(Z) 1 ( z )== Z z+ * 11.. - zz exp groupof order n. ordern. finitegroup is aa finite (iii)' r (2'11"ijn) , which is exp(2riln),which with "I1(Z) ('rt) with 1= ("11) 71(z) = )2. with "I1(Z) (iv)' AZ. (iu)' r r = ("11) (zr) with 71(z) "fz(z)= Z z+ a,nd"I2(Z) where"I1(Z) (T,72) -- r f,,T , where + r. (")' r (v)' 7 = ("11,"12) * 11 and 7{z) = zz +

=

=

= = =

= =

=

=

=

2.2.2. Construction Surfaces Covering Surfaces of Universal tJniversal Covering Construction of means aa path on surface R R means First of on aa Riemann Riemann surface definitions. A path we need need several several definitions. of all, all, we points and C(0) 1]. The continuous The points C(O) and the interval interval [0,1]. where.II is is the I -- R, R, where curve C: C: 1 continuous curve [0, also points of respectively. We We also of C, C(l) lern'inal points C, respectively. initial and and terminal to be be the the initial are said said to C(1) are confusion book, if if no no confusion the book, say path from Throughout the to C(1). from C(O) c(0) to c(1). Throughout that C is aa path say that c is same letter letter C. C. is by the the same possible,its also denoted denoted by is also its image image C(I) C(/) is is possible,

s8urraaogl"sra^ru1'Z'Z

2.2. Universal Coverings

29

'd = (ilu tlA u\ il,"r;3:"9:r|"'; A uo CWed e sI Ar uo,, qled e o't11llv 61 pre6 sl q!. q d lurod y 'U ecsJrnsuueruelg e;o Sutrerroce eq (g')L'A) p"l '0),C lurod leururrel eql pue (0)C lutod prltut aq1 qf!^A U uo ,C . C rlled e 1aBaar 'rg;o lurod FIlluI eqt qq^a C;o lutod Ieunurel eq1 Eurlcauuoc ,(q '(g),9 = (t)C lsq? qcns A ao tC Pue C sqled oarrlrog

For two paths C and C' on R such that C(1) = C'(O), by connecting the terminal point of C with the initial point of C', we get a path C . C' on R with the initial point C(O) and the terminal point C'(1).

Let (R, 'Tr, R) be a covering of a Riemann surface R. A point p in R is said to be lie over a point P in R if 'Tr(p) = p. A lift of a path C on R is a path G on R with 'TroG C. Now, by using paths we shall construct concretely a universal covering surface of a Riemann surface. Fix a base point Po on a given Riemann surface R. Let (C, p) be a pair of any point p on R and any path C on R from Po to p. These two pairs (C, p) and (C', p') are equivalent if p = p' and C is homotopic to C' on R. Denote by [C, p] the equivalence class of (C, p). Let R be the set of all these equivalence classes [C, p].

=

= 4z turl eql }eql ees err,r')Lodz xelduroc parrnba.raq1 splarf {(!z'!2)},tgurel -les 'gg ur ureurop pelcauuoc fldurrs e s\ dn l€qt qrns (or'on) pooqroqq3rau al€urprooc e e{sl 'U p [d'C] - gf lurod fue ro;'1ce; u1 'Surddeur ctqdrouroloq '1xe11 e setuof,aqA * U:1, l€qt os Ar uo alnltnrls xalduroc 3 euueP a,n 'deur e Eurra,roc Jo uorlrpuoc aql sagql"s pue U oluo Ur yo Eutddeur (uolllnrlsuof, eql ,cg 'a : (la'gl)r. rq eas o1 fsea q snonurluoc € sr l 1l 1eql ua,rr3 uorlceto.rd eql ee U * A:)L p1 'sceds 1ecr8o1odo1 JroPsneg € seuoteq g ueql 'A u\ 4 Jo spoor{roqq3rau pluaurepunJ Jo uralsfs e ouuep s^\ 'd2 aseql p;" dn uea/$leq aauapuodsauoc euo-ol-euo Isf,Iuouef, e a^€LI e,t 'uteutop ig,'h palcauuoc ,tldurrs e sr.d2 acurg 'D o1 d uroq d4 ur pauteluoc qled frerlrqre ue st d4 ';2 ul .{eirrr e qcns ul ur e sr b pw 1eq1 lutod 2I lb'bC.9] slurod II3 Jo les aqt dn pooqroqq3rau e 4p fq alouaq 'U ul ul€ruop pelceuuoc fldurrs-e $ qtul^r d lo '9, ol Peau an '1srrg e ecnPor?ul rog uo ,(3o1odol {ue e{el 'g lurod { Jo ld'Cl 's,lrolloJ se qqt leqt ees eA\ e satuoreq eceJrns Surra,roc U l"srelrun Jo B '[d'g] sasselcacuale,rrnba aseq?ileJo les eql eqU lef '(d'C) Jo ssplt acuele,unbaeqlld'CJ dq aloueq'g uo tC ol crdolouroq sr j pue d - d !\Tuapanba erc (d' ,g) pue (d'9) srted oarrl aseq; 'd o1 od uror; A uo C r{led fue pue U uo d lutod fue 3o rted e eg.(d'C) 'ecsJJns ulretuelg e 'U eosJrns uuetuell{ ualrE e uo od Jo lurod es€q 3 xld te1 'alotr1 ecsJrns Surre,rocl"srellun e i(lalarauoc lcnrlsuoc lleqs arrrsqled Sutsn fq

We see that this R becomes a universal covering surface of R as follows. First, we need to introduce a topology on k For any point p = [C,p] of R, take a neighborhood Up of p which is a simply connected domain in R. Denote by Up the set of all points [C· C q , q] in R in such a way that q is a point in Up and C q is an arbitrary path contained in Up from p to q. Since Up is a simply connected domain, we have a canonical one-to-one correspondence between Up and Up. By these Up, we define a system of fundamental neighborhoods of pin k Then R becomes a Hausdorff topological space. Let 'Tr: R -+ R be the projection given by 'Tr([C,p]) = p. By the construction, it is easy to see that 'Tr is a continuous mapping of R onto R and satisfies the condition of a covering map. Next, we define a complex structure on R so that 'Tr: R -+ R becomes a holomorphic mapping. In fact, for any point p = [C, p] of R, take a coordinate neighborhood (Up, zp) such that Up is a simply connected domain in R. Setting Zp = Zpo?!, we see that the family {(Up, zp)} yields the required complex structure on R. 'ur uo ernlcnrls

R,

'p?peuu@ fiylul'ts puD pep?uuoc st 'aaoqo pautoldxa sD peptulsuoc 'g acottns ?qJ 'T'Z Btutuarl

Lemma 2.1. The surface

constructed as explained above, is connected and

simply connected.

fdolouroq € eleq am'f11eutg 'I ) n roy (n1)"g = (n)(c't)p fq g'uo (t't)p qled e ausep a,r'7 x / f (s'?) fue ro3'uaqJ,'1 3 s due .lo;'d = (I)"f, = (0)'J pue 'C = rI 'oI - og sagsrles tl'{ ("'.)uf = 'd } leql q?ns uaql uee^.rleq rldolouroq € eq U * 1 x I ii p1 'o7' o1 crdolouoq sI C leql su€etu q?Iqtl 'l"d'Cl,tq peluasarde.r 'q1ed pasoll " $ 'fod'Cl -C ecurs lod'ollleqt epnpuoa am sl C Jo lurod leurturel eql puie 'od lutod es€q IIII^a g uo qled Pasolee sI 5t uaql 'ei" '[od'olTol crdolouroq q C tnd [od''I] lutod eseq q]p\ g uo g qled Pesop ',r,r,o11 fierta 1eq1 aes ol luelrgns fl 1I 'Paleeuuos ,(ldturs q f€qt b,rord arr,r !f '[d'g] ol 'pelceuuoc q U leql satldtut qclqar lod'oll uro+ U uo I qled € a^eq al,r '7 ) s fle,ra iog [(s)g'"C] = (s)g 3ur11ag'I ) 1ro; (ls)j = (rt"C rq A uo'C Wed e eugep'1 I s q?ea ro,{'U uo qled e Aqlod'o1f ql.I^{ pal?auuoc sl lI f [d'g] lurod i(ra^a teql A{oqsol sjcgns }l 'U Jo ssauPa}ceuuoc aq1 a,rord ol1[t'0] - I ) ?,(ue ro; oa = (t)"1,(q paugap g-uo qled eq] aq oI p"I'too.t'4

Proof. Let 10 be the path on R defined by Io(t) = Po for any t E I = [0,1]. To prove the connectedness of R, it suffices to show that every point [C, p] E R is connected with [Io,Po] by a path on k For each s E I, define a path C$ on R by C.(t) 9(st) for t E I. Setting G(s) [C.,C(s)] fo: every s E I, we have a path Con R from [Io,po] to [C,p], which implies that R is connected.

=

=

Now, we prove that R is simply connected. It is sufficient to see that every closed path G on R with base point [10 , Po] is homotopic to [Io,Po]. Put C = 'TroG. Then C is a closed path on R with base point Po, and the terminal point of G is represented by [C, Po]. Since G is a closed path, we conclude that [Io,Po] = [C, Po], which means that C is homotopic to 10 , Let F: I x I -+ R be a homotopy between them such that {F. = FC. s) }.El satisfies Fo = 10 , F 1 = C, and F.(O) F.(1) Po for any s E I. Then, for any (t, s) E I x I, we define a path C(t,.) on R by qt,.)(u) = F$(tu) for u E I. Finally, we have a homotopy

=

=

30 30

2. 2. Fricke Fricke Space Space

F1r,"; E between b"t*""n C and[Io,Po] F(t, s):; I Xx I -- t R i and by setting settingF(t, F1t,"; s) = [Get,.), F.(t)] for for llo,polby [C1r,,;,r"(t)] (r, s) s) E 1 1. Therefore, we (t, I x I. Therefore, we conclude that R is simply connected. 0B conclude E is connected. € putting these these observations observations together On putting together with the uniformization theorem, theorem, we we obtain obtain the following following theorem. theorem. ,

Theorem Theorem 2.2. For Fo'revery eoerg Riemann surface surfaceR, there there exists exislsa universal uniaersalcovering coaering surfaceR which is is biholomorphic surface R of R, which biholomorphic to to one one of the the three three Riemann surfacesC, Riemann surfaces A, C, or H. C,orH. Throughout Throughout this section, section, as 6 ao universal universal covering covering surface surface R E of aa Riemann surface R r? we we always surface always take take the one one constructed constructed above. above. From the construction of such such aa universal universal covering, covering, it is get the is easy easy to get following lemma by an an argument argument similar to that used used in the case case of analytic continuation (Ahlfors [A-4], 8). [A-4], Chapter 8). (Existence and path) For any Lemma Lemma 2.3. (Existence and uniqueness uniqueness of of aa lift path lift of of aa path) any path point p, p, on with point C on R initial and for any P R initial and for ang there exists erists a unique R oaer there unique F of Rover lifl C wilh initial initial point p. lift C of C with fi. (Litt of Theorem Ttreorem 2.4. (Lift of aa mapping) mapping) For Riemann surfaces surfaces Rand let R and S, let (R,Tn,R) (S,trs, (R, 1fR, R) and 1fs, S) be and (5, be their universal uniaersal coverings coaeringsconstructed construcledas as explained etplained preaiously,respectively. giuen an previously, respectiaely.Then Then given an arbitrary arbitrary continuous continuousmapping mapping ff:: R R-- t S, S, there exists etists a continuous there -t 5 continuousmapping mapping it R fr,--with f01fR osoi. This Thit mapping mapping S with forp== 1fSoj. - (1, where is uniquely uniquety determined tleterminedunder iI is untler the the condition conditionthat that i(pd wherePI fr. and € Rand i@) = ql, fu E = ff(1fR(PI». are such such that that 1fs(qt} rs(Q1) = S are iiI e 5 4r E brn(Fr)) Moreover, Morvooer, if if ff is differentiable differentiable or holomorphic, holomorphic, then then if is also also differentiable differentiable or holomorphic. holomorphic.

i:

i:

This mapping -t 5 f: R mappine i t RFl.ir called called a lift lift of f: ^9is .R -* t S.

= =

= =

Proof of Theorem Theorem 2·4· 2./. Setting and ql get a Setting PI we get fu = [CI,pd 4t = [DI,f(pd]' lCr,pr] and [Dt,/(pr)], we 1 .f m a p p i n g defined d e f i n e dby b v i([C,p]) p o i n t s . mapping = [D f(Cdf(C),f(p)] for all points [C,p] ( C t ) t ( C ) , / ( p ) l . a l l f(lc,pl) [C,p] _ l D tI. f Jor - ql and f01fR - 1fSoj. R. Then it it is is obvious obvious that that i(pd in k Since 1fR and 1fs nsol. zrp a.nd n5 Since /(f1) {1 fSrp are locally biholomorphic and f/ is continuous, are continuous, / must be continuous. continuous. It It is is also also that ifif f/ is differentiable j. the The uniqueness uniqueness trivial differentiable or holomorphic, then so trivial that so is /. 2.3. 0 assertion assertion follows follows from Lemma 2.3. D

i

(Uniq:reness of Remark. (Uniqueness universal covering) covering) For any two universal universal coverings of universal Rernark. (R,r,R) (R, 1f, R) and (r?1 R I ,,11,R) 1fl' R) of a Riemann Riemann surface R, there there exists exists a biholomorphic biholomorphic surface r?, mapping g cp of R to r?q RI with with 1fIOcp example, Ahlfors and Sario [A-6], Trog - 1f. n. See, See,for example, [A,-6], Theorem 18A of of Chapter l. L

l

=

Proof. To prove (i), suppose that 7I"(p) = 7I"(if) = p. Then we ~ave p = [C1 , p] and if [C2 , p] for some paths C 1 and C 2 on R. Putting Co C 2 . C 1 1 , we see that I = [Co]. satisfies if = I(P)· To see (ii), take a point pER, and set P = 7I"(p),p = [C,p]. Choose a neighborhood U of p in R which satisfies the condition in the definition of a

e Jo uorl-rugep ar{} ur uorlrpuo? aql sessrl€s qclq,$ u ur d 3o 2 pooq.roqqStau e esooqo 'ld'Cl = 4'@)" - d 1as pue 'g f gl tutod € a{€t '(rr) aes o5 " '(g),0 - i sagsp, 'l"C) -- L ', - o5l 3ur11n4 'A uo zg pue IC sqled euos rc1 t€rl? aas am IC . 7'C fd'z7l = p pue [d'rCj = 4 a,re{ er'ruaql 'd = (!)y = (4)a 1eq1esoddns '(r) a.,'o.rdoa '{oo.t4

=

=

'Q* X u ( y ) r p q l q c n s J ) L s l u a u a l a f i u o u t Q a T g u { I s o l l t?' o a . t ' oa " r ' a q y ' g sSaoi (rrr) lo y TasqnsTcoilutocfiuo.to!'st IDW:ry uo fipnonu4u@srp fr.1.r,ailo.ti1 'sTurodpat{ ou soy fiyquapt.ayy.tol Tilacxa - J ) L f r ^ r a a"ar o t Q = n U ( d L e ) D e ' " t o l n c q t o dq ' { p ! } 1 lo yueu,a1q ? D q l 1 l ? n sU u ! 4 ! o 2 p o o t l . r o q q \ n u a l q l p n s o f l e r e q l ' U ) g f . r a a a i o g ( n )

(i) For any P, if E R with 7I"(p) = 71" (if) , there exists an element I E r with if = I(P)· (ii) For every pER, there is a suitable neighborhood U of p in R such that I( U) n U = ¢ for every I E r - {id}. In particular, each element of r except for the identity has no fixed points. (iii) r acts properly discontinuously on R; that is,for any compact subset K of R, there are at most finitely many elements I E r such that I(K) n K f. ¢.

'@)L= D uo s?srseatayT'(p)tt.= (4)v q?!nU ) !'q fruo.tog o ltlln J a L \uau.ta1a

Lemma 2.6. The universal covering transformation group Riemann surface R satisfies the following properties:

:sa4.tado.td6utno11otaq7 sa{stqos g acottns uuvur?tg eqJ 'g'Z BtutrroT 6ut.r,eaoc o Io (U':r-'A) to .1 dnotf uorTout.tolsuo"tT losJeaNutu

r

of (ll, 71", R) of a

'a,rr1calrnssr acuepuodsa.rroc O srql ecueq pu€'.[C] - l, teql seqdrur7'Z ruaroeqJ'(d'A)ro Jo luetuele u€ = ["d'Cj = (l'd ''I])t reqr s^\oqs g'U €urureT snql sr [9] ecurg '(t'd'"tl).[C] 'flarrrlcadsar'g;o s1u1od pue', pue pue aql ar€ Ierlrur Ieurural ['d'oI] I'd'Cj C l€I{} Jo lJll € sl , acuaH'od lurod eseq qlu{ g. uo qled pesol? s sI Col, sarldurr y JLov uorleler aql ueql '(l"d'"tl)L ollod'oll uroq Ur rio qled e al Q 'a,rr1celrns slq] 1eq1 aaord o; sr ecuapuodsarJoc 3 ,L luaurela ,tue a1e1 'err,r1celur st eeuepuodseJJo?sltl]

where 10 homotopic to 10 , and hence [Co] is the unit element of 7I"l(R,po). It follows that this correspondence is injective. To prove that this correspondence is surjective, take any element I E r. Let C be a path on R from [Io,Po] to 1([10 , Po]). Then the relation 71"01 = 71" implies that C 7I"oC is a closed path on R with base point Po. Hence C is a lift of C, and [Io,Po] and [C,Po] are the initial and terminal points of C, respectively. Thus Lemma 2.3 shows that 1([Io,PoD [C,Po] [C].([Io,PoD. Since [C] is an element of 7I"l(R,po), Theorem 2.4 implies that I = [C]., and hence this correspondence is surjective. 0

=

=

=

Ie.I'J

l€qt sA^olloJII'ed'a)rL Jo luatuale lrun eqt q [u ef,uaq Pu€'07 o1 ctdolouroq sr op 'snq;'[t'O] = 1 ) t fue to1 od - (l)'l reqr qrns U uo qled eq] sl o1 ereq!\

= [Co, Po] = [Io,Po], is the path on R such that Io(t) = Po for any tEl = [0,1]. Thus, Co is

'lod' o " o If = fod' Cf = (l"d' tl).|' Cl a^eq a^r ueql 'J Jo lueurele lrun eql q -[?] 1eq1asoddns 'arrrlcefutq ]l ]€q] e,rord oa '.7 o1 (od(g)tv;o rusrqdroruouor{e sr ecuapuodserrocsrq} }tsqt pI^Ir} sl rI'loord

[Co].([Io,PoD

we have

r. To prove that it is injective, suppose that [Co]. is the unit element of r. Then Proof. It is trivial that this correspondence is a homomorphism of

71"1 (R,

Po) to

'(A'o'U) 0ut.taaoc losrearun n to 1 dnot'6 uotTotu.totsuo.r,T |uuaaoc losraarun?ql oluo A Io ("d'A)rv dnot6 pTuau,opunt aql uo spyaffi-['d - log)acuapuodsauocaaoqDeyJ'g'Z rraroaqJ to tustrlilrotuost.

Theorem 2.5. The above correspondence [Co] 1-+ [Co]. yields an isomorphism of the fundamental group 7I"l(R,po) of R onto the universal covering transformation group r of a universal covering (R, 71", R).

'(A'v'A) Jo uorleruJoJ -sue.r1Sur.rar'oce sr 1r 'sl leql '..,1o1 s3uolaq -[op] srql 'uo1]-ruuepeql fq 'fpea13

Clearly, by the definition, this [Co]. belongs to formation of (R, 71", R).

r, that is, it is a covering trans-

'ld'c.'c) = ([d'c1).1'c] ry> la'c) [Co].([C,pD

= [Co' C,p],

[C,p] E

R.

fq g uo -[op] uorlce eql eugep e^{'(od'U)rv>['C] ]ueuele fue rog 'f, 1o (od'g)rY eqt o1 crqd.rouroslsl J dnor3 Eurraaoclesrelrun slt l€q? aas dnorE leluaurepunJ 'p ace;rns uueuerg e II€qs alll Jo (g'.u'9,) ece;rns EutrarrocIesJeAIunuarrrEe rog

For a given universal covering surface (R, 71", R) of a Riemann surface R, we shall see that its universal covering group r is isomorphic to the fundamental group 7I"l(R,po) of R. For any element [Co] E 7I"l(R,po), we define the action [Co]. on R by 2.2.3. Universal Covering Transformation Groups sdno.rg uor+BrrrroJsuer;, EurreloC

31

lesrallun'e'Z'Z

2.2. Universal Coverings s8urra,ro3l"sra^rufl'Z'Z

IT

g2 32

2. 2. Fricke FrickeSpace Space

I covering map map in in §2.1, and denote (U) covering denote by by U the connected connected component U the component of of 7rr-r(J) $2.1, and containing p. Actually, it is containing Actually, it is sufficient to take a simply connected domain sufficient to take a simply connected domain U f. U c o n t a i n i n gp. f o r s o m e containing If I(U) n U f:. if; for some I E r, then f t h e n there p o i n t sPI, t h e r e are a r e points e.lt1(0)n0 e 0U + { 1€ , f u , fihu E with ih Since7r0 ro7 we get get 7r(pd n, we T(fr1) = 7r(lll), r(4), and with for and hence henceiiI f1 = I(PI). l(it). Since l = 7r, Q1 PI, fi1,for r7r is is biholomorphic biholomorphic on = on U. Thus we U. Thus we have have I(pd id(Ft), where where id fd is is the the identity. identity. l(Ft) = id(pd, - id. By Theorem Theorem 2.4, we conclude 2.4, we conclude that lhat I1 = By id. Finally, to to verify verify (iii), (iii), assume assumethat that there there exists Finally, exists aa sequence sequence{{ In consisting Z" }~=l }f,r consisting of mutually mutually distinct distinct elements elements of of r l- such such that of that In(I<) I{ f:. n I< for all all n. n. Then Then 7"(1() n * if;/ for .ln([n). for each each n, = In n, we we can can take take two points iin, two points for rn E€ I< (iin). Since 1{ with with rfn Since K n = [n,Fn is compact, compact, taking taking aa subsequence subsequenceif if necessary, necessary,we we may is }~=l' may assume assume that that {iin { drl1T=r, converge to to iio, + 00. = 7r, /{, respectively, respectively, as {r ro E€ I<, as nn --+ oo. Since Since 7r0 zro7, ?r,we we {i"n }~=l }Lr converge Qo,io ln = = 7r(r r(4") = r(f")n ) and and 7r(iio) o(4") = 7r(r r(i,). obtain 7r(iin) Take aa neighborhood neighborhood U [/ of 7r(iio) r(,i'") in R .R o). Take satisfyigg the the condition ofthe of the definition of aa covering satisfying coveringmap map in §2.1, and denote denote by $2.1,and and,V U and I/ the connected connectedcomponents componentsof 7rU (U) containing zr- I1(U) containing iio and rfo, respectively. o, respectively. f, and .y"(y)n0 f:. if; Since {In(iin) convergesto rfo, we ha~e hav,eInCt})nU for aa s~~cientJy sufficient-lylar~e large Sinc.e o, we { j"(q") }J:"=l }f-, con:erges t' g for (J,it n. Smce ro7"(0) Since7rO followsthat In(U) 7, namely, namely,bn+d ("tny)-|,1n(0) n. It follows 0ln(U) = U. 0. ln (U) = U, 7"(U) = V, assertion (ii), we we conclude conclude that In+l By the assertion 0! is aa contradiction. 7,.11 = In' 7,. This is

=

=

=

=

=

=

=

Exarnple 3. Here is is an a"nexample group which does 3. Here example of aa group Example does not act act properly discontindiscontinuously. Let a be be aa real real number not equal uously. equal to 27r 2r multiplied by aa rational number. number. = generated by I( Then the group generated z) = eia: z does not act properly discontinuously edoz does act discontinuously l(z) o nC C - { {O}. 0}. on

Representation of 2.2.4. Representation of Riemann Riemann Surfaces Surfaces as Quotient Quotient Spaces

r

We shall explain a way to construct a Riemann surface surface RI Rlf from fro a Riemann surface R subgroup r R and a subgroup l- of the biholomorphic automorphism group Aut(R), surface Aut(R), where f is is assumed assumed to satisfy the properties properties (ii) and (iii) in Lemma 2.6, where 2.6, that is, every element element of r f except except for the unit is, unit element element has has no fixed points in R, E, and acts properly disconti~uously discontinuously on R. acts E. -equiaalentor Two tobe Two points F,C P, ii eE RRare are said to be r or equivalent if there f -equivalent equiaalentunder uniler fr if exists exists an e-lement element I.f e E fr satisfying 4=t@). ii = I(P), Denote Denote by [p] equivalence class the equivalence class [f] of the set of of fi. p. Let R/f RI r be !he of all these these equivalence equivalence classes cl~ses [PJ, called the p], which is called quolient quotient space space of of r? R by .i-. r. Define Define the projection 7r: RI r by r(fi) 7r(p) == [p]. r'. R *--+ R/f \fl. We introduce the quotient topology "n on fr,/f RI r.. AA subset RI/fr isis said to subset U of R be open if if and only only ifif the inverse inverse image image 7r(U) of of [/ U is an open subset subset of of E. R. the The o-t I (U) p_rojection projection r7r is readily seen seen to to be a continuous continuous mapping of of E R onto tr/f. RI r. Since Since ,R R is connected, connected, so is r?/f. RI Moreover, Moreover, we we see see that that Rl RI f is a Hausdorffspace, Hausdorff space, for lr acts acts properly discontinuously discontinuously on .R. R. Now, we define define a complex structure structure o" on n/ RI fr as as follows: follows: for any point point fpER, e E, take a neighborhood. neighborhood Up Up of of Ip satisfying the property property (ii) (ii) in Lemma 2.6. 2.6. We may assume assume that that there exists a local coordinate zp zp on 0p. Up. Then, putting putting p == r(fi), 7r(p) , Uo Up =,tr(0), = 7r(Up), we see see that that r: 7r: 0O Up ---+ tJ, Up is homeomorphic. homeomorphic. Hence, Hence, setting zo zp == I zpozr-r, (Up,zp)}rrrtl , defines Zp 07r- , we conclude conclude that that {{(Up,zp)}PER/r defines a complex complex structure structure so ihut that

r

r.

r

'29-r - (z)L ,'a , -;

with

lal 2 - IW =

1. This is also written as

(2.5)

G'z)

,(z)=ei8 z-£¥, 1- £¥z

s, u?lprn oslo st slttJ 'T = "lql - "lrl ,ltp^ ) ) Q(o anqm

a, b E C

,

where

az + b =-----=-, bz +a

,p+z!_k)L

(2.4)

@'z)

9t zo

,(z) =

fi.r'aag(nr) ru.ro!o soq (V)nV Tuaurap to '0*Dql?n C)q'oanym where a, b E C with a f. O. (iii) Every element of Aut(Ll) has a form

'q+zo=(z)L

(s'z)

,(z) = az + b,

(2.3)

tuaag Q1) ut.rolo soq (g)wy to Tuaurala ' I = " q - p Dq w n ) c P ' c ' q ' oa n a y m where a, b, e, dEC with ad - be = 1. (ii) Every element of Aut(C) has a form

az + b ,Z() = --d' ez+

,P*zc _k\L

(z'z) (2.2)

Q* zo

fr"raag(r) ur"roto soq (g)7ny lo Tuaua1a

(i) Every element of Aut(C) has a form 8'Z stutuoT

Lemma 2.8 erll e^sr{ Il pue 'V 'C'Q

:sturoJ 3utmo11o; sul"tuoP l€?Iuoue?;o surstqd.rouolne erqdrouroloqrg

Biholomorphic automorphisms of canonical domains following forms:

C,

C, Ll, and H have the

crt1daoruoloqrg'1p'8'Z

2.3.1. Biholomorphic Automorphism Groups of Canonical Domains srrrerrroq [BcruorrBC go sdno.rg rusrqdrourolny

'

1 g ) t n vp u e '( v ) t n v ' ( o ) t n v ' ( q ) n y

In the preceding section, we have seen that every Riemann surface R is represented by the quotient Riemann surface RI r of a universal covering surface R by its universal covering transformation group r. Here, by virtue of the uniformization theorem, R is biholomorphic to one of the three Riemann surfaces C, C, or H. We denote by Aut(R) the group of biholomorphic automorphisms of R. From Lemma 2.6, r is a subgroup of Aut(R), consists of elements without fixed points in R except for the identity, and acts properly discontinuously on R. With this in mind, let us study the biholomorphic automorphism groups Aut(C), Aut(C), Aut(Ll), and Aut(H).

'puttu ul qql qllft sdnor3 ursrqdrourolne crqd.rouroloqlq aq1 fpnls sn 1a1 'g, uo flsnonulluo?$p fFadord eql roJ ldacxa g' ur slutod paxg E?" Pue'fllluepl '9'A €Luluel urord 'ur '19)tnv;o dnor8qns 3 sl Jo J 1noq1r,nslueruelaJo slsrsuo? 'i'Q sursrqdrourolne-clqdrouroloqtqlo dnorE aql (U)t"V,(q alouap eAyI/ to 'ureroeql uorlezrur.rd; seceJrns uuetuerg "".rq1 nq1 jo "uo o1 crqdrorioloqlq q U -run aql Jo anlnl fq 'era11 '.7 dno.r3 uolleuroJsuerl Sfrueaoc l€srollun 8ll ,(q Ur eceJrns Surra,roc lesralrun e p J/A eceJrns uusruarll luarlonb eqt fq Paluas lerdar sr U eceJlns uu"IuaIU ,(re,ra 16q1 ueas aAeq eal 'uollcas Surpeee.rdeql uI

suorleurroJsuBl,I,snlqgtrAtr'8'U

2.3. Mobius Transformations

'@)".--,W [P]

I--.

1I'(p).

r

r

?Ul repun. A o7 Tuapanba filloatydtotuoto\?q q J nq V k I /U acuapuodsa.u,oc acottns uuotu?ty Tuat1onbeqt u?qJ 'tr dno.t6uotyont"totsuntT0uuaaoc losJearun 'Z'Z rrraroaq.1, qnn A acottns uuour?rg o to |uueaoc losrearun o eq (A')L'A) pI

Theorem 2.7. Let (R, 11', R) be a universal covering of a Riemann surface R with universal covering transformation group Then the quotient Riemann surface RI of R by is biholomorphically equivalent to R under the correspondence

r.

'uorlresse Surrrrrollo;eq1 ,(lalerpeurul el"q a \ ueql 'J ,(q acopns uuouery tf 1o eW J /A eteJrns uueruelg sql ilec eM'J /A 3o Euirarroce st (tr f g';u'g) TuatTonb

r.

(R, 11', RI r) is a covering of RI r. We call this Riemann surface RI r the quotient Riemann surface of R by Then we have immediately the following assertion. suorl"rurolsu?rl snlqol I 't'z

33

2 .3. Mobius Transformations

34 34

2.2. Fricke Fricke Space Space

w h e rBeE0Re R a nadEaLl. eA. where and (iv) Every Eaeryelement elemenlof of Aut(H) Aut(H) has hasaaform (iv) form. az+b az+b = cd + d' t(z) = I(Z) cd+d'

(2.6) (2.6)

= 1. w h e r a,b,e,d ea , b , c , dEe R with w i t had a d--, be b c= I. where In (2.2), (2.2), it it is is sufficient sufficient that that complex complex numbers In numbers a, o,,b, b, e,c, and and dd satisfy satisfy the the concondition ad od -- be b" :f. However, 0. does dition O. However, I does not change when a, b, e, and d are multinot change when a, b, c, and d are multi# 7 plied by by aa common common constant. constant. Hence, Hence,-we we may plied may normalize normalize the the expression expressionof of I7 by by - 1. ad -- be bc = 1. Every Every element element of of Aut( Aut(e) ad C) isis called called aa Mobius M1bius transformation transformation or or aa linear fractional transformation. transformalion.In particula.r, an linear In particular, an element element of of Aut(H) Aut(H) is is called called aa real Mobius Miibius transformation transformation or or aa real real linear real transformation. linear fractional fractional transformalion. Proof^of Lemma 2.8. 2.8. First First of of all, all, let let us us determine Proof of Lemma determine the the form form of of an an element element I7 E € = 00, Aut(C). If,( If 7(oo) oo, then in aa neighborhood neighborhood of 00, Aut(C). 00) = the Laurent expansion oo, I7 has has expansion 00

L

I(Z) t Q )== "az, ++i

b n zn ,- n , bnz-

n=O

where a :f. 0. Then I(Z) oz is is holomorphic on C, where e , and and hence hence the maximum I O. tQ) -- az shows that I(Z) oz must be principle shows be aa constant constant function, say say b. 6. Thus we we have have lk) -- az - l/(z a z l+ b b with w i t h a:f. o + 0O. z o # @ , t h e n I(Z) If 1(00) = Zo :f. 00, then setting 11(Z) = zo), s e t t i n g t l Q l @ ) == az z"), .If z(oo) t{z) we see see that that both 11 are elements Aut(e), we are elements of Aut(C), and 1101(00) = 00. Thus oo. 11 and 1101 lpl 71o7(m) = l/(,(z) w e have h a v e 110l(Z) I / Q Q ) - - zo) z o ) = a1z a r z *+ b rb,1, where w h e r e a1,b we 0 r , 61r E w i t h a1 o , :f. 0. € C with folQ) = * O. Therefore, I7 is expressed expressedin the form (2.2). (2.2). Therefore, Next, every element element It E Aut(C) is extended Aut(C) if extended to an element element of Aut(e) e Aut(C) if we we - oo. By the above put above argument, it is obvious put 1(00) obvious that that I7 is represented represented in 7(oo) = 00. (2.3). the form (2.3). Let 7 element in Aut(A). Aut(Ll). Set Set 1(0) Then the Mcjbius Mobius transforI be an element 7(0) = {3. B. Then .y2 - 1101 also mation belongs to Aut(A). mation r(z) 11(Z) = (, (z -- {3)/(1 pz) belongs Aut(Ll). Hence 12 Hence 0)/G Bz) 1*.r also belongs belongs to Aut(A) Aut(Ll) and 92(0) 92(0) = = 0. O. Schwarz' Schwarz' lemma implies that 12 rotation 72 is a rotation i8 z, with real number d. = eiqz,with Hence, 12(Z) = e B. Hence, I expressed form (2.5). It is is expressed (2.5). in the forrn trQ) It T easy easy to to see see that that 7I is written written in the form form (2.4). Finally, for any element element 7IE Aut(H), taking taking a biholomorphic mappin mapping T(z) == e Aut(H), gT(z) , oT- 1 eE AutlA). (z-i)/(z+i) (z-i)/(z+i) of of fI H onto .4, Ll, we have have an element element lr 11 == To Aut(Ll). Thus 71 11 ToloT-r is a Mijbius Mobius transformation transformation and is represented represented in in the form form (2.2). Since Since 7I sends sends ry' H onto itself, we may may assume assume that that c, a, D, b, c, and d are real numbers, numbers, and ad bc > O. and ad- Dc ) 0. Therefore, this 7I is written written in in the form form (2.5). 0 tr

=

=

For For more more on the the fundamental properties properties of of M6bius Mobius transformations, such as transformation transformation of of circles into into circles, circles, and the the invariance of of the crms cross ratio ratio under them, them, we we refer, refer, for for instance, instance, to to Ahlfors Ahlfors [A-4], [A-4], $3 §3 of of Chapter Chapter 3; 3; and and Jones Jones and and Singerman Singerman [A-48], [A-48], Chapter Chapter 2. 2. Now, Now, for for every every 7I eE Aut(e) Aut(C) given given by by

'slueruel€?s3urno11o;eql e^€rl a,r,r'uor1e1nc1ec aldurs e ,(g 'oz - (oz)L Surr(;sr1es oz > Jo tas aqt C IIs 'r(1r1ujpreq} oz slurod pexgJo las aql eq (t)xrg 1a1 }ou sr qcrqn

which is not the identity. Let Fix(,) be the set of fixed points Zo of" that is, the set of all Zo E C satisfying ,(zo) = ZO° By a simple calculation, we have the following statements. 'sl

ler{}1'L1o

, I = c q- p D , ) ) p , c , q , o

_ az + b ,Z( ) ---, cz +d

a, b, c, dEC, ad - bc

= 1,

, ' . 1 1 , i= Q*zo

@)L

Let, be a Mobius transformation given by

.{q ue,tr3 uor?euroJsuer} snrqory e aq I 1a1 'z'e'z

2.3.2. Canonical Forms of Mobius Transformations suorlBurroJsuBrl

snlqgl trJo srrrroJ lBcruouBc

'{1aar1aadsa.r '(1'1) atnTvufts to dno.r,6fil,opun lonads eq1 '1)29 pue (U'6)79 araqirr plle 7 aa.r6ap lo dno"r6nautl yonads lDa, eqt are (1

where SL(2, R) and SU(l, 1) are the real special linear group of degree 2 and the special unitary group of signature (1,1), respectively.

' { t+} lfi 'r)ns= G't)nsa = (v)wv ~

PSU(l, 1) = SU(l, 1)/{ ±I},

pue

}/(u'?,hs= (ll'z)tsa = (n)pv

PSL(2,R) = SL(2,R)/{±I}, '{(e)t"v

~

'{ r+

Aut(Ll)

and

Aut(H)

a^eq a^{ 'flrelrurrg ) Ll 6oLor-.{ } = (ra)lny reqr pue

and that Aut(pl) = {F-Io,oF Similarly, we have

I, E Aut(C)}.

' = -, :; lz:ii';il li:1, | l Zl] [ Zo

t--+

A [Zl] Zo

CZI

+ dz o

= [azl + bZo ] ,

Remark. When the Riemann sphere C is identified with the one-dimensional complex projective space pI, an element of Aut(C) corresponds to a projective transformation of pl. Indeed, define a biholomorphic mapping F: pI --+ C by F([zo : zd) = zo/Zl, where [zo : zd is a homogeneous coordinate of pI Then we see that an element ,(z) = (az + b)/(cz + d) of Aut(C) corresponds to a projective transformation

uorleuroJsuerl a.,rr1ce ford e o1 spuodser.roc(3)tnv Jo (p + zc)l&+ zD) - (z)1, lueurale ue 1eq1 ees e/tr uaq;,'rd Jo eleulProoc snoeueSouoqe sr [tz : 0z] araqrrr'rz/02 = (ftz : ozl)g * rd :J Surddeur crqd.louroloqrqe euuep 'paepul 'rd Jo uorleruroJsu€rl f,q ? a,rrlcefo.rde o1 spuodsauoc (3)lny Jo luetuela ue '1d aceds err,rlcalo.rd xalduroc Ieuorsuaurp-euo aql qlra pagrluapl q C a.raqdsuuetuerg eql ueq1\ llrDureq 0

'(c'dts

We set PSL(2, C) SL(2, C)/{ ±I} and call it the projective special linear group of degree 2. An element A of M- I (,) for an element, E Aut(C) is called a matrix representation of ,. Note that, is represented by two elements ±A in SL(2, C).

ul yT sluauala o,rl!,1 ,tq peluasarder sI ,L 1eq1elop "L Jo uorlD?uesa.tdat rt.tTou e pelle)c$st(g)l"y 3 ,L luaruele ue rog (1,)r- W lo V lueuala wtr '4 aa.rfaplo dno.t6 = (C'Z)lSd toauq lotcads aar,Ttato.td eql tr lpc pue {1+}/6'dlS tas aIA

' { r + } / 6 ' z ) t s= ( q ) w v

of the special linear group SL(2, C). Conversely, we have a homomorphism M of SL(2, C) onto Aut(C) defined by M(A)(z) = (az+b)/(cz+d) for A E SL(2, C), where a, b, c, and d are the entries of A as above. Then the kernel of Mis {±I}, where I is the unit matrix. Hence, by the homomorphism theorem, M induces an isomorphism Aut(C) ~ SL(2, C)/{ ±I}.

ursrqdrourosrue secnpur (ruaroeq? tusrqdrouoruoqeql dq 'ecua11'xrr?Burlrun aql sI 1 ereq^r ltl '{ 'c'g 'o ereqrrl 'eloqe se y f + 1 sl W Jo loura{ aql ueqtr, Jo saulueeq?er€p pue '(C'Z)1S )>y .ro;(p+zc)/(q+zo) = (z)(y)W tq paugep(q)ny oluo (3'6)79 yo ;;4rusrqdroruoruoqe e^eqem'r(lasrarruo)'(C'dlS dnor3 reauqletcadsaq1;o

A=[~

:]

lp c1 l'^ -l =V L q D)

qll^{ j

luauele u3 aAeq e^r

= cz+ az + b, d

,I =cq-pp

we have an element D+z) (1__-i_ (z\L 9*zo

a, b, c, dEC with ad - bc

> p'?'q'D

,(z)

= 1,

suorl"urrolsu"rJ snrqol{'t'z

2.3. Mobius Transformations

35

9t

3366

2. 2. Fricke Fricke Space Space

o.

(i) The The case case where where oo 00 € E Fix(7),i.e., Fix("Y), i.e., cc == 0. (i) If a/d aid = 1, that that b, is, ca - 6f d === *1, ±1, then then 7"Y has a sole sole fixed point point oo 00 and it it is lf written in in the form form written

=

=

1'(z)=z+b, 1 e)=z*b, On the other other hand, ifif a/d aid f:j:. 1, then 7"Y where b is a non-zero complex number. On another fixed fixed point point zo, and is represented as as has another uw- -z oZo= = \ ( A(Z z - z -o )zo), ,

w =lQ), = 1'(z), and )A is a complex number equal neither to to 0 nor to to 1. where to

(ii) The The case case where oo 00 f~ Fix(7), Fix(/), i.e., i.e., cc:j:. o. (ii) f 0. point = If (a + d)2 4, then "Y has a sole fixed point Zo and it is written written as as zo and it If * d)2 7 has sole

=

1 1 W -- % Zo= ; : Z1-+ Zo w o'

---=--+0', (c*+ d)' If (a where to w = 1'(z), and a a is is a a non-zero non-zero complex complex number. number. If d)2 :j:. 4, then then l' where 7 # 4, lQ), and the form has two fixed points Zl and Z2, and it is represented in points 22, and it is represented z1 and fixed has w Z - Zl W - Z Zl t ^, Z7 ---Z' -l , --=A W Z - Z2 " , - tZ2r -

L. where A is equal neither to 0 nor to 1. is a complex number equal where W w = "Y(z), 1(z), and ,\ contobe Aut(X)'conjugate Now, E Aut(C) are said to be Aut(X)-conjugate or conare said Aut(A) elements"Y1,"Y2 Now, two elements e 7r,jz that "Y2 6o"ho6-t such that jagale in Aut(X) jugate Aut(X) if Aut(X) such ob- 1 ,, element b 6E if there exists exists an element e Aut(X) 1z = bO"Y1 where H, or ..:::1. A. C, C, H, where X X is is one one of C, This following lemma. leads us us to the following This leads two f,xed id) has has one otueor two Lemma fixed points tmnsformation "Y(:j:. Mdbius tronsformation 2,9. Every Eoery Mobius Lemma 2.9. 7(f id) tmnsformation "Yo: on following Mobius M|bius tronsformation the foltowing antl is Aut(C)-conjugate Aut(e)-conjugate to the on C, e , and 7o: , a a* 0:j:. . s o m eaa €ECC, (i) +da for z+ ( i ) If"Y t, h e n"Yo(z) T.Q)= Z h a sa sole s o l efixed I f t has f o r some f i x e i lpoint, p o i n tthen \ C , ) (ii) points, then "Yo(z) = AZ for some A E C, A :j:. 0,1. p o i n t s , ) , 2 s o m e t h e n T o ( z ) € ( i i ) If"Y h a stwo t w o fixed I f 7 has for +0,1. fited

o.

canonical representations of canonical We Matrix representations canonical form We call this "Yo form of "Y. 7. Matrix 7o aa canonical forms and (ii) correspond correspond to forms in (i) and

0] [;?], [tf..;>.o ,f.^), 11";>"

respectively. respectively. = RU{ points are fl = RU{ 00 m }} a.rein in R A whosefixed fixed points fd) whose transformation "Y(:j:. A real real Mobius Mobius transformation 7(l id) ) of a matrix a or that the entry is such that the entry a or A of a matrix such canonicalform form "Yo to aa canonical is Aut(H)-conjugate Aut(H)-conjugate to 7o representation '''Yo is real number. number. is aa real representation.To

'peuueplla,lr '1, sr sll a.renbs (1,)rr1 Tnq fq flanbrun peurrurelap olw peretlesr ec€r1slr ueql'V- f,q pacelda.r sr y 1ousr (l).r1 snql'(p+D)'.{. uorleurroJsusrlsnlqgl4le 'p ;1 Jo nD4 " pelpc q qcg/'r * o (l)r1 1nd a14

We put tr(r) = a + d, which is called a trace of a Mobius transformation r. If A is replaced by -A, then its trace is altered into -(a + d). Thus tr(r) is not determined uniquely by r, but its square tr 2 (r) is well-defined.

' T= ? q, l \ ' ^ l= , ,C PD ) p,c,q,o L 9 DJ

A =

[~ ~],

a, b, c, dEC, ad - be = 1.

Now, a Mobius transformation r which is not parabolic is Aut(C)-conjugate to a canonical form ro(z) = AZ (A E C, At 0,1). This A is called a multiplier of r. We have two choices for the multiplier of r, i.e., A and l/A. It depends on the choice of a fixed point of r corresponding to the attractive fixed point of roo Its multiplier A satisfies the equation (a + d)2 = A+ l/A + 2. Note that a + dis the trace of a matrix representation of r:

:,L;o uorleluasa,rdarxrrleru " Jo ecerl eql np+p tsql atoN 'Z+y/I+y= e , ( pa r ) u o r l e n b ea q l s e g s l l e sy r a q d r l p t u s 1 1 'ol,;o lurod pexg e^r1?€r11ea{} o1 Eurpuodsarroc,L;o lurod pexg sJo alloq? aql uo spuadep yy/I pue aqt roJ sarroqr o,lrl e eq a64'l;o 1''a'r'l,yo.rar1dr11ruu .taqd41nut, e pallsc sl y slql'0'O* y'C ) y) zy - Q)'L uroJ Iscruouer € ot apSnluoc-(g)tnv o. cqoqered lou sr rl?rqar ,L uorleurroysrrerl snrqotrtre 'alo1q 'g uo sTurodpae{ omy soq L fipo puo ctloqtedfitl s! ,t (H) ta tt, .zz = rz puV , *H 'H 'rz slutod par{ on\ sorl L ) zz ) rz pUl qtns zz tt fi1uopuo tt, ctTdqla st' L 'g uo sr L Tutod pat{ alos o soy L lr fiyuo puo lt cqoqn.r,od

is is H", (iii) r is

(r) (r)

(i) (ii)

r r

parabolic if and only if r has a sole fixed point on ft. elliptic if and only,jf r has two fixed points Zl, Z2 such that and Zl = Z2' hyperbolic if and only if r has two fixed points on ft.

Zl

E H,

Z2

E

Lemma 2.10. Let r be a real Mobius transformation which is not the identity. Then the following hold: :p7or16utmo71olaq7 uayl 'fr,7t7uapt ?ql pu s, q)rqn uotgou.t.r,olsuorl sntqory loar o eq L pT 'Ot'Z eurtuarl

's11nsaresaql Surulqurop 'uor?ross€ aqt urelqo ea,r 3ura,ro11o; /tls"a 'cqoqradr(q ro 'cr1dt11a ro 'cqoqered reqlla sI ','[1t1uapt ro (11)7ny Jo luetuale fra^a 1€ql ^\oqs ol fsea st 11 'fla,rrlredsar '1,

Note that there are Mobius transformations which are neither parabolic, elliptic, nor hyperbolic. An example is given by r(z) = Az (A E C, IAI # 1 and A fi. [0,00)). A Mobius transformation which is not the identity is said to be loxodromic if it is neither parabolic nor elliptic. In this book, we assume that the loxodromic transformations include hyperbolic ones. Let Zl and Z2 be the fixed points of a loxodromic Mobius transformation r. Suppose that Zl and Z2, respectively, correspond to the fixed points and 00 of a canonical form ro(z) = AZ for some A with IAI > 1. Then Zl and Z2 are called the repelling fixed point and the attractive fixed point of r, respectively. Denote by r-y and a-y the repelling fixed point and the attractive fixed point of r, respectively. It is easy to show that every element of Aut(H) or Aut(~), which is not the identity, is either parabolic, or elliptic, or hyperbolic. Combining these results, we obtain easily the following assertion. eql lou sr q?rq^r'(V)lnV

go paxg Suqledar eq} !o pue r; ,(q alouaq pexg a^rlrer11e a{l pue lurod lurod 'flarrleadser 'L elql pve Turotl pax{ 0ur11ada,eq} pell€, 1o yut,odpar{ aatyco.tqqo elre ez pue rz uaqJ, 'I < lVl qll^ y euos to! zY - Q)'L ruroJ l"tluouec e Jo qurod pexg eql o1 puodsarroa 'fla,rrlcadser'zz pu€ Iz 1eq1 asoddng oo pu" 6 'l, uorleur.ro;suer? snrqoq cnuorpoxol e;o slurod pexg aql aQ zz pue rz lo.l 'seuo ?rloqredr(qapnlcur suorleurroJsusrl c[uoJpoxol eql airr '4ooq slql uI 'crldqe rou ttloqersd .raqltau sl ?l JI cruoJporol l€ql eurrrnss€ eq ol pres sr flrluepr eql lou s! qrlqa uorleurroJsuerl snrqory y'((oo'O] / V zV : ue,rrEsr aldurexauy 'cqoq.redfq rou 'cr1dq1a pu€ > I f) I lVl'C Q)L.{q 'cqoqered raq?reu are qf,rqra suorl€ruroJsu"rl snlqol are eraq+ teql aloN I

°

' I + y ' 0 < y e r u o s r o Jz y = ( z ) o L u o r t e l l p e o l a l e 3 n f u o c s r l I ; r c t l o y , a d f i qq f ( U ) '(7 3 u) ttuT - (r)'L uorlelor e o1 ele3nluoc sr ycr,Ttlqla q t (I) ]I * 0'1g )f g auos tol zsp

(i) r is parabolic if it is conjugate to a translation ro(z) = z + Q' for some Q'EC,Q'#O. (ii) r is elliptic if it is conjugate to a rotation ro(z) = eie z for some () E R, () # 2mr (n E Z). (iii) r is hyperbolic if it is conjugate to a dilation ro(z) = AZ for some A> 0, At 1.

'o+n'c>p

eurrros roJ a + z -

(z)ot uorlelsuerl e o1 ele3ntuoa s! 1l y cqoqotod q f (t)

'f1r1uapr eql tou $ qcrrlAr uorl€ruJoJ -suerl snrqontr e eq ,L 1a1 'sadf1 earql otq suorl"ruroJsusrl snlqotr tr fSsselc all

We classify Mobius transformations into three types. Let r be a Mobius transformation which is not the identity.

snlqg,trtrJo uorlBcgrssBlc '8,'8'z suorlBrrrroJsrrB.LT.

2.3.3. Classification of Mobius Transformations suorl"urolsuerl snlqgl I 't'z

2.3. Mobius Transformations

37

LT

38 38

2. 2. Fricke Fricke Space Space

By a simple calculation, we we see seethat Mobius transformations transformations are are classified classifiedby trace. ' trace. Lemma be a Mobius transfonnation which which is not the the identity. Lernma 2.11. 2.LL. Let r7 be M6bius transformation idenlity. Then Then the following hold: hold: the following if and and only ift-?(r) if tf (7) = 4. a. (i) r7 is parabolic parabolic if if and only if if 0 ~ tf (1) < a. (ii) r7 is elliptic if < 4. f t-?(r) (iii) rr is hyperbolic hyperbolic if if and and only ift-?(r) if tf Q) > > 4.. (iv) (iu) rr is loxodromic lorodromic if if and and only only ift-?(r) if tf (1) E e C -- [0,4]. [0,4].

=

Finally, we we define define the axis axis of a hyperbolic real real Mobius transformation r. 7. - )z with A ,\ > 1, by a Suppose that r7 is conjugate conjugate to a canonical canonical form ro(z) > 1, Suppose that U@) = AZ = 6oroo6-1. real 6. Namely, suppose that r7 = 6oloo6-1. The half-line Mcibius transformation 6. Namely, suppose real Mobius joining 0 and = { geodesic, L = iy I 0 < y < 00 } in the upper half-plane H is the geodesic, joining oo} half-plane .Il is < < {iy | 2 I1 (see 00, I(Im Z)2 on H (see §3 of Chapter 3). with respect respect to the Poincare Poincar6 metric IdzI z)2 3 3) . oo, with $ ldzl2I [m denoted by A-y. Ar. Then The image L) of Lunder .L under 6 is called called the axis oris of r7 and is denoted image 6( 6(L) joining the fixed A-y geodesicjoining r,, and a-y a., of r, is characterized characterized fixed points r-y ,4', is the geodesic 1, which is as real axis. axis. Similarly, r., and a-y o., and is orthogonal to the real as a semi-circle semi-circle which joins T-y we Aut(A). we define axis A-y A, of a hyperbolic transformation r7 in Aut(Ll). define the axis

2.4. Fuchsian Models Fuchsian Models First, we whose universal universal covering covering surface surface is is not we show that a Riemann surface surface whose show that biholomorphic to the upper half-plane one of C, e , C, C -halfplane H f/ is is biholomorphic to one properties of discrete discrete subgroups subgroups {O}, we study some some fundamental properties { 0 }, or tori. Next, we groups. of Aut(H), i.e., Fuchsian Fuchsiangroups. Aut(H), i.e., 2.4.1. Type Surfaces of of Exceptional Exceptional Type 2.4.L. Riemann Riernann Surfaces

Let us whose universal universal covering covering surfaces surfacesare are biholosurfaceswhose us determine determine Riemann surfaces morphic to either C or C. morphic either 0 biTheorem R has a universal uniuersal covering coaering surface surface R fr. bisurface^ 2.L2. A Riemann Riemann surface R has Theorem 2.12. holomorphic to the Riemann sphere C if and only if R itself is biholomorphic is biholomorphicto holomorphic lhe Riemann sphereC if and only if R

C. C. transformation Proof. that R .E = C. e . Since Sitt"" every element r7 of its covering transformation Prool. Assume that Flom Lemma is a Mobius transformation, it should have fixed points. From group r should have fixed is M
=

=

surface biholobiholoTheorem has a universal uniaersal covering coueringsurface surfaceR Theorern 2.13. 2.L3. A Riemann Riemann surface R has lo one one of C, morphic plane C if biholonrorphicto if and and only only if if R R is biholomorphic morphic to the lhe complex complexplane C --{ 0{ }0, }, or o r ttori. ori.

'zry = tr @)rL ^q palereua3 sl J leql qcns Iy raqrunu a.Lrlrsode slsrxe a.req1'g uo flsnonurluorsrp dl.radord s+re J ecurs .y reqrunu aarllsod auos roJ zy - (z)L yr fpo pue yr o,Lqlyr e^rlelnuruoc fl (g)7ny ) L .19)WV ur uorle3nfuoc ,tq (g ^{oqs ol fsea sr lueuala ue leql < oy) zoy ,1 - (r)"L 'orToqred{q sr oL ,,no11 leql aunss€ {etu a,u uaql Wql asoddns 'rq z = (z)rL fq pale.reua3sl + J teqt q)ns rg requrnu a,rrlrsode s?srxaaraql 'g uo dlsnonurluoc$p dl.radord slce J a?urs .tI ) g auros .rogg * z = (z)L urroJ aqt ur uellrrA\ sl f ,,l1uopue gr o,L a^rl"lmuurot JI {tgl * (g)wV 3 l" luaurala u€ leqt ees ol fsea sr 14'(g)nV ur uorle3nfuoc fq > o g ) o q + z - ( z ) o L r y q 1 e r u n s s ef e u r e a ru a q l , c q o q e r e ds o , L ; 1 'cqoqrad{q

Proof. We may assume that r =1= { id}. Take an element 'Yo E r with 'Yo =1= id. Since 'Yo has no fixed points on H, Lemma 2.10 implies that 'Yo is parabolic or hyperbolic. If 'Yo is parabolic, then we may assume that 'Yo(z) = z + bo (b o E R, bo =1= 0) by conjugation in Aut(H). It is easy to see that an element 'Y E Aut(H) is commutative with 'Yo if and only if 'Y is written in the form 'Y( z) = z + b for some b E R. Since r acts properly discontinuously on H, there exists a positive number b1 such that is generated by 'Y1 (z) = z + b1 . Now, suppose that 'Yo is hyperbolic. Then we may assume that 'Yo(z) = AoZ (A o > 0) by conjugation in Aut(H). It is easy to show that an element 'Y E Aut(H) is commutative with 'Yo if and only if 'Y(z) = AZ for some positive number A. Since r acts properly discontinuously on H, there exists a positive number A1 such that r is generated by 'Y1 (z) = A1Z. 0

r

@*'q'U

'H uo slurod pexg ou seq ol, acurg o,L .ro crloqered sr saqdurr etuure1 1eq1 0I'Z 'p! oL qtl,lr '{pgl1 o,L .too.t4 luaurale ue a{€tr * J 3 * J leqt arunsss,(eu e11

'ct1cfics, u?Vl 'uoqaqos? 'H uo snonut?uocsrp lN J II fr4.rado.rd s! J to uorl?o eW llUI q?ns puv g uo sTutoilpac{ ou sly {p?} - J .VTZ BururaT lo Tuaua1e fi.raaa Toqgqcns (g)nv {o dnolfiqns D eq J pI

r be a subgroup of Aut(H) such that every element of has no fixed points on H and such that the action of r is properly discontinuous on H. If r is abe/ian, then it is cye/ic.

r - {id}

Lemma 2.14. Let

'adfi7 TouorTdnr? lo eq ol pres sr rrol ro '{ O} - C 'C 'C Jo euo o1 erqd.rouroloqlqq qcrq^\ ac€Jrnsuueuer}I V

A Riemann surface which is biholomorphic to one of C, C, C - {O}, or tori is said to be of exceptional type.

' .t / C ot cryd.toutoloqrq s! A pW q?ns J dno.tf ac47ol D slstr,? a.r,aq1'g sn"to7fr.tana"lo3r .z(.re11o.ro3

r.

Corollary. For every torus R, there exists a lattice group biholomorphic to C /

r

such that R zs

'3 o1 crqd.rouroloqlq sr snrol € Jo aceJJns D 'dno.r3erlc{c e aq pFoqs 3ur.ra.,loc I€srelrun e e)ueH J e qcns leq} slras$ qcrrl^r (p1'6 eurtual) eurural 3ur,rlo11og eqt s1?rp€rluoc srqJ .g. ;o dno.r3 FluauepunJ eq? o1 crqdrourosr sl J roJ (6 4uer;o dno.r3 uerlaqe aarJ € eq lsnur J uar{} ,.Fl o1 ctqdrouroloqlqq U JI'snrol e q g 1eq1asoddns'fleurg'C = U leqt ^rou{ a , $ ' I ' Z $ y o 1 e l d u r e f g u r u a e ss e A rs V . { O } - C = A 1 a 1 , } x a N. C = A 1 a Ba m 'palcauuoc 'C fldurrs =A fl ?eql aurnsselsrg,asra,ruoc aq1 lroils o1 C ecws 'flalrlcadsa.r 'snro1 e pue '{ '9 o1 crqdrouroloq 0} - C '(II) pue '(ll) ,(r) sasecur ,e.ro;araqa U ac€Jrnsuueuerg eql

Therefore, in cases (i), (ii), and (iii), the Riemann surface R = C/ r is biholomorphic to C, C - {O}, and a torus, respectively. To show the converse, first assume that R = C. Since C is simply connected, we get R = C. Next, let R = C - {O}. As was seen in Example 1 of §2.1, we know that R = C. Finally, suppose that R is a torus. If R is biholomorphic to H, then must be a free abelian group of rank 2, for is isomorphic to the fundamental group of R. This contradicts the following lemma (Lemma 2.14) which asserts that such a r should be a cyclic group. Hence a universal covering surface of a torus is biholomorphic to C. 0

r

-lq sl J/C

r

-

'U re^o luapuedepur fpeauq er€ qcrq^r C f r g ' o g a u t o sr o y t g * z = ( z ) r L p r r s o g* z = ( z ) o l a r a q a r ' ( r t ' . t ) = J ( l l l )

(i) r={id}. r = ('Yo), that is, r is generated by a translation 'Yo(z) = z + bo for some bo E C - {OJ. (iii) r = ('Yo,'Y1)' where 'Yo(z) = z+b o and 'Y1(Z) = z+b 1 for some bo,b 1 E C which are linearly independent over R.

'{o}-c)oq

oruos roJ oq*z

(sr - (z)ol uorlelsuerl e {q pele.reueErl ,('tl leq} J

--,t

(tr) (ii)

{ p t l = . t (r)

:(1'deq3 Jo Z$ '[t-V] sroJlqy ';c) sesecaerql 3ur,rao11og aql rncco ereql l"ql elo.rd uec e^{ ueql '(9'6 eruuel ;c) g uo flsnonurluocsrp 'g uo slurod pexg ou seq -,f > f fpadord slc€ J teql [eeer em '.ra,roaro1,1i {pl } f u e a s n e r e q ' l = D ' r a q l r n g ' ( O f " ' C f g ' o ) q + z o - ( z ) L t u r o Je q l u r uellrr^,l\sl J ;l,L fre,re g'A"uure.I fq,(g)lnV;o dnorEqnse sl J acurg.dno.rE uol+€ruroJsue.rl Sur.ra,rocl€sralrun st! eq J lel 'C = U leqt aunssy '{oot4

Proof. Assume that R = C. Let r be its universal covering transformation group. Since r is a subgroup of Aut(C), by Lemma 2.8 every 'Y E r is written in the form 'Y(z) = az + b (a, b E C, a =1= 0). Further, a = 1, because any 'Y E r - { id} has no fixed points on C. Moreover, we recall that r acts properly discontinuously on C (cf. Lemma 2.6). Then we can prove that there occur the following three cases (cf. Ahlfors [A-4], §2 of Chap.7): sIePoI{ u"rsrlf,nJ 't'z

2.4. Fuchsian Models

39

6t

2. 2. Fricke Fricke Space Space

40

2.4.2. Fuchsian Fuchsian Models Models and and f\rndamental FundaIllental Dornains Domains 2.4.2.

The following is an immediate immediate consequence consequence of of Theorems 2.I2 2.12 and 2.13. 2.13. The surface fr, Theorem 2.L5. 2.15. A A Riemann covering surface R biunioersal couering surface R has has a universal Riemann surface Theorem and only lype; that is, if holomorphic to H if and only if R of exceptional type; is, if is not exceptional if R H if holornorphic ^of if R is not biholomorphic to anyone of C, C, { 0 }, or tori. C o r t o r i . o n e o f C , C , is not biholomorphicloany if {0},

If a universal universal covering surface E R of of a Riemann Riemann surface ,R R is the the upper upper halfhalfIf plane fI, H, we call its its universal covering covering transformation transformation group Ir a Fuchsian Fuchsian rnodel model of .R. R. In In this this case, case, fr is asubgroup a subgroup of of Aut(H). Aut(H). However, However, identifying identifying I/ H with with 4, ,,1, of we sometimes sometimes consider consider a F\chsian Fuchsian model fr as as a subgroup of Aut(A). Aut(Ll). Remark 1. By By an argument similar similar to that that in the proofs of of Theorem 2.13 2.13 and Remark -1. Lemma 2.I4, 2.14, we see see that that the fundamental group of of a Riemann surface surface R is commutative ifif and only only ifif .R R is biholomorphic to to one of of C, C, C, C -- {O}, tori, { 0 }, tori, r } . unit disk orr aannuli n n u l i {{ zz E < Iz < }. i s k ,,1, . 4 , ,,1 4 - - {{ 00} }, tthe he u nit d e C 11 l z Il < | 1< , o

geometric image image of correspondence correspondence between between a Riemann In order to obtain a geometric surface R and its Fuchsian model f, r, we use use a fundamental fundamental domain domain for f. r. An An satisfies for f if F open set F of the upper half-plane H is a fundamental domain r if satisfies domain 11 of open fundamenlal the following three conditions: with 1 =f (i) ,(F) f with, every 7 E oFF = ¢ e r / for every, { id. z({) n (ii) If 11, then closure of Fin .F in H, .F is the closure If F

a H

,r(F). =U [J ,(F). 'YET 7el

with respect respect to the (iii) The relative boundary of measure zero zero with has measure 0F of F in H has two-dimensional measure. Lebesguemeasure. twodimensional Lebesgue as is considered consideredas These surface R = H //fr is us that that the Riemann surface These conditions tell us l.. points on of covering group r. under the covering dF identified under

F F with

y'. For we define definesimilarly its Example coveringgroup r l- in Example 2 in §2.1, each covering For each Emmple 4. $2.1,we give examples of fundamental (i)", ... (t)" give examplesoffundamental fundamental domain. The following (i)", ..., , (v)" 2, respectively. respectively' domains groups of (i)', ... (t)' in Example 2, . . . ,, (v), covering groups domains for covering

(i)" ( i ) "F I mzz< 2211"}. r =- {z r}. e C | 0 < 1m { , Eel - {z Ee H I| 00 < Re (ii)" ( i i ) "Fr = R ez z< 11}.} . {, (iii)" a r s z< 2211"/n}. ( i i i ) "F r/n}. F =- {z | 0 < argz { 0 } 10< { , Ee cC -- {O} (iv)" ( i v ) F" .=F {z - { zE eHH11l I << Izl l r l< foX}. }. - {z Eel 1 , 0< } bbr,r , 0< 0 < bD<< 11}.} . (v)" 0 < aa < 1, ( u ) "FF = { , e C l rz -= aa+ domain for for aa There fundamental domain canonically aa fundamental way to to construct construct canonically There is is aa simple simple way paths smooth paths along suitable suitable smooth Fuchsian cut R -Ralong r?. First, First, cut surface R. model of of aa Riemann Riemann surface Fuchsian model of component of connectedcomponent on .F be be aa connected Let F domain RRo. get aa simply simply connected connecteddomain on R ,Rto to get o . Let

sIePoII u"rsqf,nJ't'z

TV

2.4. Fuchsian Models

41

'saldurexa 'fe,u, slql uI aa,rq1 eql u-r"lqo aal Eur.nolloJ 'J IoJ uleuroP IelueuepunJ e sI d slql 1eq1flrsea aeseAr'Z'Z$ul (A'y'A) Eurraloc lesJe^IuneJo uorl?nrlsuoc eql ,tg'.u detu Sur.ranoceql rapun tA p (;A)t-! a3eurr esra^ur eql

the inverse image 11'-1 (R o ) of R o under the covering map 11'. By the construction of a universal covering (il, 11', R) in §2.2, we see easily that this F is a fundamental domain for r. In this way, we obtain the following three examples.

'6, u1 $ qcrqrr ureluop Fluarrr€punJ e ser{ J l"ql leeduroc f1a,rr1e1a.r smolloJ 'rslncur€d ul'?,'7,'3U ul patertsnll sB J roJ ureuop le]uetuepunJ e l.I sl ('U)r-l;o g luauoduoc pelf,euuoc V'oU ur€ruop pelrauuoe rtldurrse 1e3 aa,r 'lg pue fy 1yeEuop g,3ur11ng'od lurod aseq qlr&t salrn? pesolc alduns qloorus il€ arp {g pue fy trqt qans (6 l) d 6nua3 Jo Ur eceJrns uueruarg pasop " roJ (od,A)to Jo srol"reuaS;o ualsfs l"cruouec " nq,=f{ lg'lV } tet 'g a\duorg

Example 5. Let {Aj,Bj }J=l be a canonical system of generators of 1I'1(R,po) for a closed Riemann surface R of genus g (~ 2) such that Aj and Bj are all smooth simple closed curves with base point Po. Cutting R along all Aj and B j , we get a simply connected domain R o . A connected component F of 11'-1 (R o ) is a fundamental domain for r as illustrated in Fig. 2.2. In particular, it follows that r has a fundamental domain which is relatively compact in H .

.4, B,

R

€- )t

------------R Fig. 2.2.

'z'z'Btr 'g alilu,oag

Example 6.

c, I

'V Fig. 2.3.

'8'Z'8tJ

aa.rq13uo1e U ln? pue U 3 od lurod e e{e; 'tO pue 'zO 're self,rrc ea.rq1fq pepunoq 'aue1dxalduroe eql ut Ar ureuop e 'g'Z '3ld ur pal€rlsnll se 'raprsuo3

Consider, as illustrated in Fig. 2.3, a domain R in the complex plane, bounded by three circles D 1 , D 2 , and D3 • Take a point Po E R and cut R along three

2. Space 2. Fricke FrickeSpace

42 42

smooth curves same argument as as that that in example example 6, curves C 6, Ct, Cz, C3. 1, C 2 , and C 3 . By the same elements we for R tr'for .R as as is shown shown in Fig. 2.3. 2.3. The elements we have have a fundamental domain F q(R,po), give respectively,give corresponding to the elements elements [Ad, 12 E [A 2] E 7r'1(R, Po), respectively, 11, l- corresponding € r [At],[Ar] Ir,''lz generators of r. we a canonical of Chapter 3, we shall describe describe f. In §1.5 3, ca.nonicalsystem of generators $1.5 another way of cutting fundamental domain for this group. cutting R .R to get a fundamental D;i degenerate degenerateto a Example case of Example 6, each circle circle D limiting case 6, let each Example 7. 7. As a limiting pi to obtain a Riemann surface surface,R C-{n,pz,ps}. single R biholomorphic to C{PI, P2, P3 }. single point Pi A Fuchsian principal congruence congraencesubgroup subgroup r(2) f(2) .R is conjugate conjugate to the principal Fuchsian model of R - (o, + b)/(cz + such that that of level consists of all elements elements I(Z) leoel 2, 2, which consists * d) such + b)j(cz 7(z) = (az 2 . As s y s t e mof of l , and a n d a = d = 1,b 1 , 6 : ec = 00 mod m o d 2. A s a system a,b,c,dE Z , a d --b cbe = 1, a,b,e,d € Z,ad picture generatorsof r(2), we have and 11(Z) z/(22 + 1). The picture generators f(2), we have 11(Z) *2 2 and * 1). fQ) = z + fQ) = zj(2z on the left hand side of Fig. 2.4 shows Fig.2.4 shows an example of of a fundamental fundamental domain for r(2). side of this figure figure illustrates a fundamental f(2). The picture on the right hand side domain for a subgroup details, see see Ahlfols Aut(A) conjugate conjugate to r(2). l-(2). For details, subgroup of Aut(L1) §2 of Chapter 7; and Jones and Singerman [A-48], Chpater 6. [A-4], Singerman 6. 7; Jones [A-48], [A-4], $2

== ==

I

-------1:= z+2------

c,

c,

F

----)

~

Z

C,

1:= 2z+1

C,

-1I

z-i z-t

t(}=--.

Z+l

-_ ......1 ----+-0---..L-- R

rQ) a fundamental .F/ for r(2) fundamental domain in H

a fundamental fundamental domain in L1 4 for r(2) f(2)

Fig. F i g . 22.4. .4.

Remark subgroup of Aut(H) Aut(H) acting domains for a subgroup canonical fundamental domains Remark 2. 2. As canonical regions. For properly discontinuously regions and Ford regions. ff , we we have have Dirichlet regions discontinuously on H, details, Ford [A-31], we refer books such such as as Beardon [A-ll], refer to standard text books details, we [A-31], [A-11], Jones Lehner [A-66], [A-67], and Maskit [A-71]. Lehner Singerman [A-48], Jones and Singerman [A-71]. [A-66], [A-67], [A-48],

sIaPoII u"rsqsnd 't'z

t

2.4. Fuchsian Models

43

(n)pV

go sdnorEqns alarcsrq 'g'V'Z

2.4.3. Discrete Subgroups of Aut(H)

qll,!{ (U,Z)IS lo t;3{ "y } ecuenbasaq1 ,ara11.(U,Z)ZS;o f3olodol eqt,tq paenpursl (U'Z)ZSa;o fEolodol aql'I't$ ur ux\oqs $ sp uorl€rgrluepr eql repun (1g'Z)lSa dnor3 ar1 eqtJo euo eql 01 luale,rrnbasr f3o1odo1 slql'oo ol spual u * H Jo slesqns lceduroc uo I o1 .{pr.ro;run set.raauoc "L I (n)pV I I ol saEre,ruoc 1g)lnV Jo I=J{ u,L} acuenbas e teql sueeru srql '{to1odo1 uado-lcedruoc eql ''e'r.'(11)WV uo ,(Eo1odo1Iernleu e Surugap q1ral ur3aq e11

We begin with defining a natural topology on Aut(H), i.e., the compact-open topology. This means that a sequence {1'n }~1 of Aut(H) converges to l' E Aut(H) if 1'n converges uniformly to l' on compact subsets of H as n tends to 00. This topology is equivalent to the one of the Lie group PSL(2, R) under the identification as is shown in §3.1. The topology of PSL(2, R) is induced by the topology of SL(2, R). Here, the sequence {An }~1 of SL(2, R) with

l"p t t " o l =o "rl

fl il=,

ol saEre,ruoc

converges to

'dno.t6 uvrsq?ng e se ol perraJer osle fl (V)l"V l" dnor3qns elercsrp '(C)lnV rc'(3)wV '(V)l"V;o dnol3qns elercsrp€ augep V jo Jaqunu elqelunoc e lsoru leJo slsrsuoc dnorE u€rsr1rr\f eira'f1.re1turg'sluaurala e '{1tlqe1unoe tuorxe puoees eql segsr}es (U'A)ZS ecurg 'dno.r,6uvrsycnl e Jo palp? q (H)InV;o dnorEqns eterf,srp O '19)tnV Jo leql uorJ pernpu! J uo f8o1odo1 e^rl"ler aql ol lcadsar qlrar slurod pelsloflJo slsrsuoc J ''a'l'Gt)lnv Jo lasqns alansrp e s.t J ! ?pr?erp eq ol pres sr.(11)7ny ;o .7 dnol3qns y 'oo ol sPuel u se 'dle,rrlcedsar'p pr. '" 'q'o o1 aE.rarruortp pue 'uc 'ug' ro yr fluo pue (U 'Z)I rrl JI S

in S L( 2, R) if and only if an, bn , Cn , and dn converge to a, b, c, and d, respectively, as n tends to 00. A subgroup r of Aut(H) is said to be discrete if r is a discrete subset of Aut(H), i.e., r consists of isolated points with respect to the relative topology on r induced from that of Aut(H). A discrete subgroup of Aut(H) is called a Fuchsian group. Since SL(2, R) satisfies the second axiom of countability, a Fuchsian group consists of at most a countable number of elements. Similarly, we define a discrete subgroup of Aut(Ll), Aut(C), or Aut(C). A discrete subgroup of Aut(Ll) is also referred to as a Fuchsian group.

r

efitaouocWyln J to sTuau,ep purlstp fi11onynu.t to sacuanbasou ?srseereqJ 'dno.t6 uvrsqcnf, o s! J .gT.Z BrrruraT lo ,7 dnol0qns D ro4 :Tuaparnbaato 6utmo71otaq7 (g)ny

Lemma 2.16. For a subgroup

of Aut(H) the following are equivalent:

(i) r is a Fuchsian group. (ii) There exist no sequences of mutually distinct elements of r which cont'erge in Aut(H).

(s) (r)

'(11)wYuP

Proof. Clearly, if r is not Fuchsian, then the assertion (ii) does not hold, and hence (ii) implies (i). Now, assume that there exists a sequence {1'n }~1 of distinct elements of r which converges to an element l' E Aut(H). Considering their matrix representations, we see that {(-Yn)-l }~=1 converges to 1'-1 in Aut(H). Since (1'n)-l o1'n+1 E r and (1'n)-l 0 1'n+l f. id for any n, and since the unit element id of r is not an isolated point of r, r is not discrete. Hence (i) implies (ii). The proof of Lemma 2.16 is hereby complete. 0

'a1e1duocfqaraq sl gI'Z eurruerl '(I) saqdrur D ;o;oord eql (r) ecuag 'elercsrp tou sr J '; go lurod palelofl ue lou sl J Jo p? luetuela lrun eql ecws pue 'u fue .ro;pg iL r+uLor_("f) p* J J I+ul,or_(u,[) acurg .Q1)lnV ul r-f o1 saS.rerluoctfi{ ,_("t) } W{t aes e,ra'suorleluase.rda.rxrrleur neql 3ur -raplsuoC '(U)t"V ,L ue o1 saS.raauosr{rrq^\ J Jo sluauela ?cultslp I luaurela '/'roN '(l) serldurr (rr) ecuaq I=":{ ul,} acuanbas ? s}srxa ararl} leq} arunss? Jo pue 'p1oq lou seop (rr) uorl.rasseeql ueql 'uersqcng lou sr J y'f1tea1a 'too.t4 .lI.U to tr ilno"t\qnso ro4

uraroaql

r r

r

:Tuapatnbaa.ro6utmo11otayq (g)lny

Theorem 2.17. For a subgroup

of Aut(H) the following are equivalent:

.r

'g uo filsnonur?u@stp filtadaul sTco J (II) 'uDtsq)nf, st (l)

(i) (ii)

is Fuchsian. acts properly discontinuously on H.

on <- (oz)uL I;:{ 'l q?ns sluatuela ecuenbase pu€ leql lcurlsrp H ) J Jo Jo } 'I1 uo dlsnonurluocsrp dpedord oz lutod e eAeq e^l ueql H ) lce lou saop J 'f1asra,r,uo3'uor?rugap eqt ,tq snor^qo sr (r) seqdu4 (ll) teql /oota leql aurnsse

Proof. That (ii) implies (i) is obvious by the definition. Conversely, assume that does not act properly discontinuously on H. Then we have a point Zo E H and a sequence {1'n }~=1 of distinct elements of r such that 1'n(zo) -+ Wo E H

r

2. 2. Fricke Fricke Space Space

44 44

as n --+ ~ oo. 00. Since {{In };:::'=1 is is aa normal normal family, family, taking taking aa subsequence, subsequence, ifif necessary, necessary, r" }Lr we may may assume assume that that {{ 7, In }p, };:::'=1 converges converges uniformly uniformly on on compact compact subsets of of f/H to to a in 11 H . Flom From the following lemma (Lemma (Lemma 2.18), 2.18), holomorphic function function 7I defined in this 7I must be an element of of Aut(H). Aut(H). Hence Hence by by Lemma Lemma 2.16, 2.16, fr is not not F\rchsian, Fuchsian, this D hence (i) (i) implies (ii). (ii). 0 and hence Remark 1. For a subgroup fr of of Aut(d), Aut(C), the discreteness discreteness of of l-r does does not always always Remark -/. given by example is imply that it properly C. A typical properly e . A typical on discontinuously it acts imply that (

az*b

az + b .r= = t{t I(Z) Q ) = *cz+d +d

Iaa,b,c,dEZ+zZ , b , c , d . e Z +.i Z}\. .

Lemma 2.L8. 2.18. Let {{ f" 'Yn }Pr };:::'=1 be be a sequence sequence of of Aut(H) Aut(H) which which conaerges converges uniformly Lemma compact subsets of H to a holomorphic function function ff defined defined in H H.. Here, Here, ff subsels of on compact admits a constant constant function function with value a. 00. Then Then either either one one of of the the following following holds: holds: with aalue admits

(i) /f is an an element element of Aut(H). Aut(H). (ii) /f is a constant constant function function c with c E R. € R. S 1 on Proof. We consider consider the unit unit disk ..:1 instead of of I/. H. Clearly, Clearly, we we have have l/l If I ~ 4 instead Proof. ..:1. If If(zo)1 = 1 for some point Zo E ..:1, the maximum principle implies that /f maximum some zo € A, l A.If lf(2")l is constant is a constant function. Thus either If I < 1 on ..:1, or f is a constant function c 4, constant / l/l < I f a c t , t o A u t ( A ) . I n with lei = 1. If If I < 1 on ..:1, then f belongs to Aut(..:1). In fact, {(/n)-l };:::' = 1 1. If l/l < on ^4,then / belongs with lcl {(r")-t har we may assume assume that being a normal family, taking a subsequence, if necessary, necessary,we subsequence,if it converges uniformly on compact subsets of ..:1 to a holomorphic function g 4 subsets it converges uniformly we see see same argument, defined in ..:1. In particular, we have gof = id. By the same argument, we fd. pa.rticular, we have defined 4. ( = i d . t o Aut(H). Aut(H). that 0D b e l o n g sto o g = id. Hence, H e n c e ,/f belongs 4 and a n d ffog t h a t Igi n ..:1 l 9 l < 1 oon surface of Proposition model of a closed closedRiemann Riemann surface be a Fuchsian Fuchsian model Let rf be Proposition 2.19. Let

t h e r c exists e d s t s a sequence sequence genus ~22. {oo}, there p o i n t ((E R - = RU RU{m }, g e n u sgg 2 . For a r b i t r a r ypoint F o r an a n arbitrary € R point zo H. conaerges {In };:::' = 1 of r such that {'Yn(zo) };:::' = 1 converges to ( for any point Zo E H. € f for {6Q,)}f=t f" }Lppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppppp {

is closure F F of F is that the closure Proof. Take such that fundamental domain F for rf such Take aa fundamental domain, fundamental domain, compact I/ (Example 5 in §4.2). compact in H $a.2). By the definition of aa fundamental points in F F sequence{Zn pick aa sequence and aa sequence we In };:::'=1 of rl- and };:::'=1 of points we can sequence{{Zr}Lr can pick {rr}f;=, we normal family, family, we such };:::'=1 isis aa normal convergesto (. Since Since {In such that {/n(Zn) {'l'}Lr {l.Q^)}[,};:::'=1 converges ff to aa subsetsof H on compacts compacts subsets may convergesuniformly on supposethat {{f"In };:::' may suppose }Lt=1 converges must be be shows that f/ must holomorphic 2.18 shows 11. Thus Lemma 2.18 defined in H. holomorphic function ff defined aa constant 0 constant (. points of of the the set set {IlQ")lt Remark Let L( r) be I( zo) I I E all accumulation accumulationpoints e be the the set set of of all L(f) 2.Let Remark2. I(f) is is independent independent r}, implies that that L(r) point on Lemma 2.18 2.18 implies on H. f/. Lemma where Zo zo is is aa point l-], where call subset of of R. fr.. We We call is aa closed closed subset of It is that L(r) I(f) is well known known that of zoo zo.It is well of the the choice choice of = R R tells us us L(r) ,(l-) = 2.19 tells L(r) group r. ,l-. Proposition iroposition 2.19 of aa Fuchsian Fuchsian group the limit limit set setof I(f) the = H provided / r is provided that is compact. compact. that R H/f ft =

uJoJ oql ur uellrr^\ sl "y lsql aas elt 'u raEalur errrlrsod ,(ue rog - rluV pus y - Iy Eurllnd 'ol,;o uorleluasardar xrrlpru e q qclqa

which is a matrix representation of 10. Putting A 1 = A and An+! = A n A o (A n )-l for any positive integer n, we see that An is written in the form ,_(V)oV"V

' f ] 9 l= ' n Ao =

[~

n,

[I II 'I tes e1l > lal > 0 qll,lr (1g'Z)IS 3 y uorleluasardar K:
Proof Assume that there exists an element I E such that I has a matrix representation A E 8L(2, R) with 0 < lei < 1. We set

r

lei ~

1 provided that

'0

satisfies

e::f O.

t ? l"l sa{nqos # c to1t paprao.ul

' l' ^ ' ^l = ,

| - ?g- pD 1 g > p 'c'q ' o a, b, c, dE R,

ad - bc

=1

Lq D)

'l uotToTuasa.til?J ruloru o 6unoq ) L frtaaau?qJ a z - (z)oL uotyo1 ,! -suDrln 0ututoyuoc dnafi uotsycri"l o ?q pI (lt1?,] nzturrqg) .IZ.Z BrrruraT J

Lemma 2.21. (Shimizu [203]) Let r be a Fuchsian group containing a translation lO(Z) = Z + 1. Then every I E r having a matrix representation

'uorlrrpsrluoc € ol speel qcrqal'alarcsrp tr lou sr J '91'6 eurural ,(q 'aro;a.raqa'sluetuale --?{ l?qlqp Jo slsrsuor ? } leql ease,rl 'l + y p u e ' 0 < y ' 0 f g o a c u r g ' o o -t s u w C t s ? u a q l , I < y J I . o o + t s u s e

r

as n - +00. If A> 1, then Cn - C as n - -00. Since ab ::f 0, A > 0, and A ::f 1, we see that {Cn }~=-oo consists of distinct elements. Therefore, by Lemma 2.16, is not discrete, which leads to a contradiction. 0

I r o l =' I'qt'; j "t ueqt'I > y > g y'acua11 o1 sa3ra.iluoc

Hence, if 0 < A < 1, then C n converges to

o'l = ' I 'I | . ( , r ,- I ) e D; i

C_[10 n -

ab(l- A 1

2n

)]

.

- ug 3ur11as'arrolq'flalrlcadsar

respectively. Now, setting C n = BAn B- 1 A-n for any positive integer n, we get arrrlrsod{ue.ro; u-VrBuVB 1eBea,r'u.ra3a1ur

, o* q e , u ) 9 , o

A=[~ A~l]' B = [~ a~l]'

a,b E R,

ab::f 0,

,

g] -

[,;,

, r + y, o < y , l ' ; . A > 0,

A::f 1,

i]

g

_V

Proof. Assume that neither (i) nor (ii) holds. By Aut(H)-conjugation, we may assume that Fix(I) {O, 00 } and Fix(I) n Fix(6) 00 }. Then matrix representations A, B of I' 6 are given by

dq uerrr3an g'L lo g'V suorle?uas '{ -e.rda.r xr.rleur ueql oo } = (g)*1.{ U (l)xrg pue { m'O } = (l)*tg t€rll eunsse feur aar 'uorle3nfuoc-(Hhnv fg 'sp1oq (rr) rou (r) raqlrau l€ql eunssv 'loo.r,4

=

={

.0 = (g)xr.{ u (r)4.{ (r) (i) Fix(I) (ii) Fix(/)

n Fix(6) =
= Fix(6).

:sp1oy furmoilo! eyrlo euot"rl, 'plTg"r;r'::r:;"::, 'OZ'Z

hyperbolic and 6 ::f id, then one of the following holds:

s? L lI 'tr dno.r0uorsq?nf,o to sTuaualeonl ?q g puo L pI

Lemma 2.20. Let I and 6 be two elements of a Fuchsian group

r.

If I

IS

BruuraT

'ralel pesn ar€ rl?rq,a,r, 'sdnor3 u"rsqcr\{ sarlradord auros luesard all ;o

We present some properties of Fuchsian groups, which are used later.

2.4.4. Further Properties of Fuchsian Groups

sdno.rg uBrsqcr\{ ;o sarl.radord Jaq}JqlI'V'?'Z

sIePoI{ u"rsqf,nJ't'z

2.4. Fuchsian Models

45

9t

2. Fricke Space Space 2. Fricke

4466

6".l "o" -- l o n d nJ ' l"n

=

=

_d

=

=

2 - an-rcn-r, * a n -n(-1C n - tn . -l. an = Il-a bn = (an-!)2 (a n Cn = -(cn-1)2, -(Cn _l)2, and and d,. d n = Il+a where o, where n -1Cn _l, bn ,, cn n-l 2 -"'n-t * * Thus itit follows follows that that cr Cn = = _c ->- 0 0 as as nn ->- oo. 00. Next, Next, setting setting M M = Thus

=

max{ lol,l/(L lal, 1/(1 -- lcl) leD}, we obtain obtain inductively inductively lo"l lanl S~ M M for for any any n. n. Thus Thus each each max{ }, we an, bn, bn , and and d, d n converges converges to to 11 as as nn +->- oo. 00. Hence, Hence, .Ar An converges converges to to Ao, A o , which which ant tr the discreteness discreteness of of l-. r. 0 contradicts the contradicts

Theorem 2.22. Eaery Every element element of of a Fuchsian Fuchsian model model of of a closed closed Riemann surface surface Theorem of genus genus S g (|=2) (~ 2) consists of the the identity and hyperbolic hyperbolic elements. elements. consislsonly of of

r-{

Proof Since Since every element 7I eE f - { id id}} has no fixed fixed points points on I/, H, itit is parabolic parabolic Proof. that f contains contains a pa"rabolic parabolic element element 10. By Aut(H)Aut(H)Assume that or hyperbolic. Assume lo.By suppose that 10(Z) -= z*1. z+1. From Lemma2.20, Lemma 2.20, any element element lhatT"Q) conjugation, we may suppose .y I (+ (:f id) of of .i- with with f(m) 1(00) = oo 00 is parabolic, which is written writ ten in in the form form = oo} = is a j f = 6. Hence, ]-I(Z) Z + b for some b. Hence, {I E 11(00) oo} number e some real z r(*) *b I { ilz) assume we may with element, if necessary, we assume if necessary' element, with another cyclic group. Replacing 10 7, - ((az L, a a z *+bb)/(cz generator Since every I(Z) add - b cbc= 1, v e r y7 e n e r a t o rffor or ft. S i n c ee ) l @ z * d+) d), , tthat h a t 10 (z) 7 o iiss a g we Thus 1. that 2 shows belonging to l0, 2.21 shows that lei ~ 1. we 2.21 Lemma 0, ,i-- satisfies satisfies c :f belonging lcl +

r

=

r

r - roo

obtain

=

roo =

=

roo.

Iml(z) rmTQ)~S

r

=

=

1

1 ~511 (Imz)1e1 1r-;pp2 -

distinct any two distinct 2}. Then any 1m z > 2}. for all zz with 1m 1. Set Uo e H Illmz Imzz > Set U ) 1. o = {z {z E - roo. l--. Thus the quotient points on U 0 are element of r f are not equivalent equivalent under any element [/o corresponds r?. Since space Ro since 10 = U space Do = Uo/f* o/ roo is biholomorphic to a domain R o in R. 7o corresponds closure R R,o .r?,the closure to aa non-trivial E o of R element of the fundamental group of R, non-trivial element disk in R is Do is is biholomorphic to the punctured disk connected. Since Since Do is not simply connected. ( I}, t o {z € C |0 < b e homeomorphic h o m e o m o r p h i cto {z must w e infer that R 1 } , we i n f e r that o u < Izi 4m s t be e C |0 < { z Eel l"l < { z Eel Izi ~ I}. This contradicts that R is compact. 0tr is compact. contradicts | ltl S ). geometry disdisRemark. using the hyperbolic geometry obtained by using is also also obtained theorem is Remark. This theorem 2 = 2/(lmz)2 be be present dsz cussed of Chapter 3. We present its outline. Let ds = IdzI its outline. 3. We Chapter cussedin §1 ldzl2l(Imz)2 $1 - H/r. H/f . on R = the hyperbolic metric on which induces induces the hyperbolic on H, 11, which the Poincare Poincar6 metric on positive number a, o, 1. For For any any positive Assume f has has aa translation 10(Z) * 1. that r Assume that 7o(z) = zz + fo segment La tro joining ia denote of the segment which is is the the image image of ,Rwhich on R closedpath on by C denote by C"a au closed of -.R. length of the hyperbolic hyperbolic length and (ia) by R. Let C a ) be the projection 11": Let f( l(C")be r: H H ->by the the projection and10 1o(ia) we see seethat that Then we C metric. Then to the the Poincare Poincar6metric. with respect respectto of La .Lowith length of the length a, i.e., the Co,i.e., we have sequence have aa sequence f(C being compact, compact, we + 00. .Rbeing - 00 as other hand, hand, R oo. On the other On the as nn ->t(C") a ) ->+ 00, + Po + po oor as r(l'o") oo and ) {an };;"=l of positive numbers such that an ->00 and 1I"(ia ->as nr, ->that dn positive such numbers of o" n }L[r { which connecteddomain domain U u which where simply connected we take take aa simply point on Hence,if if we po is on R. rt. Hence, where Po is aa point n. large n. for sufficiently sufficiently large path Can in U contains is included included in [/ for C,. is po, then the closed closedpath then the contains Po, - id, This = a contradiction. id, a contradiction. implies 10 This implies lo

L?

acedg arpug '9'7

47

2.5. Fricke Space

oreds a{rl{,{ '9'U 2.5. Fricke Space

:suorlrpuoc uorlezrleurou aq1 esodurl e,!\ ,U uo 3 3ur4.reurue,rr3e o1 J Ieporu uersqrnd e ,,{lanbrunu8rsse o} rapro uI .ile,lr se Ur aures eqt Jo lapour uersqrnd e sl dnor3 eq1 '(g)wv fue ro; ,s1 ryql :(11)7ny go sursrqd ) ,.7 9 r_9J9 -rouoln€ .reuur{q pasnec ,{lrn8rqure egl seq ,lop A Io J Iapour uersqcnd e '6'"''Z'I = ! q c e e r o J ' f 1 a , r r 1 c e d s,e( .ord , A ) t o q [fg] pue [fy] o1 SurpuodsarrocJ Jo stueuela oq1 ld pue fo fq alouap ,9.6 utrreroeqlur pelels A lo J Iepou u€tsqcnd e pue (od,U)Iz uae.nleqursrqd.rourosr aql repun 'f snuaS;o U e)eJJnsuuetuerg pesol) e go (oa,g') t.u dnor3 plueur€punJ t=f{ aql srolerauaS;o uralsds q ''e'l 'U uo 3urr1.reu n rl Jo Ierruouec [lS],[!V]] 'f snuaE sec"Jrns uu€urarg pesol) pa{retu s}srsuo) 3' eraq^r Jo [3''U] II€ }o tg aceds rellnuqrrel eqt ,I raldeq3 ur peugap se^.rsv kZ) 0 snuaS;o Jo t$ 'QZ) 0 snue3go se)eJrnsuueuarg pesolc r; ecedsrefintuq)rel eql uo sel€urprooc e{]r.U pall€c Jo 'slapour u€rsqcr\{ sroleraueS;o ue1s.{s -os eugap fteqs ellr Jo l€)ruorr€c e Bursn fg

By using a canonical system of generators of Fuchsian models, we shall define socalled Fricke coordinates on the Teichmiiller space Tg of closed Riemann surfaces of genus 9 (~ 2). As was defined in §3 of Chapter 1, the Teichmiiller space T g of genus 9 (~ 2) consists of all marked closed Riemann surfaces [R, E] of genus g, where E = {[A j ], [Bj] }]=1 is a marking on R, i.e., a canonical system of generators of the fundamental group 11"1 (R, Po) of a closed Riemann surface R of genus g. Under the isomorphism between 11"1 (R, Po) and a Fuchsian model r of R stated in Theorem 2.5, denote by OJ and j3j the elements of r corresponding to [A j ] and [B j ] in 1I"1(R,po), respectively, for each j = 1,2, ... ,g. Now, a Fuchsian model of R has the ambiguity caused by inner automorphisms of Aut(H); that is, for any 8 E Aut(H), the group r' = 8r8- 1 is a Fuchsian model of the same R as well. In order to assign uniquely a Fuchsian to a given marking E on R, we impose the normalization conditions: model

r

r

'1 1e lurod paxg a^rlcerl?e sll ser{ t^a (I) ',{1anr1cadsa.r 'oo pue 6 1e slurod pexg a^r}ceJ}le pue 3ur11eda:s1 seq td (l)

(i) j3g has its repelling and attractive fixed points at 0 and (ii) a g has its attractive fixed point at 1.

00,

respectively.

'(tt) p u n s u o r l r p u oc ( r ) uorlezrl€rurou aq1 f;sr1es ud pun to lr.I? etunsse.r[eura,ll ,r(ressaceu y,(g)Wy ul Surle3nfuor'.raq1.rng'Q = (6d)xllU(to)xrg seqdrur0U.Z€unueT ,arrrlelmuuroc ..,ir ud pun to acurg 'cr1oq.red,tq arc 6j pue to q1oq,Z7-(,tuaroeqtr dq ,1cegu1 lou ere 'suotllpuoc uor?ezrlsruJouaql segsrlss rIJlqA\u Jo lepo{u u"rsq)nJ 3 s}srxesferrrle araql 'f snuaS;o a)eJrns uu€urerg pesop e uo 3' 3ur>1reuuarrr3e Jod .tlJDurey

Remark. For a given marking E on a closed Riemann surface of genus g, there always exists a Fuchsian model of R which satisfies the normalization conditions. In fact, by Theorem 2.22, both a g and j3g are hyperbolic. Since a g and j3g are not commutative, Lemma 2.20 implies Fix(a g )nFix(j3g) =
,(27)6 snuaf ppolu uvzstlcngo lo t=ou[!d'ln] s.tolo.tauaf lo ua7sfrslD?tuouD)p uant6 D rol .gZ.Z uorlrsodo.r4 lo g acottns uuDuery pesop D uo g 6utt1.tou,

Proposition 2.23. For a given marking E on a closed Riemann surface R of genus 9 (~ 2), a canonical system of generators {aj, j3j }]=1 of a Fuchsian model of R which satisfies the normalization conditions with respect to E is uniquely determined by the point [R, E] in T g •

fi.anbrun s?K otTcailsa.r-q7rm suortzpuor ,"n frriJE":Hti:'{:ri:::r"'r:;'Y;::

r

r

u uorrelaJ ,s.ro1o"raua6 uta7sfr,s Ieluau"punJ alos eq? sagsrl€s qcrq.lr to IoJ.uouDJ slr se ol parreJarrl I=;{ ld ' lo s.roleraua3yo ualsfs eql .[3' ,g] acey.rnsuueruerll pesoll } pe{r€ru e Io pporu uorsqrnl peztlDturou eq1 .7 dno.r3 u€rsqrnd srq} ilec e11\

We call this Fuchsian group the normalized Fuchsian model of a marked closed Riemann surface [R, E]. The system of generators {aj, j3j }}=1 is referred to as its canonical system of generators, which satisfies the sole fundamental relation 9

'P?= 't"]L[ lld

II[aj,j3j] j=l

= id,

r=! = lld 'fo] eraqm

where [aj,j3j] = ajoj3joa;l oj3j-1. 'rldorloolgoln

'/G ol tuale^rnba sr (,<)Y r"qr q?ns /U * A : ! Surdderu crqdlouoloqrq / Utt V € slsrxe ereqt ueqtr '6Jul[g',A) = qcns uo,3 Bur4.retu€ pue l€ql lr [3',Ur] 6 snua3 Jo /Ar ec€Jrns uueruarg pesolc raqloue e{€I .6A.A uoxllsoilo.t4 lo !oo.t,4

Proof of Proposition 2.23. Take another closed Riemann surface R' of genus 9 and a marking E' on it such that [R, E] = [R', E'] in Tg • Then there exists a biholomorphic mapping f: R --+ R' such that f. (E) is equivalent to E'. A lift j of f to H, which is an element of Aut(H), is taken to satisfy

,Hor{ro ,,u",u^ -,;,j"ii 1.j.,,i,",,; i";:;"Jn

2. 2. Fricke Fricke Space Space

48 48

where {ot;,01 {aj, {3j }f=, H=l itis the the canonical canonical system system of of generators generators of of aa Fuchsian Fuchsian model model of of where condito ^D'. From R' which satisfies the normalization conditions with respect to E'. From condiwith respect conditions .R' which iatisfiLs the normalization tion (i), (i), we we have have iQ) j(z) == )z AZ for for some some lA )> 0. 0. Further, Further, by by condition condition (ii), (ii), a, a g and and tion = = a~ have have the the common common fixed fixed point point at at L, 1, and and hence hence )A = 1, 1, i.e, i.e, ij = id. Thus Thus we we get get a! tr aj = ati aj and Bi {3j Fi. {3i. 0 ai

=

=

Lemma 2.24. 2.24. Let Let {oi,gi}o;=, {aj, {3j U=l b" be the canonical canonical system system of of generators generators of of the Lemma for aa point lR, [R, El E] in in To. T g. AIf an element element t(z) ,(z) = normalized Fuchsian Fachsian model i for normalized + b)/(cz ++ d) d) of of {o;, { aj, {3j H=l do"t does not not coincide coincide with Bo, {3g, then bc bc }i= 0. (az + |i}t=,

r

= = =

=

=

Proof. ln In the case case where where 6b = c = 0, we have have Fix(7) Fixb) = Fix(Be) Fix({3g) - {0,*}, {O, oo}, Proof. case where the Next, in hence, and {3g in case a contradiction. are commutative, and and hence Bn 7 and Bo 0, we get Fix(7) Fix(,) = Fix(Be) Fix({3g) = {0}. {O}. Thus, 7, and {3g being non6b = 0 and c *i= Q,we commutative, Lemma Lemma 2.20 implies implies that that f is not not F\rchsian. Fuchsian. Hence we have a commutative, By the same same argument, in in the case case where b Ii= 0 and c == 0, we contradiction. By tr obtain a contradiction. 0 obtain

=

r

=

°

11=1

By this lemma, the canonical canonical system system {loi,gi aj, {3j )f=t of generators generators of of the norBy written uniquely in the malized Fuchsian Fuchsian model lr for a point point [.R, [R, E] E] in TTog is written form form

a·z 1 +b·1 , CjZ + dj T#, a'.z b,, atrz + * b~ 1 {3 9ij --= c',1 z + d'· ffa 6i,'

~._

*, -= "'1

1

ai di, ci € R,

ci > 0, ) 0, Cj

aidi- bici= I,

dili "'i> d'. -- b'· c'· = 1| 0' a'· o'i,our,C, ) 0, eF., cj fiCi 1 1 1 1

1

=

, i ,2,. .... . , 0, 9- -, 1. for f o r each e a c jh= 1 1, 6 -* R Now, we define g- 6 by R6e-6 coorilinates:F Fo:Tn Fricke coordinates g : Tg --+ we define the Fricke d ' s- r ) . a ' o- 1 ,C~_l' c ' n -r , d~_l)' ce - t , dgdg - 1r ,, a~_l' . . . ,,ag-l, as- | t Cg_l, a l , c~, c \ , d~, d 1 ,... ( 4 1 ,Cl, c 1 ,d 11, ,a~, :Fg([R, l ] ) = (al, f o ( [ R , E])

=

surfaces closed Riemann surfaces The image F space of closed the Fricke Fricke space is called the g(Tg ) is fo(To) g = :F Fn Fo topology of genus g. The topology of F is introduced by the relative topology of F Fo genus 9. g in g is introduced connected R 6g 6. In §2 of Chapter 5, we shall verify that F is a simply connected domain is asimply that.F, 5, we shall R6c-0. g $2 6 g- 6 • By the following theorem (Theorem 2.25), :F is a in R 2.25), Fsg is a bijective mapping theorem R6g-0. with F -Q ?,g with of identifying T on T g. Hence we define define aa topology on g to FFo. g under Hence we ?o of T Qg by identifying Riemann a closed of :F Therefore, a topology of the Teichmiiller space T(R) of a closed Riemann • space ?(.R) Teichmiiller of d.g Therefore, a we book, we rest of of this book, In the rest surface of T ?n. from that of genus 9g is is induced induced from of genus -R of surface R g • In assume that T and T(R) are equipped with these topologies. topologies. with these are equipped "(.R) assume that Tn g and 6 ---+R is injective. injectiae. Theorem g- 6 is R6'-6 coorilinales:F fo: g --+ Fricke coordinates g : TTo 2.25. The The Fricke Theorern 2.25.

=

= (al,cl,dl, ( o r , " t , d v , .... ..,a 's-r, p o i n t :Fg([R,E]) Proof. ,a~_l' e v e r y point fo(lR,t]) t o show s h o w that t h a t every W e need n e e d to P r o o f . We genof system canonical dg-l' d~_l) in F determines uniquely the canonical system {aj, {3j } of genthe uniquely determines Co-r,ils-1) in F,g {oi,0i } point E)eTo. erators ofthe normalized Fuchsian model r for the point [R, E] E T • the Jfor model F\rchsian normalized eiatoriof the g [8, : 1t a i dji --bjcj bici = r e l a t i o najd = 1,2, t h erelation For , g-1), o b t a i n e dfrom f r o m the i sobtained (j = I , 2 , .... ..,9 - l ) , 6 i bj is e a c hjj (j F o reach the same BV by with 0, and hence aj is determined uniquely by :Fg([R, E]). By the same fo(1R,4)' uniquely oi is determined hence and with Cj cr' > 0, ) argument, = 1,2, is also alsodetermined. determined. 1) is 1,2,..... .,,gg -- 1) argument,{3j 0i (j U=

6'

2.5. Fricke Space

aredg e:1crrg'g'g

49

What remains to show is that both O:g and {3g are determined by .rg([R, ED. By the normalization condition (i) for r, we have {3g(z) = ,\z with ,\ > 1. By the normalization condition (ii) for r, O:g has its attractive fixed point at 1, and hence

eeuaq pue'I 1e lutod pexu e^rlcertle sll mq to'J roJ (ll) uorlrpuoc uorl€zllsruJou aql {g'I y q}l^{ zy = (z)6! el"q a.&t'J.loJ (r) uorlrpuoc uorl€zrl€ruroueq} /tg < '(g'Ul)ot,{q paururralap ers 6d p* to qloq }sq} q ^roqs ol sureuer leq1\

Q'z)

(2.7)

=

'lld'fall=[U IeS'rldo6po6g= 6noLaltsqe/$ '.raq1rng lfd'!plr=!6lJ uolleler pluatuepunJ aql urorJ

id, putting ,

=

Set

, P + z c_ Q ) L 9tzo

a, b, c, d E R,

ad - bc = 1.

' I - ? q- p D

,(z)=az+b, CZ+ d

.

,L 3ur11nd 'pl

nJ;:;[O:j,{3j], we have ,00:g = {3goO:go{3"9

1

=

'opa6c=6q+6o

Further, from the fundamental relation rU=l[O:j,{3j]

'g)

p'e'q'o

:ploq suorlenba 3uu*o11o;aql 1eql aurnsse .[eur 'p- po* '"- 'q- 'r- ,(q p pue 'a 'g 'o Surceldag arrr 'fressacau ;t '.,f1arr1cadser

Replacing a, b, c, and d by -a, -b, -c, and -d, respectively, if necessary, we may assume that the following equations hold: (a - l)ag + bCg = 0, ca g + (d - ,\-l)cg = 0,

(orz) (o'e) (s'a)

(2.8)

'0=6?q+6e(t-o)

(2.9)

'0=ta(r_y-p)*6ec

cbg + (d - l)dg = O.

(2.10)

.0=tp(I-p)+6qc

'(Ot'Z) pue (9'6) uro.rg 1aBaan '16l pauru.ra?ap e^eqa^,recuag '(p - i/0 - o) - y pue 'I 'ZZ'ZvrarceqJq?rpsrluoc sSlJ 'cqoqeredu ,l * p'l * e ryql s^rolloJll snql qclq/'{'}- (f)zrl ecuaqpue'I = p ueq}'I = DJI'(p - I)y = I - p 1eq1sarldurr ol€q a^\ (O'Z) p* (9'6) uro.g 'qsrue,rlou saop 6c ro 6o Jo euo lseal le aculs

Since at least one of a g or cg does not vanish, from (2.8) and (2.9) we have a-I ,\(1 - d). If a 1, then d 1, and hence tr 2 (,) 4, which implies that , is parabolic. This contradicts Theorem 2.22. Thus it follows that ai-I, d i- 1, and ,\ = (a - 1)/(1 - d). Hence we have determined {3g. From (2.8) and (2.10), we get

=

ag

=

bCg = --, I-a , D _ I - ocq -

=

(rrz)

=

(2.11)

cb g

(zrd_

-(2.12)

= 1 _ d'

.-P\ -al - -ro ^ dg

san€ (2'6) otq (UI'Z) pu€ (II'Z) Jo uorlnlrlsqns

Substitution of (2.11) and (2.12) into (2.7) gives I-q+D

a+b-l _c+d-l 1 - a cg 1 - d bg. .onP-I 'I-p+c

(erz)

(2.13)

-0"

o-L

Here, if c + d = 1, then we have a + b = 1, because cg i- 0 by Lemma 2.24. Thus, from the relation ad - bc 1, we find that a + d 2, and hence , is parabolic. Again this contradicts Theorem 2.22. Therefore, we have determined O:g by .rg([R, ED. 0

paururalap a^eq ellt.'aro;araq; 'ZZ'Z ureroeqtr slcrperluoc srql ureSy 'cqoqered sr ,L acuaq pu€ '6 = p D pug e,lr 'I = cq - pD uorlelel aq1 uror; 'snq; + leql ' V 6 ' Z e : U u . U u i e l f q 6 c a s n e c a q ' I = q + p a ^ e q e ^ , ru e q ? ' I = p a a OI ;r'are11

=

n

=

'(ls'al)U rq,u"

5500

2. 2. Fricke Fricke Space Space

Notes Notes For historical historical and and expository expository accounts accounts of of the the uniformization uniformization theorem, theorem, we we refer refer to to For universal covering Abikoff [2], [2], and and Bers Bers [29] [29] and and [36]. [36]. The The original original idea idea of of using using universal covering Abikoff surfaces is is due due to to H. H. A. A. Schwarz Schwarz (cf. (cf. Bers Bers [29], [29], pp.264-265). pp.264-265). Complete Complete details details of of surfaces covering surfaces surfaces are are contained contained in in the the books books on on Riemann Riemann surfaces surfaces listed listed in in the the covering notes of of Chapter Chapter 1. notes The notion notion of of a F\rchsian Fuchsian group was was first first introduced by by Fuchs Fuchs in in the the study study of of The ofthe equations analytic continuation continuation ofsolutions of solutions ofcertain of certain ordinary ordinary differential of the analytic second order (cf. Ford [A-31], [A-31], Chapter XI). XI). See See also also Yoshida [A-113]. [A-113]. For more second Fuchsian groups, groups, we refer to to Jones Jones and Singerman Singerman [A-48], [A-48], and Lehner details on F\rchsian called Kletnian [A-66] [A-67]. subgroups of PSL(2, C) called Kleinian groups, gro 1J,ps, are of PSL(2,C) subgroups Discrete and [A-67]. [A-66] intimately related to to the theory theory of of Teichmiiller spaces. spaces. It It is most which are intimbtely that this interesting subject cannot be covered. covered. Concerning Kleinian Kleinian regrettable that groups, see see Beardon Beardon [A-11], [A-ll], Berset Bers et al. al. [A-15], [A-15], Ford [A-31], [A-31], Krushkal" Krushkal', Apanasov Apanasov groups, GusevskiI [A-61], [A-61], Lehner [A-66], [A-66], Magnus Magnus [A-70], [A-70], and Maskit Maskit [A-71]. [A-71]. For and Gusevskil between Kleinia"n Kleinian gloups groups and 3-manifolds, 3-manifolds, we we also also refer refer to Epstein [A[Arelation between Bass and Morgan 25] and [A-26], Laudenbach Poenaru [A-29], Bass [A-76], Po6naru and Laudenbach Fathi, 25] [A-76], [A-29], [A-26], McMullen [154], Thurston [231]. Poincare [A-90] collected works works on McMullen [A-90] is his collected [231]. Poinca"r6 [154], and Thurston Fuchsian groups functions. groups and automorphic functions. Fuchsian groups, see see Nicholls For the interaction between between ergodic Nicholls discrete groups, ergodic theory and discrete Velling [A-86], Bowen and Series [47], Morosawa [158], Series [195], and VeIling and Series Morosawa Bowen Series [195], [158], [47], [A-86],

Matsuzaki Matsuzaki [241]. [241]. Fricke and Klein [A-33]. Fricke and appeared in Fricke spaces first appeared Fricke spaces [A-33]. For modern treatBers and Gardiner [42], ments, Goldman and Magid [A-36], Bers and Abikoff [A-1], see Abikoff ments, see [42], [A-36], [A-1], Keen Saito [186], and Weil [243]. and Keen [110], Saito [243]. [186], [110], group .9cfioltky group we can can use use aa Schottky For aa representation surface' we Riemann surface, representation of aa Riemann space instead instead of aa instead we obtain aa Schottky schottky space and we F\rchsian group, and instead of aa Fuchsian and Sato Sato Teichmiiller space. Bers [35], is discussed discussedin Bers space. This topic is [98], and [35], Hejhal [98], [188] and [189]. and [18e]. [188]

sa{s4os V V '1'g uorlrsodor;

Proposition 3.1. (Schwarz-Pick's lemma) Every holomorphic mapping f : ..1 ----+ ..1 satisfies : t |utddoru ctryl"toutoloyfr"taag (wlo ru:al s6{crd-z.re,*qcg)

'zz pup- Iz uee/r leq ??uDlslp?rDourod eq1 (zz'rz)d llet e^\.aclr€lslp Jo $uorxe aql seuslles d ryqI u^roqs sr.Il'zz pue rz ?ceuuoct{clq,$ 7 ut se^rnf, elqeul}leJ II€ sa^ou 3 'arag

Here, C moves all rectifiable curves in ..1 which connect Zl and Z2. It is shown that p satisfies the axioms of distance. We call p(Zl, Z2) the Poincare distance between Zl and Z2. , l 'zi l' - t 'o i'

rf c / - l u' = l (zz'tz)d

lzplz J

'V tlz slu-tod orr,r.1fue log ) zr las alrlr eJo lapotu e lrnrlsuoc ol clrletu stt{l pesn ?J€culod 'H

H. Poincare used this metric to construct a model of a non-Euclidean geometry. For any two points Zl, Z2 E ..1, we set 'drlauroaS ueaprlcng-uou

. z Q l '-l i = "tP "1'P1tr

ds = (1 _ Iz12)2 . 2

41dzl 2

The unit disk ..1 = {z E C Ilzl < I} has several "natural" metrics. One of them is the Euclidean metric ds 2 Idzl 2 dx 2 + d y 2, and another important one is the Poincare metric

xuleu 9rv?utod eq} sr euo luelrodrur leqloue pue 'rftp * "*p = ,ltpl -- es'pclrletu ueepqcng aql sl ureqlJo auo'srrrla{u ..letrnleu,,Iere^es seq {I > lrl I C ) zI = 7 }tslP }Iun eqtr

=

=

crrlatr l ?rBcurod 'T'I'8

3.1.1. Poincare Metric 'I'g

3.1. Poincare Metric and Hyperbolic Geometry l(llaruoag

pue rlrlatr tr ?rerutod

rrloqrod/tH

'uolsrnql

In this chapter, we shall discuss some aspects of the hyperbolic geometry on Riemann surfaces which is induced by the Poincare metric on the unit disk. First, in Section 1, we define the Poincare metric and study basic properties, especially those concerning geodesics. Using hyperbolic geometry, in Section 2 we define a system of coordinates, called Fenchel-Nielsen coordinates, on the Teichmiiller space of a closed Riemann surface. In Section 3, we discuss an embedding of the Teichmiiller space into an Euclidean space by means of geodesic lengths, which has its origin in classical investigations of Fricke and Klein. Finally, in Section 4, we give a sketch of the construction of a notable compactification of the Teichmiiller space, which was recently proposed by W. Thurston.

'A\ fq pasodorddlluacars€^rqcrq/!\'acedsrellnurq)Ial aqt uotlecyrlceduroe Jo q)le{s e e,rr3 aaa.'p uorlcag ut 'fleutg elq€1oue uorlf,nrlsuot eql Jo Jo 'sq1Eua1 'urely pup a{?tJd suorle3tlsa,rul Jo Ieclsselcut ur3trosl! seq qctq,lr crsepoe3go sueeru,(q acedsueaprlcngue olur acedsrallntuqclel eql JoSutppequa ue ssncsrpa,n 'g uorlces uI 'e?€Jrnsuu€ureru pesoll e 3o ecedsrallnuq?Ial 'saleurprooc uralsdse auuaPe1'r aql uo 'saleurp.roolueslerNleq?uad ;o Pallec 'd.rlauroaE esoql fletcadsa crloqradfq 3urs11'scrsapoe3 Sutu.recuof, 6 uorlces ur 'serlredordcrseq{pn1spue elrlau ereculodaql eugepaiu.'1uorlcagut'1s.rtg '{srp uuetuelg }run eql uo crr}eruersculodaql dq pecnpulfl tlclq^rseceJrns uo frleuroa3 cqoqredfq aq1;o slcedseaurosssnc$pII€qsa^\ 'reldeqc sql uI

salBurProoc uoslarN-Iaqruad puB rt.rlauroa.D rrloq.radfll

Hyperbolic Geometry and Fenchel-Nielsen Coordinates

t raldBrlc

Chapter 3

52

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

1f'(z)1 lf'.(:)l

1

=.~ 1+, -lzI - l " r2-' I1--If(z)12 lf(r)l' =

z €E a. Ll.

point in Moreover, ifif the equality equality holds holds at one one point in A, Ll, then then ff is a biholomorphic biholomorphic Moroaer, automorphism of A, Ll, and the equalitg equality holds at any point point of of A. Ll. aulornorphism of

Fix a point point z in in A Ll arbitrarily, arbitrarily, and set Proof. Fix w *z w+z .tt\w) =TlZw, 'n (w) = 1 + zw' / \ w-f(z) . l12(W) z t w t == ,W- f- z f(z) 1w' 1- f(z)w

11 and 72 12 belong to to Aut(A), Aut(Ll), and ,F(to) F(w) == J2 12 o 0 ff "0 lr(w) 11(W) is a holomorphic Then 71 Ll into into 4. Ll. Since Since .F(0) F(O) == g0 attd and mapping of 4 ' l - l , l 22

'( ) 1 - Izl f'() F,(o)= F 0 = 1 _ If(z)12 ffif'(r), z,

otr

we have have the assertion assertion by Schwarz' lemma. Schwarz'lemma. we

denote by /-(ds2) j*(ds 2) the pull-back of the Poincard Poincare metric ds2 ds 2 = we denote When we 2 2 implies that 4IdzI 3.1 implies Proposition 3,1 aldzl2/(1 /, Proposition /(l - Iz1 lrl')')2 by f, f* (ds2) < ds2

2 if - ds and that f*(ds 2) = Aut(Ll). ro Aut(A). if /f belongs belongs to dsz if and only if that I.(dt2) ------Ll Corollary. A satisfies satisfies mapping ff : Ll A --+ holomorphic mapping Eaery holomorphic Corollary. Every p(f (zt), f (zzD 1 p(21, z2),

21,22 € A.

Remark. /{(h) of aa Riemannian Riemannian metric general, the curaalure K(h) the Gaussian Gaussian curvature Remark. In general,

h(z)2IdzI2 g i v e n by by ( h ( r ) >> 0) i s given o ) is h ( z ) 2 l d z l z(h(z) 2 4 fl2logh g K(h) , h ( h=) _~ = - F d10 h.

h2

a azaz

Poincar6 metric A simple curvature of the Poincare showsthat the Gaussian Gaussian curvature simple computation shows -1 on is on Ll. 4. equal to -1 is identically identically equal under the is invariant under Moreover, )2Idz\2 is h(z)2ldzl2 when aa metric h(z we can can see see that, when Moreover, we constant up to to aa constant metric, up action with the the Poincare Poincar6 metric, coincideni with it is is coincident by Aut(Ll), Aut(A), it action by factor. factor.

t9

,{r1auroag cuoqradifll pu" f,rrlel{ gr"f,u-Iod 'I't

3.1. Poincare Metric and Hyperbolic Geometry

53

scrsaPoaD 'z'T't

3.1.2. Geodesics

Ie

eleq e^r JI '9z ul zz pve Iz $ullcauuoc (cr.r1eru gr€oulod aq1 o1 lcadsar q1ra,r)ctsapoe0e 'V ul zz pue rz Surlcauuot'g cre pasolc elq€Urlcar€ IIef, e \'V ) zr 'rz s?ulod orrr1fue rod'(r)/ fq 1r elouap pu€'C p y76ua1cqoqtadfr,tlaql sp "[ lV, ell.'V ur , ]re pasolc alqegrlcer fre,ra rog

For every rectifiable closed arc C in .d, we call ds the hyperbolic length of C, and denote it by f( C). For any two points Zl, Z2 E .d, we call a rectifiable closed arc C, connecting Zl and Z2 in .d, a geodesic (with respect to the Poincare metric) connecting Zl and Z2 in .d, if we have '(C)l = (zz'rr)d

sl puD zz puo rz q0norqt sassodt1cn1m7uau,6as Vg fi.topunoqeql oI 1ouo0or17.to euq eql ro el?Jr?eyyto ctoqns o s, puD anbtun st 7r |teaoanory'V ul zz puo rz |utTcauuoc crsapoa0o slsNaeere1l 'V ) zz'rz fi^to4tq.toro,I 'Z'g uorlrsodor6

Proposition 3.2. For arbitrary Zl, Z2 E .d, there exists a geodesic connecting Zl and Z2 in .d. Moreover, it is unique and is a subarc of the circle or the line segment which passes through Zl and Z2 and is orthogonal to the boundary {).d of .d.

'v lo

'0 1 zz pue = rz luauela ue fq tuaql Sururrogsuert /tq 1eq1 etunsse deur 0 en'(V)1ny fq uorlce ar{l repun lu€rrelur sr crrleru ar€curod eq1 acurg /oo.l4,

Proof Since the Poincare metric is invariant under the action by Aut(.d), we may assume that Zl = and Z2 > 0, by transforming them by an element

°

i9 Z -

Zl

zlz -t

= Q) L ()

#eP

,Z = e

1-

ZlZ

eleq era 'zz pue g Surlcauuoo trts pesolc frala ro;'r".II C 'U f d elq€1lns qYal. (V)7nY P

of Aut(.d) with suitable 0 E R. Then, for every closed arc C connecting

°

and

Z2,

we have

,W"[ ",ol

-:-2-=-ld-;-z > t' _2_dx_. e 1 -lzl 2 - 10 1 - x 2

= f(C)

7 luaur3es euq eq? qlr^r lueprcurof, sr Cyr fluo pueJI (rh

if and only if C is coincident with the line segment L =

-

tr

Hence p(O, Z2)

,:;:-1

.2, I xp7,

1 = (zz'1)d acuag

[0, Z2].

0

'lzz'o)

''V LV uo slurod oall i(ue Eurlcauuoc crsapoe3fre,ra'6'9 Jo rr"qns e q uorlrsodor4 ,tq 'teql a?ou eJaH ',0 fq uorlce eql rapun luerrslur sr f,y uxe aq; 'L p'V sDr€eql palpc sr Ve oI 1euo3oq1.ro sr pue slurod asaql q3no.rq1sassed qcrqa,llluaur3as auq eql ro alcrr? eql Jo y u1 lred aql l"ql Ip?eU 'Vg uo Lo prte L.r, slurod paxg Irurlsrp otrl seq l, 'crloqredfq 4 (VhnV 3 ,L uaqal '1eq1 lpcag

Recall that, when, E Aut(.d) is hyperbolic, , has two distinct fixed points r-y and a"( on {).d. Recall that the part in .d of the circle or the line segment which passes through these points and is orthogonal to {).d is called the axis A"( of ,. The axis A-y is invariant under the action by ,. Here note that, by Proposition 3.2, every geodesic connecting any two points on A"( is a subarc of A,,(. 3ur11as,(qpaugap sr g aueld-geq .raddnaql uo {sp crrlaur ar"curod eq&'tlrout?[

Remark. The Poincare metric dSk on the upper half-plane H is defined by setting

"l'Pl

2

IdzI 2

, z(lu'l) = ,rrp dS H = (Imz)2'

'y otuo H lo (! + z)/(? - ,) = (z)1,uorleurroJsuert snlqgl i eqt ,,(qv uo zspcrrleru ?r"curod eql Jo {req11nd aql 1nq3mq1ouq qclq,tr

which is nothing but the pull-back of the Poincare metric ds 2 on .d by the Mobius transformation ,(z) = (z - i)f(z + i) of H onto .d.

Hyperbolic Geometry Geometry and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3. Hyperbolic

54 54

3.1.3. Hyperbolic Hyperbolic Metric Metric on on a Riemann Riemann Surface Surface 3.L.3.

Let itR be a Riemann lliemann surface whose universal universal covering surface is biholomorphibiholomorphiLet L1. Consider Consider a Fhchsian Fuchsian model I of of ,R R acting on /.L1. Let Let cally equivalent to 4. 11": A L1 -----+ r? R be the projection of of 4L1 onto R = = A/f. L1/ Since Since the Poincar6 Poincare metric r: ds 2 is invariant under the action by .l-, we obtain obtain a Riemannian metric ds2p dsh on ds2 R which satisfies satisfies r- (dszp)= d,s2 1I"*(dsh) ds 2 .

r,

r.

r

We call this dsft dsh the Poincar6 Poincare metric, or the hyperbolic hyperbolic metric on R. E f corresponds corresponds to an element element [Cr] [G,] of the fundamental group Now, every, every 1 e 1I"1(R,po) R (Theorem 2.5). 2.5). In particular, ,7 determines determines the free free homotopy r{R,po) of ,R class say that of the class G" where G, representative of class [G,]. say that, covers We of where is a representative class C7, C,, 7 coaers [Cr]. the closed curve G,. curue the closed Cr. j € -t, - A-, When, E f is hyperbolic, it it is seen that the closed closed curve L, A,/I < ), seen that 1l , >, When image on .R R of the axis axis A, A, by 11", geodesic (with (with respect respect to the zr, is the unique geodesic the image hyperbolic metric metric on ,R R )) belonging belonging to the free homotopy homotopy class class of of G,. L, Ct. We call L-, hyperbolic geodesic to corresponding to or the closed "7, G,. closed geodesic corresponding C.,.

r

r

=

coueringsurface surface Proposition be a Riemann surface with with universal uniaersal covering Lel R be Riemann surface Proposition 3.3. Let H, and r be a Fuchsian model of R acting on H. Let model acting on Let H , and 11l be Fuchsian

+ = ---, ,Z tk) cz+ ( )

a z * bb _ az c z * dd'

a , b , c , dE a,b,c,d e PR, -,

a d -- bc b c = 1, 7, ad

r

geodesic be element on and L, bethe the closed closedgeodesic on R corresponding hyperbolic elemenlof 11, L, be R corresponding beaa hyperbolic I , and to lenglhl(L,) I(Lr) of L, L, satisfies satisfies to ,. Then the the hyperbolic hyperboliclength 7. Then

t.'(r) - @+d)2= 4cosh2 e) 2 an {t) are Proof. tr2(7) are invariant under the conjugation of, of 7 by an t(L-r) and and tr Prool. Since Since l(L,) - AZ )z (.\ 1). also element = (A > 1). We may also we may assume assume that ,(z) element of ol Aut(H), Aut(H), we 7(z) we have this case, case,we have and d = 1/-/>.. I/\5.In In this assume t5, b = c =0,0, and assumethat ao - -/>.,

=

= =

=

dy ((L.t) =fIr^ = 210ga. ) = l o gA 2log a. = -y = + = log A

l(L,)

1

Hence we have have the assertion. assertion. Hence we

o!

3.1.4. 3 . 1 . 4 , Pants Pants

Consider which admits the hyperbolic metric by aa surface R r? which Riemann surface Consider cutting aa Riemann family of mutually geodesicson R. Let P be be aa relatively simple closed closed geodesics mutually disjoint simple compact subsurfaces.If If P contains contains connectedcomponent component of the resulting union of subsurfaces. compact connected no more geodesicof R, be triply triply connected, connected, i.e., i.e., .r?,then P should should be more simple simple closed closed geodesic homeomorphic region, say say homeomorphic to aa planar region,

cc

55

frlauroa.g rqoqraddll

'I't Pu€ f,rrlel{ grsf,urod

3.1. Poincare Metric and Hyperbolic Geometry

({i t rr- ",}^{i >rr+,r})- { z > l z l } = 0 4 Po =

({ Iz

{izi < 2} -

+ 11 ~ ~} U {Iz - 11 ~ ~}) .

'U uo rlsapoe3 pasolc elduns € sl 2I ul d Jo frepunoq e^rleler eqt Jo luauoduroc palrauuoc frarra 3r Pue Palceuuoc i(1dt.r1q d JI g' 'g e IIef, e^\ '.re1;eara11 1o sTuod;o .rted e U Jo d eD€Jrnsqnslcedtuoc f1errr1e1e.r Surppnqa.r.ro; secerdlseilerus eql Jo auo s€ pereplsuoc aq ue? d et"Jrnsqns e qcns

Such a subsurface P can be considered as one of the smallest pieces for rebuilding R. Hereafter, we call a relatively compact subsurface P of R a pair of pants of R if P is triply connected and if every connected component of the relative boundary of P in R is a simple closed geodesic on R.

4J/V:d

Fig. 3.1.

'r'8'tIJ

Fix a pair of pants P of R arbitrarily. Let r be a Fuchsian model of R acting on ..1, and 1r : ..1 ---+ R = ..1/ r be the projection. Let P be a connected component of 1r- 1 (P). Denote by rp the subgroup of r consisting of all elements ,of such that ,(P) = P. Then p is a free group generated by two hyperbolic transformations, and P = P/ p. Set P = ..1/ rp. Then P is a surface obtained from P by attaching a suitable doubly connected region along each boundary component. Hence, P is again triply connected, and rp is a Fuchsian model of P (see Fig. 3.1). Clearly, P is considered as a subsurface of P, which is the unique pair of pants of P. In other words, P is uniquely determined by rp. Habitually, P is called the Nielsen kernel of P, and P is called the Nielsen extension of P.

'd louoNeuelseueslerNaql Pellet sr d Pu€'d Jolaweq ueslerN erll Pall€l 'd st 2r 'f11en1rq€H'-dJ fq paurunalap flanbiun ,, j '.pto^ reqto uI Jo slued ',,(pee13 '2' 3o aee;rnsqns € s'e pereplsuoc sl d ;o rred anbrun eql s! plq^{ 'pelcauuoc ,{1dr.r1 '(t'g '31.{ eas) ; Jo lapour uelsqcqil e q d.7 pue 'acue11'lueuodtu6r ,t.repunoq qcee Suop uorSar palceuuof, .{1qnop ure3e sr 2' paul€tqo areJrns€ sI uaql 'a,t/V - d las elq€lrns e Surqcelp fq dr 4'uro.r; ql/d d pue'suotleurrogsuerl crloq.rad,tqom1fq pele.reue3dno.r3aa.ge s1 d.7 uaql 'd = ({)t }€tI} q?ns J Jo L dnor3qns eqt dJ fq alouaq '(d) r -! Jo lueuoduroc sluetuele yo Surlsrsuoc J Jo 1e = U aq1 aq pelceuuoce aQ V i )L pve'7 uo 3ur1ce JIV d 1e1'uorlcelord 'fpre.r1rqr" g 3o 7 slued ;o .rted e xtg U Jo lapour uersqcqE e aq J 1e1

r

r

r

56

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

3.1.5. Existence Existence and and Uniqueness Uniqueness of of P'-ts Pants 3.1.5.

shall discuss the the relationship relationship between the the complex complex structure structure of of a triply triply conWe shall domain J? Q and and the the hyperbolic hyperbolic structure structure of of P, P, the the unique unique pair pair of of pants pants of of nected domain Q, induced by by the hyperbolic hyperbolic metric metric on O. Q. O, Let L1,L2, L 1 , L 2 , and L 3 be the boundary boundary components, components, which a.re are simple closed closed and..L3 Let geodesics, of of the the pair pair of of pants pants P. P. Let Let J-e model of of the domain domain geodesics, o be a Fuchsian model Q acting on A. Ll. Then'i-s Then'ro is a free free group generated generated by two two hyperbolic transO that 1'1 and may assume that 1'1 and 1'2 cover .L1 L 1 and L2, formations, say, say, 7r may assume and. 1'2. We formations, 72 71 72. respectively.

r

Theorem 3.4. 3.4. For given triple triple (ayaz,as) (a1, a2, a3) of of positiae positive numbers, numbers, For an arbitrarily giaen Theorem planar Riemann connected planar surface Q such such that there exists Riemann surfoce triply connected lhere erists a triply

=

t£(L ( L ji )) = a aj, 1,

=

ji - 1 ,1,2,3. 2,3.

it by constructing constructing O Q explicitly. explicitly. Proof. We prove it say C C2, Let Cr C 1 be the part part of of the imaginary imaginary axis in A. Ll. Fix Fix another geodesic, geodesic, say 2, 2. On the Ll such that the Poincar6 Poincare distance distance between between C1 C 1 and C2 C 2 is equal equal to a1f at/2. such that on 4 from which which the Poincar6 Poincare distance to C1 C 1 are equal other hand, geodesics geodesics on 4Ll from other a3/2 form form a real one-parameter one-parameter family family (i.e., (i.e., the family family of circular arcs arcs Cl C~ to oBf2 geodesic, Hence there exists exists a geodesic, tangent to the broken circular arc in Fig. 3.2-i)). Hence and C Cg Cz2 and say that the Poincare Poincard distance between C such that family such say C 3 is Cs, 3 , in this family equal a2f2. equal to a2l2.

i)

ii)

Fig. 3.2. Fig.3 .2.

determined by the condition Next, let Zl 4 uniquely determined z2 be be the points in Ll 21 and arrd Z2 a1

p(rr,r) p(Zl,Z2)

== 2' ?,

zzEe C2· zr Ee CCt, Cz. 1 ,Z2 Zl

and {zs, let {Z3, 22. Similarly, Let Z4} and Z6} geodesicconnecting 21 and and Z2. Similarly, let connecting Zl L\ be be the the geodesic Let L~ {rs,re]1 {z3,za} be by the the conditions conditions pairs of points uniquely determined by uniquely determined of points be the the pairs

L9

pu" rrrlel l ?r"3ulod 'I't

3.1. Poincare Metric and Hyperbolic Geometry i(rlauroag ruoqradi(g

a2 = 2'

57

Z9

' 8 8 8 8 8 C ) v z ( z C ) ,E zz - ( v z ' e s ) 6 p(Z3' Z4)

Z3

a3 tp

= 2'

E C3 , Z6 E C ll

'rC)sz'eC)sz

, 7 , = (sz(s2)i p(zs, Z6)

E C2 , Z4 E C3 ,

Zs

'eslrlurod fg Eur,rraserd les 3 go ursrqdrour ''e'l'f^2 -o1ne crqd,rouroloq-rlu" aqt qlr^r o1 uorl?auer eql;q (g'Z't = f) laadsar 't=[{!,1'!c} lh p1 pepunoq uo3exaq cqoqrad,(q pesolt eq? eq o p"r fq ('(fa'g 'ft9 eag) 'ez Pue ez Pue'?z pue 8z Eurlceuuoc scrsapoaSeq1 'f1a,rr1cadse.r'!7 pve 1I ,{q alouaq 'flelrlcedser

respectively. Denote by L~ and L~, respectively, the geodesics connecting Z3 and Z4, and Zs and Z6. (See Fig. 3.2-ii).) Let D be the closed hyperbolic hexagon bounded by {Cj, Lj }J=1' Let 'TJj (j = 1,2,3) be the reflection with respect to Cj, i.e., the anti-holomorphic automorphism of C preserving Cj pointwise. Set 'TJ2,

tebsV)-rl0

1'2 = 'TJ3

'TJ1·

'Vtoeb-zL

1'1 = 'TJ1

0

Then 1'1 and 1'2 are hyperbolic elements of Aut(Ll). Let ro be the group generated by these 1'1 and 1'2' pele.rauaEdno.rEaq1 aq o.ir1e1 '(V)l"V

.z,L pue rl, aseqt fq t" sluauele arloqrad,tu are zl, pue rl, ueq,L

Q=L1/ro Fig. 3.3.

'8'8'ttJ

It is clear that il = Ll/ ro is triply connected, and that the unique pair of pants P of il is the interior of the set obtained by identifying the boundary of DU'TJ1(D) under the action by roo Thus il is a desired surface (see Fig. 3.3). 0 '(g'g '81.{ eas) ace;rns peilsap " sr snqtr '0J ,tq uorlce eql rapun (O)tbnO tr {Ji go ,trepunoq eq1 Surf;rluapl fq paurclqo ?as eql Jo rorrelur eql sl (J ;o 2, slued yo rpd anbrun arl? lstll pue 'pelceuuoc f1du1 sr.oJ /V = U leq+ realc s-rlt

'4 perepro aq1lo st176ua7 cqoqtedfrq aqy fiq lo sTuauoduoefi".topunoq p?aunrepp fryanbrunsa 4 syuodto .ttorl o to ernlrtuls aelilutoc ?ttJ .g.g ruoJoaql

Theorem 3.5. The complex structure of a pair of pants P is uniquely determined by the hyperbolic lengths of the ordered boundary components of P.

feur e,r. 'r(.ressacau.l uorleEntuot-(g)7ny ue 3ur:1et 'asodrnd srql lsrll eutrrnssp rog'r={{{o},(q peururralap,{lanbrun are zL pue I,L }eql ^\oqs o} seclsns U 'e? 'I zL) eL sraloc o teql pue (e = 1) 17 s.rairoc{1, ?sqt arunsss feur aa,r ,_(tf 'ara11'0.7 srolereuaS;o rualsfs e aq 'I.L} 'g eueld-y1eqraddn eq} uo leT Jo {zt 3ur1ce Jo lepour uersqr\{ e eq 0J pu€ 'd Jo uorsuelxe ueslarN eql eq d ?c,-I'd d Jo (g'Z'l = f) f7 lueuoduroc frepunoq aq1 ;o q1Eua1crloqlad,(q eql eq lp p"I 'fy.rer1rq.reuaarEsr sluedgo ned s 'too.r4 teql esoddng ('IIt'[Ott] uaay'93) 2,

Proof. (Cf. Keen [110], III.) Suppose that a pair of pants P is given arbitrarily. Let aj be the hyperbolic length of the boundary component Lj (j 1,2,3) of P. Let P be the Nielsen extension of P, and ro be a Fuchsian model of P acting on the upper half-plane H. Let h1,I'2} be a system of generators of roo Here, we may assume that I'k covers L k (k = 1, 2) and that 1'3 = (1'2 0 I'd -1 covers L 3. It suffices to show that 1'1 and 1'2 are uniquely determined by {aj }1=1' For this purpose, taking an Aut(H)-conjugation if necessary, we may assume that

=

58 58

3. Hyperbolic HyperbolicGeometry 3. Coordinates Geometryand and Fenchel-Nielsen Penchel-Nielsen Coordinates

11(Z) 1 ,1, I t ( z )=,x2 = \ 2z2, , 00 < ),x< < a z * b az + b / \ .r2(z) ad-- be bc- 1,I,c 0, ) 0, 12(Z) =;ii,, ez + d' ad e>

=

=

and that that 11 is is the the attractive attractive fixed point of fixed point and of 12, or equivalently, equivalently, 12,or

=

a l b = e+d, sai, a+b

b

O <-- ! <
Then we we see see that that 12(00) a/c > and aa + Then ) 00 and since the 0, since the middle-point middle-point > 0, + dd > fz(m) == a/e (a -- d)/(2e) d)/(2c) of points of of two two fixed fixed points of 12 has aa value value less (a lessthan than 12(00). 72 has Zz(oo). Next, write write Next,

az -1,, -u+ u Ab, ad-be =) -_-_, (, 7 3\ -1 ) -(z) ' ( z= a d - b z= - 1.1 (,3) dz*b

ez+d

(73)-l = 12 j2o0 ,7r, we may Since(,3)-1 may assume assumethat Since 1> we

6 , = a \ , b=b/>', 6 = b 1 \ , e=e>., E - c \ , d=d/>.. d=a1>,. a=a,x, particular,e> d > O. Moreover,the In particular, of the 0. Moreover, the middle-point middle-point(a (6 - d)/(2e) a11pe1of points the fixed fixed points (73)-1 has greaterthan hasaa value valuegreater than (,3)-1(00) (73)-r(x) = a/c. of (,3)-1 d < O. d/8. Hence, Hence,a a+ O. + the other On the other hand, hand, by by Proposition On Proposition3.3, we have 3.3,we have

=4cooh2 () + 1/>.)2 l/r)' =4cosh2 (>' (~1) (+) ,, 2 =4cosh (a+ d)2 d)2=4 cosh2 (a (~2) f +) ,, \ 2 / '

(a;) . \ 2 /

=4cosh2 (+) (a J:2=4 cosh2 @+ d)2

j2 are Therefore,'y1 and 12 are uniquely uniquely determined determined by {a1, Therefore, 11 and a3 }. az,as}. {or, a2,

D tr

proved that, for any triple have proved We have triple of positive numbers, numbers, there exists exists a pair of pants admitting admitting a reflection reflection (induced, (induced, for example, example, by "7d that the hyperrlr) such such that bolic lengths of of the ordered ordered boundary components components are are the given given triple (Theorem 3.4), and that 3.4), that it it is uniquely determined by the given given triple triple (Theorem 3.5). Thus we (see also we have have the following corollary (see also Fig. 3.4). 3.4). Corollary. Corollary. Eaerg Every pair pair of of pants P has has an anti-holornorphic anti-holomorphie automorphism Jp J p of of order two. two. M o r e o a e r ,the L h esset e t F Jr rp = {{z E P Il JJpp( off a all points off J pp consists Moreover, l l ffixed ixedp oinlso consists z e ( r z) ) = zz}} o of satisfyingthe of thrce three geodesics geodesics {Di}|=, {Dj }1=1 in P satisfying the following following condition: condition: For euery L,2,3), Di every j (j (j = 1,2,3), D j has has the the endpoints endpoints on, on, and is orThogonal orthogonal to, to, both both LL ij aand n d LL 1j +; r1,, w h e r eLL +4 = LL t1.. where

=

=

=

=

We call "Ip J p described described in this corollary the rc,fl,ection reflection of of P. P.

.6J

Let R be a closed Riemann surface of genus g (~ 2). As before, consider cutting R along mutually disjoint simple closed geodesics with respect to the hyperbolic metric ds1t on R. When there are no more simple closed geodesics of R contained in the remaining open set, then every piece should be a pair of pants of R. Recall that the complex structure of each pair of pants of R is uniquely determined by the triple of the hyperbolic lengths of boundary geodesics of it by Theorem 3.5. It is clear that R is reconstructed by gluing all resulting pieces suitably. Hence, we can consider, as a system of coordinates for the Teichmiiller space T g , the pair of the set of lengths of all geodesics used in the above decomposition into pants and the set of the so-called twisting parameters used to glue the pieces. Such a system of coordinates is called Fenchel-Nie/sen coordinates on Tg .

uo selDuzpron ueslely-leq?ury pelle;. sI sel€ulprooc;o rua1s.{s€ qcns 'sacerdaql an13o1 pasn sralatuered 3ut1stall pellec-os arll Jo las aql pue slued olur uorysod -ruocep e^oqe eql ur pesn scrsapoe311e;o sqlSualJo les aq1;o.ued aq1 'rg aceds rellnurqcral aql roJ sel€urproo?;o ue1s.{s e se '.reptsuocue, a^\ 'ecue11'f1qe1tns secard3ur11nsarIIe 3urn13fq palcnrlsuocer sI U leql r€elc sI lI 'g'g ura.roaq; ,{q 1r Jo scrsepoe3,{.repunoqyo sqlEual cqoqradfq eql Jo a1dtr1aq1 fq paunurelep .{lanbrun sl g. slued ;o rred qcea Jo ernlf,nrls xelduroc eq} }eql II€reU Jo '1as uado Surureurereql ul 'p. yo qued 3o .rrede eq plnoqs ecard drarraueql paureluoc U Jo s)rsepoa3pasolc alduns eroru ou are areql uaq1yg, uo {sp f,Ir}aur flenlnur 3uo1ey crloqrad.rtqaq1 o1 laadser qllr* srtsepoa3pesolc aldurts 1u1ofs1p 3ur11ncraprsuoc'eroyaq sy'(e {) f snua3Jo af,eJrnsuu"ruerg pesol?e aq Ur lerl

3.2.1. Pants Decomposition uorlrsodtuocaq

slusd'1''Z'e,

'.re1deqc$r{} Jo rapulqurer aq1 ut fleeq suollJasseeql esn 'g .ra1deq3 lr]un rueroer.{lsqtJo;oord e Sur,rr3auodlsod a6 [eqs e,lr q3noql 'g-fgll o7 ctr1d.r,ou.to?uo?! puD e-oell a! uzD'tuopD s! 6l acods aqu.r,treqJ (g1'g uraroaqa) 'uraroaql s.rallnurqtraJ

We postpone giving a proof of this theorem until Chapter 5, though we shall use the assertions freely in the remainder of this chapter.

Teichmiiller's theorem. (Theorem 5.15) The Fricke space Fg is a domain in R 6g-6 and homeomorphic to R 6g-6.

'.{1e,rr1rn1ur reql€r 1nq 'o3e aurrl 3uo1e pelrecuof, sehruaJoaql Euro,o11o; eq1 're,roarohtr'e-6etlJolasqnse sl tdr ecedse4crrgaql'flsnoue.rd pa1e1ssy '.{rlaruoa3crloqred,tq Sursn ,,lq 6g o1 saleurproo) Jo ad.{1raqloue acnpor}ur a,r. 'uotlces slt{l uI 'saceJrnsJo slepotu uelsqcnd Sursn ,{q 'eceds ueeprpng tg lasqns l€uorsueurp-(S-0g) learJo (eceds e{clq aq1 parueu) (Z ?) f snua3 ;o tg aceds rellnuqtlal aql peluasardar am,'6 reldeqS u1

In Chapter 2, we represented the Teichmiiller space T g of genus g (~ 2) as a subset F g (named the Fricke space) ofreal (6g-6)-dimensional Euclidean space, by using Fuchsian models of surfaces. In this section, we introduce another type of coordinates to T g by using hyperbolic geometry. As stated previously, the Fricke space F g is a subset of R 6g- 6 • Moreover, the following theorem was conceived a long time ago, but rather intuitively. * *

sa +B ur P r o o cu a sla r N{ a q r u a J' z' 8

3.2. Fenchel-Nielsen Coordinates '?'8'ttJ

Fig. 3.4. 3.2. Fenchel-Nielsen Coordinates

59

sat"urProoc uaslarNlaqruad'z'8

60 60

3. 3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-NielsenCoordinates Coordinates

grve a precise precise definition Now,. these coordinates, and verify verify that that they they Now,. we give definition of of these give a system of global coordinates on T To. g• point [R, For this purpose, first E] of first fix a point of T ?r. mutually disjoint disjoint 4 of mutually g • A set £[.R,.D] geodesics on R simple closed closed geodesics .R is termed maximal matirnal if if there is no set £' 4' which properly. We call a maximal maximal set £.C = {Lj mutually disjoint disjoint includes £4 properly. {fi}j!,If=l of mutually geodesics on R simple closed syslem of closed geodesics rt a system of decomposing decomposing curves, curaes, and the family family pants of all connected components of of R -- Uf=lLj the pants p consisting of the UltLi P = {Pd~l {PxW' decomposition of R corresponding corresponding to £-. decompositionof L.

Example. Emmple. When When 9g = 2, there are the two kinds of pants decompositions shown in Fig. 3.5. 3.5.

Fig. F i g . 33.5. .5. system of decomposing decomposing an arbitrary arbitrary system Proposition 3.6. Let £If=l bebe an Proposition L = {Lj {li}l=, genas 9C (~ (>-2), letPP = {Pkl~l 2), and let curves curaes on a closed closed Riemann surface R of genus Riemann surface lPxlf=t

be pants decomposition and N satisfy satisfy decomposition of of R corresponding concsponding to £-. Then M and be the lhe pants t,. Then N ==3 3g g - -3 3

and a n d M ==2 2g g - -2 .2.

Proof. Cut of connected element Ly1 of £-. n1 be the number of r? along an elementL L. Let n1 Cut R genera components be the sum of genera of all connected components connected components components of R .R-- L .t1, 1, and gl 91 of R -- L we have have tr1. 1 . Then we

gl 9 r-- nn1r ==( (g g - -1 )1)- 1- . 1.

Clearly, components of R ,R-- L -t1 Clearly, the number of boundary components 1 is two. Moreover, we can whenever we we add a cut along along a new can see see inductively that, whenever Moreover, we element increasesby two, and the components increases element of £-, 4, the number of boundary components connected sum of genera genera of all connected connected components components minus the number of connected we have components by one. one. Hence, Hence, we have components decreases decrea^ses _ _2N 2 N and gM = = ((gg _- 11)) _- NN, , 3M a n d 0 -_ M =

I9

u3,{'z't seleuProoc uaslarN-Iaqf,

3.2. Fenchel-Nielsen Coordina.tes

61

o

which imply the assertion.

'uorlrasse aql ,(1dun qcqaa

suorlcur\{ q1Eua1 crsapoaD 'Z'Z't

3.2.2. Geodesic Length Functions Fix a point [R, 17] of Tg , and a system £ = {L j }f=l of decomposing curves on R. For every t in the Fricke space F g , we denote by [Rt,17t ] the point in T g corresponding to t. Then, we can determine uniquely a system £t = {Lj (t)}f=,l of decomposing curves on Rto Namely, take a marking-preserving homeomorphism ft : R ---+ R t (cf. Theorem 1.4). For every Lj in £, let Lj(t) be the unique closed geodesic in the free homotopy class of the closed curve ft (Lj) on Rt. It is not difficult to show that Lj(t) is simple, and that Lj(t) and Ljl(t) are mutually disjoint when i :f; i'· Hence, £t {Lj(tnf=l is a system of decomposing curves on Rto Let r t be the Fuchsian model of R t represented by t. Note that Lj(t) is the projection of the axis of an element of rt which covers ft(L j ) for every i. Now, for every tin Fg and every i, we denote the hyperbolic length l(Lj(t)) of Lj(t) simply by lj(t). We consider lj(t) as a function on Fg (or equivalently, on T g ) and call it the geodesic length function for Lj. Then Proposition 3.3 implies the following lemma.

uo 'flluelerrrnbe.ro) 6g uo uorlcunJ € se (1)f7 raprsuoc eM'(ib fq fldrurs Ol7 p qfual cqoqrad{q eq} elouep eiu'f l(ra,repue td ur 3 itre,re ro;'aro11 ((l)!l)l 'f, fra,ra q (!1)tg sreloc qcq/rt ,J Jo luatuala u€ Jo slx€ eq1 ;o uotlcafo,rd aq1 sr (1)f7 leq? eloN'1fq paluesardarrgrJo lepow u"rsqcr\{ eqt aq tJ ta1 .rar uo salrnc Sursodruocap;o uralsfs e sr t--;f{11;t1\ = rj'eoue11 ',1 + | uaqar lurofsrp .r(11en1nur ere (7),17 pue (3)f7 l€rl? pue 'eldurrs sr (1)f7 ?€rll aorls ot llnclgrp tou '? uo (!7)r1 a^rne pasole arlt q ss€p fdolouroq eerJaql ur crsapoe3pesolc lI Jo enbrun aql eq (l)fZ f"t 'J ul !7 {ra,re rog '(y'1 uraroaq; 'Jc) tU A : tt 'tgr uo selrnf, Sutsodurooap ursrqdrouroauroq Eur,rrasard-Eur4retue alel tr(1arue11 I=J{(t) l?} = ,J urelsfs e ,tlanbrun aunurelap uec er$'t".II '? o1 Eurpuodserroc Jo t; ur "U] ,tq alouep e$'6,tr a*ds aqcrq4eqt ul t fra,ra rog '3. uo aqt lurod lt3' '6J selrnc Eursodurocepf" tlf{fZ} 7 ure1s.{se pue lo [g'g,] lurod e xld

=

serTdur ueqy t7 rc1uot1cun! 6'9uorlrsodord vtfuq ?*?po;fflttftTopH iU

Lemma 3.7. Every geodesic length function lj (t) is real-analytic on Fg •

'6d uo cr1fi1ouo1oa.r, st (l)h uo4cunt y76ua7nsapoa| fi.taag 'Z'g BuruxaT sralaruerBd Eullqar;'g'Z'g

3.2.3. Twisting Parameters alouep pu" '(e'I o1 fre11oro3 eql 'z'f4 leql II€lsU g" f7 Eul\eq d

Next, for every i, let Pj,l and Pj,2 be two pairs of pants in P having Lj as a boundary component. Here we allow the case where Pj,l = Pj,2' Recall that Pj,l and Pj,2 admit the reflection J l and J 2, respectively, by the Corollary to Theorem 3.5. Take a fixed point of h, on Lj for each Pj,k (k = 1,2), and denote it by Cj,k. Fix also an orientation on Lj (see Fig. 3.6). As before, let [R t , 17t ] be the point of Tg corresponding to t for every t in Fg • For every t and i, let Pj, 1 (t) and Pj ,2 (t) be the connected components of Rt Uf=,lLj(t) (which are pants of Rt) corresponding to Pj,l and Pj,2, respectively. Recall that each Cj,k (k = 1,2) is the end point on Lj of the geodesic Dj,k joining Lj and another boundary component, say Lj,k' in Pj,k' Let Lj,k(t) be the boundary component of Pj,k(t) corresponding to Lj,k' Denote by Dj,k(t) the geodesic joining Lj(t) and Lj,k(t) in Pj,k(t) with minimal length, and by Cj,k(t) the point of Dj,k(t) on Lj(t). Then each Cj,k(t) (k = 1,2) is a fixed point of the reflection of Pj,k(t). Let 1j(t) be the oriented arc on Lj(t) from Cj,l(t) to Cj,2(t). Since Lj(t) has the natural orientation determined from that of L j , we can define the signed hyperbolic length Tj(t) of 1j(t) (so that Tj(t) is positive or negative according to whether the orientation of 1j(t) is compatible with that of Lj(t) or not). Set

1ag'(1ou rc (7)17 Jo 1eql qlrar alqrleduroc s1 (3)fuJo uorteluerro eql raqleqa ol Eurproace a.,rrleEauro a,rrlrsod sg (3).r.r.1eq1 os) (l)l,l,l" (l)f, qt3n"t cqoqred,tq pau3rs eql eusap ust e^l 'lI lo l"ql uro+ peururalep uorleluarro l€rnl"u eql seq (l)fZ acurg'(3)z'fc o1 (3)t'fa urory (3)f7 uo f,trepatuerro eqt aq (t)fap1 .(ic'ta Jo uorlcagar aqlgo lurod pexss sl (U'I = t) (1)t'!c qcee uaql'(l)f7 uo Ot't1;o lurod aq1 (ic'ft fq pue '{t3ua1 prurunu qtyn (1)r'12,ur (7)t'!7 pue (1)f7 Eururof crsapoa3 aq1 (1)r'fc, i(q alouaq 't'17 ol Eurpuodsarro) (i't'!d;o luauodruoc frepunoq eq1 eq (7)t'!7 p"l'1'14 u1 tr'!7 fes 'luauoduroc f.repunoq raqloue pue !7 Sururof r'f6. crsepoe3 aq1 yo f7 uo lurod pua a{} sl (Z,I - q) t'fc qcee }€ql IIeceU 'f1a,rr1cadsa.r 'z'!4 pue r'14 o1 Surpuodsa.uo?(tU;o slued a.re qcrq,u,)(l)fZt=Jn - ? Jo sluauoduroc palcauuo? aql aq (iz'fa pue (1)t'f4' 1a1'f, pue l fra,re rog ''d.rl 'a.ro;aq ';;o sy I {rara.rog 1o1 Eurpuodserroc lurod aql aq l,3'rA) 1a1 '(g'g'3t.{ aas) f7 uo uol}sluerro us osls t1 "g'r'fa,tq = {) r't, qcea rc1 !7 uo Y Jo lurod paxg " e{"I 'g'g ureroaqJ ^q 'flarrrlcadse.r 'zf pue r/ uorlcager aql lrurpe z'!4 pue t'14 - r'.r2'a.raq,r,r es€l aql ,&roll"e^r areH'lueuoduroc drepunoq e ul slued;o s.rred oiu.1aq z'!4 pue r'fd 1a1'/ fra,re roy 'lxaN

Tj (t) Bj(t) = 211" lj(t)' (t)!t-"

.l.r!t!"^ -= (7)!6

62 62

3. Hyperbolic 3. Hyperbolic Geometry and Fenchel-Nielsen Fenchel-Nielsen Coordinates

Dt,z

D;,r

(P.;,1: Pi,z)

Fig. F i g . 3.6. 3.6.

Then Bj(t) Bj(t) the twisting parameler parameter with 0i(f) is well-defined well-defined modulo 271". hr. We call caII01(t) respect respect to Lj. L1. Lemma j, exp(iBj(t)) Lemma 3.8. For every eueryj, exp(id1(t)) is well-defined well-definedand and real-analytic real-analylicon on F Fn. g• Proof j. For every Proof. Fix Ftx 1. every tf in F .Fr, Fuchsian group represented represented by t. fi t be the Fuchsian g , let r Take which covers Lj(t), and denote it by 'Yj(t). Take an element element of r covers tr;(l), denote each 4t 71(t). Next, for each 1,2),Iet be the element of which k&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&(= (= 1,2), let 'Yj,k(t) be the element of which covers Lj,k(t) and satisfies that covers L1,x(t) and satisfes that [ t 7i,x(t) geodesicDj,k(t), Di,n(t), connecting connecting Aj(t) the geodesic ,41(f) and and Aj,k(t) A1,*Q) with the minimal length, length, is is projected onto Dj,k(t), Di,*(t), where where,4i(t) and Aj,k(t) A1,r(t) are are the axes axesof 'Yj projected Aj (t) and (t) and and 'Yj,k(t), 7i(l) T,x(t), (see Fig. respectively respectively(see Fig. 3.7). 3.7). points of 'Yj(t), we may assume Here, we assumethat the fixed fixed points Here, and 'Yj,2(t) move 7i(t), 'Yj,l(t), ti,t(t), and 7i,2(t) move • Hence, real-analytically on F when we take a conjugation of r by an element Hence, when we take aconjugation ,Q an element g t d. \(t) > j ( r ) .. zz((Aj(t) o f Aut(H) s o that g o e sto points of A u t ( H ) so t h a t 'Yj(t) t o 'Yj(t)(z) 1 ) , the t h e fixed f i x e d points ) 1), 7 1 ( t ) goes f u ( t ) ( z ) = . \Aj(t) corresponding to 'Yj,k(t) move also of 'Yj,k(t) also real-analytically on F Fo each k. #. g for each 7i,1(t) move fu,*(t) corresponding projection of the end c1,1(t) is the projection Now, Cj,k(t) end point Cj,k(t) Zi,r(t) of "t Dj,k(t) Now, to Aj(t). A1Q). fu,x(t) Ifence, if we show Hence, if we show that Cj,k(t) 6i,x(t) moves moves real-analytically on F Fr, assertionfollows follows g , the assertion by the definition definition of Tj(t) 11(t) and and Lemma 3.7. 3.7. p1 and P2 p2 be show this, fix k, /c, and let PI To show be the fixed points of 'Yj,k(t). Set ii,i(l). Set (v* > 0). c13(t) = iYk iv* (Yk 0). Since Since Cj,k(t)

r

+ (PI - P2) = + P2 ) , =(PI '7* (ry)' (o'to')' 2 2' 2

Yk

we see we seereal-analyticity real-analyticity of Cj,k(t). ci,r(l).

2

2

o

';c) ursrqd.rouroe$lpe sl g_ogtl x ,_rg(ag)

-

.([tqz] pue ,h-yl 1.rad1o44 6l : 'dgenlcy 'IrDur?A 4i

----+

(R+?g-3

X

R 3g- 3 is a diffeomorphism (cf. Abikoff

.Uo{lqy

[A-l), and Wolpert [251]). Remark. Actually, .p : F g

'(sa,r.rncEursodurocep;o uralsfs eql qtyr\ 'to) d uotltsodtuocap slued aql 7 rl?-rrA paler)osse t;;o selouxprooxueslery-Ieq)uef ^seler;rprooc asaql II€r elA

We call these coordinates.p Fenchel-Nielsen coordinates ofTg associated with the pants decomposition P (or, with the system I:- of decomposing curves).

.6a uo uI 'e-oellX nuaq puD'6,I uo sa?Durproo? 7oqo16 lo ue7sfrro saat64i'.to7nc4.r,od 'OI'8 tuaroaql 6g to rusttld;oruoeuoq e s? 4 6utddnu, s?ttJ e-re(+U) oTuo

Theorem 3.10. This mapping .p is a homeomorphism of F g onto (R+)3 g-3 xR3g- 3 . In particular, .p gives a system of global coordinates on F g , and hence on T g . 'le,roelotr41 'uaroaql 3uralo1o; eq1 erro.rd,r.ou ilBqs aru

Moreover, we shall now prove the following theorem.

'uf uo ctTfipuo-pat st

is real-analytic on F g . ( ( l ) " - u " a ' . . . , ( t ) r0 , ( t ) e - u e T. . . , ( t ) r t ) = ( t )r t

ueqJ '! tuaaa ut 6g uo (7)lg nTatuo.tod 6ut7stm7aqyto qcun"tqsnonurluo? panlna-e16utsD ottr '6'g eurtrrarl

Lemma 3.9. Fix a single-valued continuous branch of the twisting parameter OJ(t) on F g for every j. Then

:3ur,r.lo11o; eql e^€r{ e,lr snqtr '(uraroaql ftuo.rpouour eql) t/ uo r{ou€rq snonurluoc panp,r,-e13urs € s€rl (3){6 frarra',{13urp.roccy'uleurop paltauuof, .{ldurrs (pueq raqlo aqt uO e s-rtJ l"q1 sale?s(91'g ura.roaql) ura.roaqls(rellnuqclatr

Here we set R+ {x E R \ x > O} and Sl {z E C \ \z\ l}. On the other hand, Teichmiiller's theorem (Theorem 5.15) states that Fg is a simply connected domain. Accordingly, every OJ(t) has a single-valued continuous branch on Fg (the monodromy theorem). Thus we have the following:

'{f = Itl I C > t} = rS pue tO < t lu>

t} - +lI 13seA\ereg

=

=

=

' (((r)t-rtOl)dxa'. . .' ((1)t6r)dxa'(t)e-aq . .'' (t)rt) = (t)'t

tJt(t) = (£l(t), .. ·£3g_3(t),exp(iB 1 (t)), ... ,exp(i03g _ 3(t))).

:e-re(rS) x e-oe(+tI) -

6tr :

e Paugepeleq ax\ '.re;o5 4i Surddeu cr1.,{1eue-1ear

So far, we have defined a real-analytic mapping tJt : F g

----+

(R+)3 g-3

X

(51 )3 g-3;

3.2.4. Fenchel-Nielsen Coordinates uaslalN-IaqcuoJ'v'z'

salBurPlooc

I

't'8'ttd

Fig. 3.7. Q|'r,

(t)z''g

(t1z''t

U7t'rt

63

sal"urprooc uaslarN-leqf,uaJ'z't

3.2. Fenchel-Nielsen Coordinates

t9

64 64

Ilyperbolic Geometry 3.3. Hyperbolic Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates

Prool. First, First, we we show show that thatitrr injective. Suppose Proof ~ isis injective. ~(t2) for Supposethat that ~(td fr(tr) ==rir(t) for some some t1 and t2 in Fo. Let be the Ra, the Riemann Riemann surface surface represented t 1 and t 2 in Fg- Let Rt. be representedby by tit; (with (with the the natural marking), marking), and and Pi pa^ntsdecomposition natural b" the th" pants decomposition of of R A == {Pk(i)}~~;;3 {P}(i)};s=;" be &,t • correspondingtoP, for each eachif (= (= 1,2). 1,2).BV corresponding to P, for By Theorem Theorem 3.5 3.5 and and the the assumption, assumption, there is is aa conformal conformal mapping, mapping, say g&, of say gk, of Pk(l) P7,(1)onto there onto Pk(2) P1(2) which which respects respectsthe the boundary correspondence correspondencefor for every every k. /c.Moreover, proof of boundary Moreover,the the proof of Theorem rheorem 3.5 implies B.bimplies that that dsl== (gk)*(ds~). (si4.@sl). dsi = 1,2) Here,ds; dsl (i (f = 1,2) is is the the hyperbolic hyperbolicmetric metric on on Rt pa.rticular,every Here, •. In particular, every gk is aa &,.In 9r is hyperbolic isometry isometry of of the the closure closureof of Pk(l) Pi(l) onto hyperbolic onto the the closure closureof of Pk(2). Pp(2). Since Since |j(tr) = ?i(tr),

j = 1,...,39- 3,

g1 can all gk glued together can be be glued together into into aa marking-preserving all ma^rking-preservinghomeomorphism, homeomorphism, say say h, h, of R rt1, onto R -R1r. is holomorphic on Since hlr is of on R r?1, except for for aa finite finite number of of t2 . Since t1 except t1 onto analytic curves, curves, so so is is hh on on the entire entire Rt .R1, analytic by Painleve's theorem. Hence, h is aa Painlev6's theorem. Hence, h is 1 biholo-morphicmapping of R -R1,, which implies implies that t1 biholomorphic t1 = t2. lz. Thus we we have proved have proved t1 , which P is is injective. injective. that ~ we show show that ~ f is is surjective. surjective. For purpose, we Next, we For this purpose, we begin begin with fixing aa g -3 x R 3 g-3 arbitrarily. For every .. . ,a3g-3,0'1,'" (4a1, (R+;sc-a point (a1,'" R3s-3 arbitra.rily.For every ,eas-s,e1t. . . ,0'3g-3) ,ass-z) of (R+)3 P3 decomposition P of R, denote denote by {L P (C .C) the boundary k,j}1=1 (C,C) k in the pants decomposition {Il,i}i=, components of P Pp. there is components 1.5, there is aa unique unique pair of pants, pants, say say P~, Pto,such such that $1.5, k . From § lengths of the boundary components the triple of the hyperbolic lengths components of P~ Pf is is equal equal pil pi,2 given triple {ak,j}1=1' to the given Set pi = {pk}~~~3. As before, let Pj,l and Pj,2 be and be {ou,i}i=r.SetPt= {P;}ir=lt.As before, let pj,xbe elements of P neighboring j, neighboring each each other along the elements Lj for every j, and let PIl be along .Li every element of P/ correspondingto Pj,l Pip (f (l = 1,2). the element pi corresponding be the point on I,2). Let cj,l Ci,2be oii the th" j Pjp corresponding correspondingto ci,t for every boundary of Pj,l to Cj,l and f. every and l.Now, by gluing Pj,1 suitablV along Pj,l and P/,2 Pj,2 suitably L j for along curves curves corresponding corresponding to ,Li every j, we obtain a Riemann surface, surface, say every j, we R . We need to choose a suitable say r?'. need choose ' gluing (and aa-suitable suitable marking of .R') R ' ) so R ' corresponds so that that R/ corresponds to a point tt,' of F u c htthat h a t f~(t') ( t / ) iiss e i v e n((a1' q u a ltto o tthe he g a 1 ,... . . . ,,aa3g-3, 1 s _ B 0'1,' , d l , . .. . . , ,a0'3g-3)' ss_a).T Fg, ssuch equal given This h i s ccan an be achieved gluing by Pr{,1 and Pj,2so achieved Pj,l Pj,2 so that that the twisting twisting parameter becomes becomes the given ai 0' j for every j. j. We shall explain this procedure procedure more rigorously by using Fuchsian Fuchsian models. models. pi,z, In the proof, rest of this we consider In of consider only only the case case where where Pj,l Pil t'"I Pj,2, for the other case case can be considered considered similarly. Fix Fix j,j, and let 4,r rj,k be a Fuchsian Fuchsian model of of the Nielsen Nielsen extension extension Pj,k of fj,t Pj,k P|,r "t for each each &. k. Here we assume assume that that every 4.,r rj,k acts on the upper half-ptanl half-plane fH,, aird and that that the transformation transformation

=

AZ, l,(Z) (z)=\2 ,

=

)A- e xexpaj p a i ) L> 1

belongs li,z, and, and 7 covers belongs to to both both 4,r r j ,l and rj,2, covers the the boundary boundary component, say Lj,k' of of say Ll1r, Pj,r Pj,k corresponding to.Li to Lj for each each & k (& (k == 1,2). We We also assume assume that that the the nilural natural orientation orientation of of the the axis axis ,4 A == {z {z €E Hl, HI Z -= 'iu,y iy, y )> 0} O} of of 7, corresponds corresponds to to the the prescribed prescribed orientation orientation of of .ti, Lj, and and that that the the point point ii €E /lH lies lies over cj,k with with respect respect over cl,x to to li,* Ij ,k for for each each /c. k.

99

sal"urProoc uaslarN-Iaqf,uad'z't

3.2. Fenchel-Nielsen Coordinates

65

luauale eql raplsuoopue'(r,6f lofo)dxa = fp 1eg

Set di = exp(aiG:i/27r), and consider the element 0 < lp

'ztp - (z)g

of Aut(H). Identify every z on the axis A of'Y considered as an element of rj ,2 with b(z) on the axis A of'Y considered as an element of ri,l' (See Fig. 3.8.) ('g'g'q.{ aag) 't'1Jo tuetuelerrc se pareprsuoo,LJoy srxe aq} uo (")g qft^ z'!tr p lueutueleue se pereprsuoc,LJo y srx€ eql uo z f.tete,tg11uap1'(n)WV lo |;e

'nJ

Fig. 3.8.

'8'8'EtJ

'eroJeq "'fa r'{" 3urn13 * pue l€tll q?ns ,U oluo A p rt tusqd.rouroauoqe xrJ fq paurclqoec"Jrnsuueuerg eql sq ,A 1eI"tr;o dlrrrrlcaf.rns;oyoord er{l Jod ('r 'dtq3 '[tz-f] tl{se4 acu€tsulroJeas'sruaroaql uorl€urquo) .rog'(saaeg.rns om1Surlaauuor.rog)uuoeql uorlDurquocs(uNelNse ulrou{ flpcrsselc q slqtr) '!/A Jo uorsuelxeueslarNeql Jo Iapou u€rsqcr\{ e sr I'lJr {q paleraueEdnorSuersqf,ndaq1'sprornraqlo uI'{ rl"€a roJ ,-gz'!J9, l ' pue ! , tt'!4 uo '?"p ,It!^ {.raaaEuole'}'I$ ul s (a,unaf.repunoqEurureura.r lueprcuror r'la p uorsuelxe uaslarN aqlJo uorlrnrlsuor eql ur pasnse^rqf,rq^\'ureuropEurr elq"lrns e 3urqce11e fq lnt urorJpeurelqoec"Jrnsuusuarg eql ''e'l) fr14ureurop ?ql uorsuelxeueslerNeql uo f,rrlerucqoqred,(q"ql'(H)?nV 3 9 acurg 3ur11nsar Jo 'fo r'fa uorJ fp3o1 olnpou ot lenba sr z'!p oI {fua1 uor}elsusrlarlt }erll qcns t'l,l z'!4 pue t'fa p 3urn13e e^eqe^{ueqJ to uorlecgrluapleql ,(q) 1z'!,7pue

Then we have a gluing of Pj,l and Pj,2 (by the identification of Lj,l and Lj,2) such that the translation length from Cj,l to cj,2 is equal to log di modulo ai' Since b E A ut( H), the hyperbolic metric on the Nielsen extension of the resulting domain Wi (i.e., the Riemann surface obtained from Wi by attaching a suitable ring domain, which was used in the construction of the Nielsen extension of P;,k in §1.4, along every remaining boundary curve) is coincident with dS~/,, on Pj,k 1,

for each k. In other words, the Fuchsian group generated by ri,l and bri,2b-1 is a Fuchsian model of the Nielsen extension of Wi' (This is classically known as Klein's combination theorem (for connecting two surfaces). For combination theorems, see for instance Maskit [A-71], Chap. 7.) For the proof of surjectivity of .j" let R' be the Riemann surface obtained by gluing Pj,l and Pj,2 as before. Fix a homeomorphism h of R onto R' such that 'Z-6?,'...'I = {

"la =(qa)rl

h(Pk)=P£,

k=1,···,2g-2.

66 66

3. 3. Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates

point ofT Then Then [R',h.(E)] determinesaa point be the point of of To.Let l'be the corresponding correspondingpoint g • Let t/ [rR',h-(t)] determines preceding .Fo. • F From the preceding construction, construction, it is is clear clear that g

=

=

, " ' ,3g l i ( t ' ) = aj, a t , ji = I1,··· fj(t') , 3 9-- 33,, and that and ).t

0 1 ( t ' 1 : ^ : I " e d 1 p o 1 ) ( m o d2 r ) , uj

j = t , . . ' , 3 9- 3 .

preseni, we we cannot (Note that, at present, cannot say say that OJ(t/) 01(t') = aj, ai, because because the choice choice of 0i is is not unique.) unique.) branch of OJ ------R' Hence,letting 7j Ti; : R' ---+ R'be the Dehn Dehnlwist Hence, be the twist with respect respectto Lj, trl , the the curve curve on.R'corresponding we can on R' corresponding to to.ti, Lj, we can find find integers integersn1, ... ,n3g-3 such that ftrt.'. such ,fl3s-3

[R :,(T i 'o...orti :i" o h) .( t) ] corresponds correspondsto aa point, say say t", ttt, which satisfies satisfies , i r 1 t ,= , 1( (a1' o r , .... . . , ,a3g-3, a s s _ s ,a1, o t r... . . . ,a3g-3). ,osc_s). !it(t") we have have shown that that j, fr is is surjective. Thus we Here, the Dehn twist 7j with Here, with respect respect to to.Lj Lj is, is, by definition, aa homeomorphism homeomorphism Q of R' onto R' ,? corresponding to the following following surgery: cut R' along along Lj L! and a.ndreglue rotation of 211" (see Fig. 3.9). 2zr (see after aa rotation 3.9). Note that applying T Ti, we can can make make the j , we value of OJ value di increase increase by 211" 2r while every every other OJ j) remains unchanged. 0i (i/ remains unchanged. U'+j)

t

o0 o twist F Fig. i g . 33.9. .9.

g-3 xX R3r-3 -------+ we have have proved that fr j, : F (R+)3 R 3g- 3 is bijective. By Now, we Fo (R+)sr-a g ---+ j, is also also continuous. continuous. On the other hand, Teichmiiller's theorem Lemma 3.9, 3.9, tit 6 6 (Theorem 5.15) homeomorphic to R g- • Hence Hence the following states that F 5.15) states Fo R6c-0. g is homeomorphic theorem, Brouwer's theorem on invariance theorem, invariance of of domains, domains, implies that thatV!it is actually n a homeomorphism. 0 homeomorphism.

aldurrs (anbrun) aql aq lV m 'raqtrr\{ 'IV tq tr alouap pue '17 s])asre]ur qcrq^r fr14ur crsapoaEpasolc aldurs e xrg 'z'f4, n lI nt'!4 - lr14 ps'f fre,ra rog as"r eq} ^{olls e,lr'ure3y) 'luauoduroc,trepunoq e se lI Eur,teq ('7"{d = I'fd teqf '7 o1 Surpuodserroc 'J > lI {rarra rog dJo slueuele eq1 a'f4, pue I'ld' fq elouep U Jo uorlrsodurocapslued aql aq d +aI'U uo selrnc Sursodurocapgo 7 uelsds e pue (6 f' snua3 ersJrns uu€ualu e xg 'uorlces snor,rerd eql q sY l) Ur Jo '21'9 uorlrs 'uorl€zrJlerue.red;o pur{ raqloue arrrS leqs aira. -odo.r4 ur 'ra1e1 'suorlrunJ {lEua1 crsapoa3o^\} qll^r sa}eurproof,ueslerN-leqrueg paxg q relaurered 3ur1sral1qcea Surcelde.rfq '6g go slurod aleredas suoll?unJ 'a.re11 {}8ua1 esor{^\ scrsapoa3 pesolc aldurrs 6 - d6 Jo }es e lcn.rlsuoc aa,r '(FOt] IIBAToSpue eleddag ';c) elqrssodurr q slql '.rala,no11'oJuo selsurprooc pqo13 a,rrEsuorlcuny qfual asoql( srrsapoa3 posolt aldurs g - 69 Jo les € 6?ilrererll ;t elqsrrsap lsoru eq plnoa lI 'areJrns eql eururalep sqfual crloqradfq asoq^\ k ?) 0 snuaS;o af,eJrnsuuetuarg pasol, e uo scrsapoe3pesop eldurrs;o 1ese Surpug;o uralqo.rdeql replsuor e,n 'uotlaas qql uI

In this section, we consider the problem of finding a set of simple closed geodesics on a closed Riemann surface of genus 9 (~ 2) whose hyperbolic lengths determine the surface. It would be most desirable if there was a set of 6g - 6 simple closed geodesics whose length functions give global coordinates on Tg- However, this is impossible (cf. Seppala and Sorvali [194]). Here, we construct a set of 9g - 9 simple closed geodesics whose length functions separate points of T g , by replacing each twisting parameter in fixed Fenchel-Nielsen coordinates with two geodesic length functions. Later, in Proposition 6.17, we shall give another kind of parametrization. As in the previous section, fix a Riemann surface R of genus 9 (~ 2) and a system .c of decomposing curves on R. Let P be the pants decomposition of R corresponding to .c. For every Lj E .c, denote by Pj,l and Pj,2 the elements of P having Lj as a boundary component. (Again, we allow the case that Pj,l = Pj,2.) For every j, set Wj = Pj ,l U L j U P j ,2. Fix a simple closed geodesic in Wj which intersects L j , and denote it by ,1J. Further, let ,1} be the (unique) simple

3.3. Fricke-Klein Embedding tu rp p e q tu g u la lx- a { r l4 4 ' g ' g

e. ')

corresponds to a continuous curve in T g • The variation of a Riemann surface represented by such a curve is called a Fenchel-Nielsen deformation. We shall investigate the deformation of this type in Chapter 8 by using quasiconformal mappings. We shall also give a direct proof of continuity of ~-1 with respect to

o1 lcadsar {1a r_4;o flmurluor;o;oo.rd }cerrp € arlr3 osle lleqs a1ys3urddeu IeuroJuocrsenb Sursn fq g reldeq3 ur adfl $rll Jo uorleruroJep eq1 ele3rlsaaut Upqs a1yuotTotu.totap u?slerN-Ieycuadre pelpr sr elrnf, " qf,ns ,tq pelueselder ac€Jrns uu"tuerg e Jo uorlerJel eql 'tJ ur eAJnf,snonulluot e o1 spuodsauoc ( { U > I | ( € - 6 s o t. . . r r * l n , J ' t - ! n , . . . , I r 2 , 8 - 6 t D. ,. . , I p ) } ) r _ 4

ljf

- -1

({(a1,··· ,a3g-3,0'1,··· ,O'j-1,e,0'i+1,··· ,0'3g-3) leER})

eqt yo fg e3eurre.rdeql '0I'g ueroeqtr dg 'sralaure.red3ur1sr,ra1 e u oe { € l p u e ' n - r s ? I x e - n e ( + u ) ; o ( e - E e o ' . . . ' r D ' s - 6 8 D' . ' . ' 1 o ) l u r o d e x r g (}j

Fix a point (a1, ... , a3g-3, 0'1, ... , 0'3g-3) of (R+)3 g-3 x R 3g- 3 , and take one of the twisting parameters. By Theorem 3.10, the preimage

'uollJasse aql a^eq arrr "{rerlrq.re ! sr f erurg'IO uo snonurluocsl r-d pue'O;o lurod rorJelur ue sr f '.re1nct1.red uI'IO qlr^r luaprf,uroceq plnoqs d dq g Jo g lul rorrelur eqf Jo (g' 1u1)d e3eur 'pueq retllo eq} uO -,lI aq1 'acue11'pelrauuoc s\ u^roqs eq u"? ll ler{} fI '/s' Jo uleluop rolrelxe eql pue ureruop rorJalur eql 'fla^llredsar 'zO pue IO {q alouaq 'slueuoduroc pelcauuoc o,r.l s€q ,S - .rll (uraloeql s(u€pJof leuorsueurp-, "ql fq 'acue11 'rll ul 'fle.rrrlcedser'ereqds pcrSolodol e pu€ IIeq pesol) 1ecr3o1odo1 e erc (gg)dt - rS pue,g're1ncr1redu1 '(g)dt = ,g oluo €r go ursrqdrouroeuoq € s.r g uo o1yo uorlculsar aq1 'lceduot sr Br aculs 'r reluar qll/rl - o pue'f1t.re.r1tqre pesop e f1tre.r1tq.re xg'(f)r-d las 5' ileq '[ZI-y] sreg 'y3) '(qc1aqs y) too.r4 O q n lurod e xIJ ('[98-y] ueu^\eN pue

Proof (A sketch). (Cf. Bers [A-12], and Newman [A-85].) Fix a point y in D arbitrarily, and set x = ep-1(y). Fix arbitrarily a closed ball B with center x. Since B is compact, the restriction of ep on B is a homeomorphism of B onto B' ep( B). In particular, B' and S' ep( BB) are a topological closed ball and a topological sphere, respectively, in R n . Hence, by the n-dimensional Jordan's theorem, R n - S' has two connected components. Denote by D 1 and D 2 , respectively, the interior domain and the exterior domain of S'. On the other hand, it can be shown that R n - B' is connected. Hence, the image ep(Int B) of the interior Int B of B by ep should be coincident with D 1 . In particular, y is an interior point of D, and ep -1 is continuous on D 1 . Since y is 0 arbitrary, we have the assertion.

=

=

'o oluo 'utotuop o s.t ("g)dt ueqJ ' iy olu! uE O ;g to tustrliltoutoeuoU o st d) puo 'onl uDlI ssq lou ta,a\ut. uo to uo4catut snonur?uo?p eg uE {_ ull : 6 7a7 eq u pI (sureurop Jo acuBrJBlur uo uraroaql s6rar*no.rg) 'tt'B uraroatlJ

Theorem 3.11. (Brouwer's theorem on invariance of domains) Let n be an integer not less than two. Let ep : R n ---+ R n be a continuous injection of R n into R n . Then D = ep(Rn ) is a domain, and ep is a homeomorphism of Rn onto D. turppaqurg uralx-a{f,rrJ't't

3.3. Fricke-Klein Embedding

67

L9

68 68

3. Coordinates 3. Hyperbolic HyperbolicGeometry Geometryand and Fenchel-Nielsen Fenchel-Nielsen Coordinates

geodesicwhich is closed closed geodesic is freely freely homotopic to the simple simple curve curve obtained from L1J 4! (seeFig. by applying L j (see 3.10). applying the Dehn twist with respect respectto to,ti Fig.3.10).

(Pi.r: Pi,z)

Fig. F i g . 3.10. 3.10.

?0. For every every For Fc,let b" the corresponding correspondingpoint of T For every every t E e F t , L't] be g • For g , let [R [it1,&] geodesicon geodesicL on R, we closed geodesic closed we express express as as L(t) I(t) the corresponding corresponding closed closed geodesic R f(L(t)) the hyperbolic length of L(t). I(t). Set Set denote by l(L(t)) ft1, t , and denote

fj(t) = t(Lj(t)), f(Lj(t)), tuo_"+i(t) f 3g - 3+j (t) = t(al(t)), f(L1J(t)), ftao-a+i(t) f(L1}(t)) ti(t) 6g - 6+j(t) = t(Aj(t)) j, and set for every set every j,

I 1 t 1= ( r ( t ) , . . . , / g o - s ( r ) ) . We the following: following: Wehave havethe -s. Theorem inlo (R+)9 (n+;seg-9. mapping 1 L is it a o proper embedding embeddingof F Fo Theorem 3.12. 3.L2. The The mapping g into

(That is, preimage of any and the the preimage any onto the the image irnage l(F t(F),g ), and is, 1 I is a homeomorphism homeomorphism onto g-9 under compact (R+)ee-s L is compact.) compact.) sel in (R+)9 under 1 compactset To prove prove this theorem, we fix a point to ts of F Fs theorem, first we g arbitrarily, and write

3 c -33 . .Tr( = (( a1,'" ( R + ; a9 s-3 -t x R R 3go r , . . . ,a3g-3,lll,'" f r 1tot )o 1 € (R+)3 '£' -_ X . , e 3 s - J , e r t . . . ,ll3g-3 , o e g - s) )E t(s) Fix j, and for every s) of F every Ss E define a point t( Fix j, € R, define d g by --1

t ( s ) = f r - ' ((a1,··· o r , " ' ,a3g-3,lll,'" t(s)=1Jf ,asc-e). , a ! - r t o i + s ' o . i + r , " ' ,ll3g-3). , a 3 s - s , Q r t " ' ,llj_1,llj+S,llj+1,'"

Then we we have have the following Proposition.

alsq era acueg 'f1a,rr1cadse.r ',? pue og ol pue 0z spues pue '? slt<eeq? qll,r{ crloq.redfq e s rLoeL luetuele ,z uorltsoduror aql uaqf 'slurod paxg se'f1a^rrrleadse.r'0rn pue oz er.ie-q qcrrl^it,olrr1 raPro Jo {H)?nV Jo slueruala arldqla eql eq zl pue Il, }al 'pueq reqto aql uO

On the other hand, let /1 and /2 be the elliptic elements of Aut(H) of order two which have Zo and wo, respectively, as fixed points. Then the composition /2°/1 is a hyperbolic element with the axis L, and sends Zo and Zl to io and i', respectively. Hence we have '(,V'

(' ') +pw,z. (' ") pz,Z _pz,w ( I ") <

, m ) d| ( , m ' , z ) d ) ( , 7 ' , z ) A

'(,n',?)d - (n'z)d 'suorlrugapeql ,(g teql r"elc s! ll prr€

By the definitions, p(z, w) = p(Z', w'), and it is clear that .II'8'IIJ

Fig. 3.11. "I m om

Zo, respectively, with respect to Wo (in the sense of the hyperbolic geometry). (See Fig. 3.11, where we replace H by the unit disk and assume that Wo = 0.)

aunss€ puts{srp }run aql fq p aceldar era eraq^a,11.g.3rg aag) ('0 = or leql '(rtrlatuoa3 cqoq.radfq erll ,f1e,rr1cadse.l ,02 Jo asues eql W) orn o1 lcadsar qlral pue',2'z o1 crrlatutufs slutod aQ 0z pue '?',V Ie.I '0o1pu€ oz r{SnorqtSurssedg uo crsapoe3aqt aq I Ie.I'F f z ueqll. dlqenbeur aq? ilroqs ol seclgns 11'loo-t4

Proof. It suffices to show the inequality when z f. Zl. Let L be the geodesic on H passing through Zo and woo Let i', i, and i o be points symmetric to z, Zl, and

'tm=ffLpuo - z fi1uopuD spyoyfrp1onba ay1t.taaoa.to141 lt fi F 'fi4ao4ceilsat'(acuoTstpcqoqtadf,tl ayl o7 Tcadsa"t p puo m to puo F puo z to sTutod-aypptut?Ul ^tD om pup oz ?reqn 'splo1

=

=

holds, where Zo and Wo are the middle-points of z and Zl and of wand w' (with respect to the hyperbolic distance), respectively. Moreover, the equality holds if and only if z Zl and w w'.

\lln)

p(ZI,W')+p(Z,w) 2

(m'z)d * (,m',z)d

P( Zo,Wo ) ~

) (on'o116

Lemma 3.14. Let two mutually disjoint geodesics L 1 and L 2 in H be given. Then p(z, w) is strictly convex on {(z, w) E c 2 I z E L1, wE L 2 }. Equivalently, for every z, Zl E L 1 and every w, w' E L 2 , the inequality

f r T q o n b a u?tU l ' z I ) , 0 1 , ' t fnr . t a a a puo rI ) ,z'z fi,.taae.tot 'frpualoatnbg'{"1> m'r7 sr (m,z)d uat17 ) z I zC) (n'z)} uo &?auo?fi17cVr1s 'uaa$ aq .7I.g BurrrraT H q zI puo rI eusepoa| yu.to[sppQpnTnut onl prl 'ur.ro; SurmolloJ eqt ur pe?"ls sr urn+ ur qcrqaa 'd ecu"lsrp ?Jecurod eq1;o {1xa,ruoc Eursn fq pa,rr.rapaq ileqs uorlrsodo.rd srq;

p, which in turn is stated in the following form.

This proposition shall be derived by using convexity of the Poincare distance

'urnunutur sp sulnllo { t1cryn 7o s '.r,o1nc4,toil uI '(E o7 taulilout o so) g uo ..tailo..til puo to aqoa anbtun o fl ?r?Vl uauor fi17cg"r1s q (((s)l)jill = G)t uotTcun! aatTtsoil?ttJ .tT.e uog11sodo.r4

Proposition 3.13. The positive function f(s) = t'(.~J(t(s))) is strictly convex and proper on R (as a mapping to R). In particular, there is a unique value of s at which f attains its minimum.

3.3. Fricke-Klein Embedding

tutppaqurg u-ralx-a{rlrd't't

69

70 70

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

p(zo,2s)<< p(z', 2'). 2p(zo,wo) ws)==p(zo,zo) p(z',Z'). 2p(zs, Thus we we obtain obtain Thus

w') ++ p(z,w). p(z',w') p(z,w). 2p(zo,wo) 1< p(r', 2p(ro,*o)

tro Proof of Proposition 3.13. We consider the the case case where where P1,r Pj,l *:f:. Pi,, Pj ,2 and and ^4! £1J interinterProposition3.19.Weconsider Proofof similarly. L at two points. The other cases can be treated similarly. sects be treated cines can points. other The sects ,Lij at two Let l's Fa be be the the Fuchsian Fuchsian group group represented represented by by ts, to, and and 7i 'Yj be be an an element element of of l-e Fa Let projected to an point which is zs a which covers Lj(to). On the axis A of 'Yj , fix a point Zo which is projected to an which covers Li(to). On the axis,4;j of 1, fix point, say p, p, of of Li(t6) Lj (to) and al(to).Let £1J(to). Let 6l 6J be be the the element of of l-o Fo which which intersection point, ar'd Bi covers £1J(to) whose axis passes passes thiough through zs, Zo, and B j be be the the ax-is axis of of 6f 6J. .By By Al(til and whose covers the assumption, assumption, the projection projection of of the geodesic geodesic .II contained contained in in Bi B j and connecting the Zo to to zto z~ = = 6l 6J(zo) L j (to) at at some some point point gq other than than p. p. Let z1 Zl be (zo) should intersect Li(to) zs lift of of qbn q on 1, I, a.nd,4j and Aj be the lift lift of of Ii(ts) Lj(to) passing passing through z1 Zl (see (see Fig.3.12). Fig. 3.12). the lift

A;

Fig. F i g .3.12. 3.12.

let z(a)L z(a)r R",let any aa E€ R, and any on H I/ and geodesicL-t on For oriented geodesic point Zz on on an an oriented any point For any bY along positive direction the be the point on L obtained by translating z in the positive direction along L.Lby be the point on tr obtained by translating z in

Eurdderu € eleq elvrueql ', o1 Surpuodsa.r.rortgr uo sselc fdoloruoq ea+ eql ul t, crsapoaSanbrun oql Jo (t2)l 'S g {raaa rog '1 dq peluaserder ec€Jrns r{l3ue1crloqrad{q aql aq (d!)-l I }el uueutrrarlleq? eq ,A tel'6,4:l l r(rar'aJod '(6 {) f snue3Jo Ar er"Jrns pesolc € uo selrn? pasolcelduns sassel?,fdolouoq ae.r;II€ Jo Surlsrsuoc1asa{} g ,(q alouaq Jo 'srrlolloJsp pegrpour q t1r;o Surppequraue '1srrg UI'g rueroaql ur sp euo qcns '[OZ-V] nreueod pue qcequepn"T 'lq1€d 'acuelsur 'punoy 's;oord 'uolsrnrlf ul ro; eq ueo qcrqrrr 1p lsourp lpro all 'A{ o? enp sr qclq^a '(e tg aceds rallnurrlrrel eq} roJ f.repunoq ?) d snua3 3o (te"pt) InJasne Jo uorl?nJlsuoc aql Jo auqlno ue aar8 llsr{s a^\ 'uorlcas slq} uI

In this section, we shall give an outline of the construction of a useful (ideal) boundary for the Teichmiiller space Tg of genus 9 (~ 2), which is due to W. Thurston. We omit almost all proofs, which can be found, for instance, in Fathi, Laudenbach and Poenaru [A-29]. First, an embedding of F g such one as in Theorem 3.12 is modified as follows. Denote by S the set consisting of all free homotopy classes of simple closed curves on a closed surface R of genus 9 (~ 2). For every t E Fg , let Rt be the Riemann surface represented by t. For every C E S, let £.(t)(C) be the hyperbolic length £( Ct ) of the unique geodesic Ct in the free homotopy class on R t corresponding to C. Then we have a mapping

uollBrullredruoc s6uolsrnqJ .7.s

3.4. Thurston's Compactification

'gl't tueroeql uroq s^rolloJuorlresss eq1 'drc.r1rqre u_f aourg ! 'a(+tI) olur U Jo Surppaquraradord e sr lr acueg 'snonurluo) pue 'aarlcafur 'redo.rd sr zll otur ?I Jo

ofR into R 2 is proper, injective, and continuous. Hence it is a proper embedding of R into (R+)2. Since j is arbitrary, the assertion follows from Theorem 3.10. 0

(f(s),f(s

+ 211"))

("2 + s)rr'(s)rr) --+ s S 1---+

Since f(s) is strictly convex and proper on R by Proposition 3.13, the mapping Surdderuaqt'tI't

uorysodor4 fq 11 uo.redo.rdpue xeluor.{11cr.r1s sr (s)/ eeurg

'@z+s)/ = (((16 +s)i[v)t = (((")r)fv)a £(L1](t(s)))

= £(L1J(t(s + 211"))) = f(s + 211").

e^€q e1'r'f, .,(.ra,ra ry (7)17 Euole 1srm1uqeq eqt Surfldde dq (l)jy uo.ry paurelqo e^rn, eql o1 crdolouroq flaery q (l){V &urs'AI'6 uruoeqJ lo !oo.r,4

Proof of Theorem 3.12. Since L1J(t) is freely homotopic to the curve obtained from L1J(t) by applying the Dehn twist along Lj(t) for every j, we have

'uorlress€ eql aPnlcuo) a^{ 'eroJeq peuorsuetusB tr (g)- = (s)rf acurg '11 uo .rado.rdpue xaruoc ,{11cr,r1s st (g)u, ?eql ^\oqs o1 f,sea sr 1r 'arourJeqlrng '1urod auo fllcexa te (g)u/ urnturunu eq? su.rep€ ll 'relncrlred uI'l,V y ly uo rado.rdpu" xaluo? rf11cr.l1s sl (g'rn'r),I'g paxg,{ue roJ snql 'g yo drepunoq aql o1 spuel ol ro z raqlra s" oo+ (rr"'z)d Wql tceJ aql ruorJ uaes.{lsea eq u€f, s€ 'rado.rd sr g 'os1y 'xeluoc ,t11crr1s$ J teql aas uec ar\{'tl't eurural ,{g

By Lemma 3.14, we can see that F is strictly convex. Also, F is proper, as can be easily seen from the fact that p( z, w) - - +00 as either z or w tends to the boundary of H. Thus for any fixed S, F(z, w, s) is strictly convex and proper on Aj x Aj. In particular, it attains the minimum m( s) at exactly one point. Furthermore, it is easy to show that m( s) is strictly convex and proper on R. Since f( s) = m( s) as mensioned before, we conclude the assertion. 0 .

( r r v r l e l s ; 1 2 ) j e, . ) a q ( , v 1 s - 1 m , 2 ) d- ( q , ^ ' r ) , 4

F(z, w, s)

= p(z, w( -S)A) + P ( w, 6J(z)(S)oj(Aj)) .

lVuo (s'rn'z)g uorlcun; ? eugap alr 'acue11

Hence, we define a function F(z, w, s) on A j x Aj x R by setting 3ur11as,tq U x |V *

J 1

J

J

.

= ZEAj,WEAj inf {p(z,w)+p(w(s)A

,6J(z)(S)OO(A-))}.

o ,,y]i"iu't"' { ( t r r r f r l s ) 1 2 7 [ s , i v ( s 1 n* )@

= (r)! f(s)

we can see that

leql

aas ue? a/r{

2~ £(Lj(to))s = S,

'g = s((07)t7)7!3 " r hyperbolic length a. Since t( s) represents the marked surface obtained from the surface represented by to by twisting along L j (to) by hyperbolic length

,(q 0? fq palueserder e?eJrns {fua1 crloqladfq fq (01)f7 3uo1e3ur1sr,r,r1 eql ruo.t; peur€lqo er€Jrns pe{reur aq1 sluasardar (s)1 acurs 'la q13ua1cqoqraddq 71

3.4. Thurston's Compaetification

uorleryrpeduroC s.uolsrnqJ'?'t

TL

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

72 72

/.(1):S-R*

for every every tt eE Fs, F g , or or equivalently, equivalently, we we have have aa mapping mapping for

i. :Fg~(R+t Fo -'--* (R+)t [: of ,F'o F g into into (R+)s. (R+)s. Ilence Hence Theorem Theorem 3.12 3.12 implies implies the the following: following: of

Corollary. The mapping mapping I*: i. : Fo Fg -~ (R+)s (R+t is aa proper proper embedding. embedding. Corollary. Moreover, this this mapping mapping remains remains an embedding' embedding, even when we take the the quoMoreover, tient space space P(n+;s P(R+t == (R+)s/R+ (R+)S /R+ as the target space. space. In In fact, letting letting Pl* Pi. be tient composed mapping mapping of of /* i. with with the projection projection n7f of of (R+)t (R+)S into into the projective the composed space P(R+)s, P(R+)S, we have have the following theorem. space

Theorem 3.15. 3.15. (cf. [A-29], [A-29], Expos6 Expose 7) The The mapping Pi. : Fo Fg -~ P(R+)s P(R+t is IS mapping Pl, Theorem an ernbedding. embedding. an natural mapping of of 5S into into (F;s, (R+t, which is defined defined by Next, there is another natural of curves. curves. Here Here and in the remainder using the geometric intersection numbers of of this section, R+ = {x x~ O}. section, we set [T € R II e 2 0}. {o E f(o1,42) For any two a1 geometric intersection a2) of interseclion number i(al, o2 E o1 and a2 € S, the geometric intersection points a1 infimum of the number of intersection o2 is, by definition, the infimum a1 and a2 each j.j. In cla^ssof of aj ai for each of .L1 L 1 and tr2, L 2 , where ,Li L j moves homotopy class moves in the free homotopy every a o E f(o'o) a) = 0 for every particular, € S. 4lro, note note that i(a, i(oz,ar). Also, particular, i(a1,a2) i(ay,a2) = i(a2,aI). -* (R Define S ~ by setting bV (F)"+ )S -- {O} i* ::.9 Define a mapping i. {0}

i.(a)(.) i * ( o X . )= - i (i(a, o , . .), ),

ao €ESS, ,

(cf. [A-29], Expos6 3) 3) that where we can can show show (cf. 0. Then we we denote where we denote {O}S [A-29], Expose {0}s simply by O. - zro i* :S~P(R+t Pi. ;.9 -----*P(F)s Pi* =7foi.

is " an injection. bv into (R+)S (R+)t -- {O} Here, note that to of R+ R+ xx S5 into to aa mapping mappingof is extended extended that i. i* is i:3:""T: {0} by setting setting - aa . ii(a,') (a,-) i.(a, i - ( c ,a)(·) a ) ( . )= for is clear clear that It is (o, a) a) E every (a, for every € R+ x S. It

PLIs). x S) - {O}) " (i.(R+ (am;O{o})==Pi.(S).

7f

Now, following: weknow the following: know the Now,we is homehomeo/P(R+)s Theorem Pi.(S) P(R+)S is Fr](S of Thesubset subset 4) The (cf. [A-29], Expos64) 3.16. (cf. Theorem 3.16. [A-29],Expose 6g - 7 =- {x 6 = omorphic to 5 E R g-6 Ilxl = 1}. l}. R6s-6 b omorphic SGs-7 {s € | lol written herehere(andisis written with (and Remark. be identified identifiedwith canbe I.IFD x S) -- {O} The set seti.(R+ Remark.The {0} can precisely, (or more more precisely, after classes(or equivalenceclasses of all all Whitehead Whitehead equivalence as) the the set set MF after as) Mf of

TL

uorleogrlaeduroC s.uolsrnqJ't't

3.4. Thurston's Compactification

73

se (S)-la ssardxaoE" uec e^{ ecueg ('g gsodxg'[OZ-V]aes)'U uo suorlsrloJ parnseatu;o (suorleredos,peaqalrq1\pu€ fdolosr repun sesselcecuep,rrnba equivalence classes under isotopy and Whitehead's operations) of measured foliations on R. (See [A-29], Expose 6.) Hence we can also express Pi.(S) as P M:F 1r(M:F).

=

'Gw)" -- Jwd

'g to1 rl arn6?er.ul"sraAsrr"rl e pue 'sarlrreln3urs pelslo$ q?ns qll^ar uo uorl€rloJ€ .rrede 'uorlrugap ,fq 'q (r/'d) uo4otlol p?Jnsoeu y U dr Jo '(!)d = (n)r1 uaql ,4 yo 'ddolosr ue fq ;ea1 a13urs€ ur paurcluoc sr qcrqar Jo llqro qtea ,/ euo Jerllou€ ol '1eql sauslles qclq/'\ pe^oru aq u3? u dr Jo sJ€al h'0] , la cJ? lesre^susrl e;1 Jo scJ"qns lesrelsuerl II€ Jo las aql uo arnsseru e sl 3, .ro; r/ e.rnseatulesJelsu€r? e '1xa11 '0 = z .{lrreln3urs eq} r"eu '1 re3elur a,rrlrsod etuos q}ua errllrsod sr '{O} C uo uorlelloJ e Jo teq} s? eJn}cnJ}s zzp,tz q)qM Jo Jeel ,f.razra3uo1e elqellueraJlp l€ool atu€s eql e eq ppoqs qolq^rJo qcea 'sarlr.repSurs pelelosr qlr^r '(g '[OZ-V] ';c) g gsodxg eceyrns lceduroc e uo uorlerloJ A' uo uo.rlsgoJe eq d la1 'ereH 'e?uarue^uoc Perns"au s Jo uorlluuep aql lle?er e^\ Jo e{es eql roJ

Here, for the sake of convenience, we recall the definition of a measured foliation on a compact surface R (cf. [A-29], Expose 5). Let F be a foliation on R with isolated singularities, each of which should have the same local differentiable structure as that of a foliation on C - {O}, along every leaf of which z k dz 2 is positive with some positive integer k, near the singularity z = O. Next, a transversal measure J-L for F is a measure on the set of all transversal subarcs of leafs of F which satisfies that, if a transversal arc 0' : [0,1] --+ R can be moved to another one (3 by an isotopy, each orbit of which is contained in a single leaf of F, then

J-L(O') = J-L({3).

A measured foliation (F, J-L) is, by definition, a pair of a foliation F on R with such isolated singularities, and a transversal measure J-L for F.

uollelloJperns€eru " qlr^irpeglueprq (s 3 p,+lr f p) (,,,r;-, .rffi.t:;y;f]

Remark 1. When i.(a, 0') (a E R+, 0' E S) is identified with a measured foliation (F, J-L), we have J-L[{3] = i.(a, 0')({3), (3 E S,

'g>d

'@)("'D).!-ldlrl

' lW ) (rt',i) = d leraua3 e rog ua,ra (g ) d) @)(d)., se uatlrra{ s\ [d]d 'uo ereq uroq{ '(7)r/ d)'Ilur = ldlrl areq^r

where J-L[{3] = infLE,B J-L(L). From here on, J-L[{3] is written as i.(p)({3) ({3 E S) even for a general p = (F, J-L) E M:F. arns?erul"sralsu"rl lecruouec aql qlpt paddrnba '(7 raldeq3Jo U$ 'Jc) e,rrlrsod fl U uo d prlua.legrp crle.rpenb crqd.rouroloq orez-uou paqtrasard e qrrqra Jo J"?l f.raaa Euop 'uorleqo; e $ 'arnlcnrls xalduroc e qty( paddtnbe fl U uaq^{ 'g uo uorlerloJ parns?eu e yo aldurexa pcrdr(1 v 'A IJoueA

Remark 2. A typical example of a measured foliation on R, when R is equipped with a complex structure, is a foliation, along every leaf of which a prescribed non-zero holomorphic quadratic differential cp on R is positive (cf. §2 of Chapter 4), equipped with the canonical transversal measure

IIm0?l·

'lfutv4l - rt J-L =

'g-rsll o1 atqdlouroeuoq eq ol uaou{ $ urnl uI q?Iq^,r '(e.rn1cnr1s xalduroc ua,rrE aq1 o1 lcedsar q?l,rl) U uo sl€rlueraJrp crle.rpenb erqd -rourolorlJo aceds eq1 ql!/{ pagrluepr aq rrcc uo arnlcn.rls xalduroc {0}nJW'g ua,rr3 fue .rog'pq1 pe^roqs [191] lnsery pus preqqnH 'pueq reqlo eqt uO

On the other hand, Hubbard and Masur [101] showed that, for any given complex structure on R, M:F U {O} can be identified with the space of holomorphic quadratic differentials on R (with respect to the given complex structure), which in turn is known to be homeomorphic to R 6g- 6 . ul lW

u-ro.{7arctnp q (6[).1a6our

Proposition 3.17. The image £.(Fg ) is disjoint from M:F in (R+)s. '"(tg)

ay1 'zr'g

uor+rsodor4

'uorlJesse aq1 saqdurr 0 qarqn 'r > (l)rt leql qcns ? a rnc pasolo alduns € eleq e^\ acuaH ',| 1e spua pue q}-I/'astlsls qcrq^r,Jr Jo J3el 3 Jo cr€qns e sr areql 'uraroaql ecuerrncer s (L)rl ryq1 qcns I cr€ l"srelsrrerl e a{el'0 < I frane.ro;'t:eIul'JW >(rl',1)-dfra.rlaroJ0otBurEra,ruocecuanbasesureluoc{S>ol["]rl]]esaql 'uraroaql acuerrncers(gr"curod Sursn ,(q (pu€q Jaqlo eq} uO +eql aoqs u?c e^{ (3 uo Sutpuedap) luelsuoc e,ttlrsod e sanlel aql 't4 3 3 {ra,re rcg.too.r4

fq rrolaq uorJ pepunoq ars (S 3 o) (o)(l)-/

Proof For every t E Fg , the values £.(t)(O') (0' E S) are bounded from below by a positive constant (depending on t). On the other hand, by using Poincare's recurrence theorem, we can show that the set {Jl[O'] 10' E S} contains a sequence converging to 0 for every p = (F, J-L) E M:F. In fact, for every £ > 0, take a transversal arc 'Y such that J-L('Y) < £. By Poincare's recurrence theorem, there is a subarc of a leaf ofF which starts with and ends at 'Y. Hence we have a simple closed curve L such that J-L(L) < £, which 0 implies the assertion.

74 74

3. Hyperbolic Hyperbolic Geometry Geometry and and Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordinates 3.

Now, the the crucial crucial fact fact to to construct construct Thurston's Thurston's compactification compactification is is the the following following Now, theorem. theorem. Theorem 3.18. 3.18. (cf. (cf. [A-29], [A-29], Exposd Expose 8) 8) For For eaery every system system LI:- == {Lj}?s-13 {Lj}J;~3 of of d'edeTheorem composing curues curves of of R, R, there there is is aa nalural natural homeomorphism homeomorphism composing qx : l*(F.) -.

U(L) C Mf

,

where where

= =

R + ) " ) lIpJL(Lj) U(I:-) MF(c F ( c ((R+)8) U ( L ) = {{p ( L i ) >>0 0 p = ( F(F,JL) , p , ) eE M

ffor o r eevery v e r y LL ij eE LI:-}. }.

The construction of of qa qt:. is as follows. follows. Here and in the remainder of of this disThe same notation notation both both for a cutve curve and for the free free homotopy cussion, we use use the same cussion, class of of it. it. For every t €E Fs, Fg , we can construct a measured measured foliation Pt = (fi,p'r1 (Ft , JLt) class that F1 F t is transversal transversal to Li Lj for every j,j, and that that such that

=

l,(t)(Lj)=i.(p1)(L1), i = 1,"''3s -3,

or equivalently, equivalently,

=

=

3 9 --3 3, , , " ' ,,3g l(Lj(t)) l ( L j ( t ) ) = JLtlLj], t t r l L i l , ij = 11,··· represented by t, surface represented Lj(t) is the geodesic, where geodesic, on the marked Riemann surface where,Ll(f) which corresponds Lj as before. We set as before. We set corresponds to Li q2(1.(t)) = Pr' onLo U(I:-). U(L). ga is actually aa homeomorphism homeomorphism onto It that this mapping qt:. It is is known that "projections" qL, we Using following: *u can can derive derive the following: these natural "projections" qt:., Using these

Theorem subsel The subset (cf. [A-29], Expos6 8) 8) The Theorem 3.19. (cf. [A-29], Expose Pt.(Fs)U PMT boundary. of P(R +)8 with the wilh boundary. compacl manifold with topology is is aa compact relatiue topology the relative o/P(FF)s (69-l)-dimensional the real Moreover, Pl. (F )uP MF is homeomorphic to the real (6g 6)-dimensional to is homeomorphic Moreoaer,Pt.(Fs)UPMf g 6 6 PMf wilh PMF is coincident coinciilent with closed gIlxl the boundary boundaryis l}, and and the Rog-o closedball ball {x € R S I}, I l"l ~ {c E 6g 7 (which is homeomorphic to 5 ). (which is homeomorphicto Soc-7). an of an neighborhood of We in aa neighborhood local coordinates coordinates in construct local how to to construct shall show show how we shall arbitrary point of the boundary PMF. . PMf point of the boundary arbitrary 1 = is aa system system I:I = there is Fix MF and (po) arbitrarily. p[' E ps E arbitrarily. Then Then there "-t(p[.) and Po PMF Fix Po e 1re P {lfij|5" L j } J;~ 3 of decomposing curves of R such that that of ,R such of decomposing curves

i.(po)(Lj f . ( p s ) ( . L)t )>0, > 0,

j=I,···,3g-3. i = I, "' ,39 - 3.

"t (See this 1:-, family {L we construct constructaafamily 4, we Fromthis (See[A-29], Expos66.) 6.) From j , L1J, L1J} J;~3 of {Li,A?,A}}}n=1" [A-29],Expose g as in (n+;s0-e simple closed curves which gives an embedding of F into (R+)9 -9 as in §3.3. Fo into gives an embeddingof which curves simple closed $3.3. g For set For every every fe > 0, set ) 0,

saloN

Notes

75

= {x E l.(F I x(Lj) >

, t < ( f 7 ) a l ( u , t ) . 1 fc } = ( r , 7 ) 1 g)

f,

= 1""

'{g-fg, ...,1= !

U(.c.,f)

j

,3g - 3},

'f1rea13 osl€uec eM'.1A= (Q'7)n)tb o I pue '(u,I)*ld ur uados\ Q'J)t)a '(0)d" - rA pue(("1)n)" = Q'7)nd

PU(.c., f) = 7r(U(.c., f)) and W = 7r(U(.c.)). Clearly, PU(.c.,f) is open in Pl. (Fg ), and 7roqJ;(U(.c.,f)) = W. We can also show that W U PU(.c., f) is open in the closure of Pl. (Fg ). Recalling that 7r is injective on l.(Fg ), we define a mapping t.p of W U PU(.c.,f) into W x [0,1) as follows: We set t.p(p) = (p,O), PEW.

$srr, l"ql 3ul1ecag '(ol)Va Jo ernsop eql ur uado sr (r'l)fla

n1t4 lerlt raoqs

se(t'ol x l4 olur(r'f,)2a n /A lool Eurddeur e eugep "'1urf'"j;f""tfi:i[i 'M)q

'(o'd)=(4)o1

'(t'7)n 1nd e,rll,

3 c ,(rarra .rog

For every x E U(.c., f), we put ((a)t)d

t.p( 7r( x))

3g-3

f; {i.(qJ;(x))(Lj ) + i.(q.c(x))(..1J) + i.(qJ;(X))(..1J)}) ).

'(4'n",)= (ivx(')'b)'t + (D(@)qb)*r) ((rtl"lrr 47b)*t+ 3 -) o*" = (7r 0 qJ;(x),exp ( -

l€q} qcns y ?uelsuo) e sr eraql 'S g ,c ,(rale ro; '1eq1 sa1e1sqcrqu, '.{lqenbaur pluaur€punJ e Sursn fq 'rn.{I

Then, by using a fundamental inequality, which states that, for every a E S, there is a constant A such that ~

i.(qJ;(x))(a) + A,

'H+(o)((c)7b).!)

(")r > (o)((c)20).r

x(a)

x E u(.c., f),

a

~

'(t'7)p)

i.(qJ;(x))(a)

:3ur,uo1o3aql ,(q u^roqs aq feur uorlecgrlcedruoc $ql yo ecuelrodurl eql '6a p fi"topunoqspolsrn?J pell€c s\ ,(repunoq slr pue '6a 1o uorToc{9ycod lWd -ntoo s.uo?srnqJ peIV) sr tg aceds rallnuqclel erll Jo uorlecgrlceduroe srq; 'od Jo pooqroqq3rau e uI seleurproor l€col se,u3 F ecuaq pue 'a3eurr slr oluo ursrqdrotuoauoq € sr d Surddeur slql leql a\oqs ue) e^r

we can show that this mapping t.p is a homeomorphism onto its image, and hence it gives local coordinates in a neighborhood of Po.

This compactification of the Teichmiiller space T g is called Thurston's compactijication of Tg , and its boundary P M:F is called Thurston's boundary of Tg . The importance of this compactification may be shown by the following:

'l1asp oTuo D saonp n(6g).ru = (i)*ld IWd lo u.rsnld.tou.toau,oq .f.tu11otog t1asyzopo A lo utstrltLtnuoeuoq fun.taserd-uorlDlueuo fi^teag

Corollary. Every orientation-preserving homeomorphism of R onto itself in-ut

duces a homeomorphism of Pl.(Fg ) = Pl. (Fg ) U P M:F onto itself.

'r-o1 tuor; peurclqo sl @Jo asre^ul aqJ tr 'Surddeur-;1es flrll repun ?uerrelur f1.rea1c arc IWd pun (61)*ld'a.reg '"(+U)2, ;o p Surdderu-Jlassnonurluo? s sacnpur ef,uaqpue J1as1rotuo StJo uorlce I€rnleu e sa?npur Jleslr o?uo A Jo d ursrqd.rouroatuoqSurrrraserd-uorleluarro fra,rg '{oo.t4

Proof. Every orientation-preserving homeomorphism t.p of R onto itself induces a natural action of S onto itself, and hence induces a continuous self-mapping tJj of P(R+)8. Here, Pl. (Fg ) and P M:F are clearly invariant under this self-mapping. The inverse of tJj is obtained from t.p-l. D

saloN

Notes ad{1 ;o ef,€Jrns uu"rualg e ;o areds rellnuqcrel aq? l€q} ^{ou{ a^it ,.re1ncr1.red ut ('[t-V] .Uo{lqv 'acuelsur ro; 'aag) '(g)g aceds rellnuqcratr eql yo s3urppequra s(ulaly-a{?rrd pue sa?surprool uaslarNlaqcuad eqt 'g$ pu" U$ ul suorlrugep ol dl.repurrs 'augap uer ear 'arlleur cqoq.radfq aql slFupe U ereJrns e q?ns Jl '(u'0) adfiy aTru{ fr11ocryfr1ouo uI'(ur'u'0) ailfiy Jo eq o? pres sr (0'u'6) edfl;o aceJrnsuueruerg e'.re1ncr1.red 'srlsrp pesolc u, pu€ slurod u lo acoltns uuourerq e lurolsrp ,(1en1nur3ur1a1apfq f snua3 Jo ar€Jrns uueruerg pasolc s ruoq peur€lqo 'ace;lns uu€tuerll e IIel e \

We call a Riemann surface, obtained from a closed Riemann surface of genus g by deleting mutually disjoint n points and m closed disks, a Riemann surface of type (g, n, m). In particular, a Riemann surface of type (g, n, 0) is said to be of

analytically jinite type (g, n). If such a surface R admits the hyperbolic metric, we can define, similarly to definitions in §2 and §3, the Fenchel-Nielsen coordinates and Fricke-Klein's embeddings of the Teichmiiller space T(R). (See, for instance, Abikoff [A-I].) In particular, we know that the Teichmiiller space of a Riemann surface of type

76 76

3. Hyperbolic Hyperbolic Geometry Geometry and a.nd Fenchel-Nielsen Fenchel-Nielsen Coordinates Coordina.tes 3.

(g, n, m) is is homeomorphic homeomorphic to to g6s-6*2n*3m. R6g-6+2n+3m. Abo, Also, Thurston's Thurston's compactification compactification (g,n,^) for such a surface (cf. Fathi, Fathi, Laudenbach Laudenbach and and Po6naru Poenaru [A-29]). [A-29]). is considered for to $3, §3, we recall recall the the inverse problem, problem, of of whether whether we can determine determine Relating to Relating the length spectra, i.e., the the set of of the hyperbolic hyperbolic closed Riemann surface surface by by the a closed lengths, of of all all simple simple closed geodesics geodesics on the the surface. This This problem problem is equivalent equivalent lengths, to the the problem problem of of M. M. Gel'fand: Gel1and: whether whether a closed surface is determined determined by by the the to eigenvalue spectra, i.e., the set of of all eigenvalues, eigenvalues, of of the Laplace-Beltrami opereigenvalue ator on the surface surface with with respect respect to to the hyperbolic metric. On On this problem, see see ator and,Wolpert [A-lID]' McKean McKean [152], [152], Sunada [218], [218], Vign6ras Vigneras 12421, [242], and Wolpert [246]. [246]. Venkov [A-110], embeddings, we further further refer to to Keen [110] [110] and Okumura Okumura For Fricke-Klein's embeddings, and Pc€naru [173]. The in §3 that Laudenbach Poenaru [A-29], that Fathi, Laudenbach in follows in The argument [A-29], $3 [173]. Expose 7, which which is due to to A. A. Douady. As for for more advanced investigations investigations on Expos6 of geodesic geodesic length functions, see see Kerckhotr Kerckhoff [112] [112] and Wolpert Wolpert [256]. [256]. convexity of A survey on Thurston's Thurston's compactification compactification by Thurston Thurston himself himself is given in in A Thurston [234]. [234]. Fathi, Laudenbach Laudenbach and Po6naru Poenaru [A-29] [A-29] is a good introduction introduction to to Thurston foliations and Thurston's Thurston's bounda,ry. boundary. See See also Gardiner Gardiner [A-34] [A-34] Chapter Chapter measured foliations 11, [101], Marden Strebel [137], Marden and Strebel and Masur Hubbard 11, Strebel Strebel [A-102]' [137], and [101], [A-102], Masur [144]. [144]. generalization of of Fenchel-Nielsen deformations, deformations, the earthquake deforAs a generalization have been been considered. considered. See Thurston [233]. See Kerckhoff [112] mations have [233]. [112] and [114], [114], and Thurston also cite Bonahon [45]. We also [a5]. Finally, Teichmiiller's theorem stated in this chapter. Finally, there are many proofs of Teichmiiller's are also proofs by For example, see that chapter. There are see Chapter Chapter 5 and Notes of that tomba [74] using geodesic length functions, and by Fischer and Tromba Wolpert [256] geodesic Fischer functions, [74] [256] from viewpoint. differential geometric viewpoint. from a differential

? raldBrlc

Chapter 4

sturddetr I ler.rrroJuorrsunb

Quasiconformal Mappings

'sJoord 'qooq 1noq1t^/l, lxal PJsPU€lsuI PunoJ eq lil/'l qcrqrn p.r3alur anEsaqal ;o froaql aql ul $uaroaql plueurepunJ lereles esn e \ 'sJaB '-I pu" sJoJIqy 'rI ol anp 's3urddeur roy elnuroJ ltsuroJuoersenb 'g uorlcas uI 'uorl"nba Ieuoll"rrel l"lueurepunJ e arrr3 aal lerlua.ragrp rurerllag erlt Jo uorlnlos aql Jo ruaroeql ecuel$xe eql errordean'7 uorlcag uI 'suor?ruUepaql ruo.r; fpsea Suurrollogsll€J crs?q elels pue 'suorlrugap esorll Jo ecuele,rrnba aloqs 'sSutddeur '1 leuroJuocrs?nb p.raua3Jo suor?ruUapIeJaAasalr 3 ea,r uorlceg u1 'sur"urop .reueld uaearlaq s3urddeur 'raldeqc qq} ul 'f13urp.roccy IeruroJuocrsenb;o asea ar{t ol sellesrno lcrrlser am 'sur€ruop .reueld uee/\{}aq esoql Jo esec eq} ol sareJrns uustuerg uaarralaqs3urd -deur go esrc aql ecnpar uec e^\ 'uraroaql uorl"zrurroJrun eql fq '1eq1 elop 'aser alqerluere$p eql ur se erues aql sureruar s3urddeur qcns Jo lueul€arl 'suorl"nlrs I€tuJoJ ayqal IeJeueE arou flqe.raprsuoc w Iool e sr sEurddetu Ietu 'uorlruUep eql ul -roguocrsenb asn ol sn sluolle luauralordurr Iscruqcel " qcns uorlrpuos ,(lrpqerluaragrp eql ua{ee^r e,n 'era11 's3urddeu leruroJuocrstsnba1qer1 -ueJeJIp paugap a^eq ell.r'1 raldeq3 u1 's.raldeqcJa1eIaql ur pepaeu arc q)Iq^r sEurddeu leuroJuocrsenb;o sarl.radordcrseq ureldxa II€qs e^{ '.ra1deqcslq} uI

In this chapter, we shall explain basic properties of quasiconformal mappings which are needed in the later chapters. In Chapter 1, we have defined differentiable quasiconformal mappings. Here, we weaken the differentiability condition in the definition. Such a technical improvement allows us to use quasiconformal mappings as a tool in considerably more general situations, while formal treatment of such mappings remains the same as in the differentiable case. Note that, by the uniformization theorem, we can reduce the case of mappings between Riemann surfaces to the case of those between planar domains. Accordingly, in this chapter, we restrict ourselves to the case of quasiconformal mappings between planar domains. In Section 1, we give several definitions of general quasiconformal mappings, show equivalence of those definitions, and state basic facts following easily from the definitions. In Section 2, we prove the existence theorem of the solution of the Beltrami differential equation. In Section 3, we give a fundamental variational formula for quasiconformal mappings, due to L. Ahlfors and L. Bers. We use several fundamental theorems in the theory of Lebesgue integral without proofs, which will be found in standard text books.

serlredor4 r(.reluawalg pu€ suorl.ruuac 'I'?

4.1. Definitions and Elementary Properties sturddel4l luurroJxrootsen$ ;o y uorlrugaq

4.1.1. Analytic Definition A of Quasiconformal Mappings

c11,(puy .l.t?

'[g'r] >cfraralsorul" .rog uo snonurluocflelnlosqe sr (/i'*)t 'fr Jo uor]cunJe s€ pue 'ftt'cl ) fi [p'a] f.ra,ra1sour1e.rol [g'o] uo snonurluorflalnlosqe x (fi'a)l'c;o uorlcunge sy :srxe f.reurEeurreql ol ro srxe I"eJ eqt ol reqlle lalered ar" seprsasoq^{ ul '"] x = e13ue1car f.ra,re.rogsploq uorlrpuocSuraaol O A W [g'rf loJ eql y (saut1uo snonurluocfi1a7n1osqo) nV aq o1 pr"s $ O ur€ruop.reueld e uo (rt'r)l = (z)/ uorlcunge 'araq :lCy q / ter{t Eururnsse ,(q 'eldurexaro; 'paalue.ren3 'a'e alqerlua q s1ql'uretuop q pereprsuoc eql (areqal{rarra lsorule) 'esuasalqe?rns -ragrp i(ler1red '1sea11e'aq ppoqs 'luaurarrnberqqt 3f Jo esn€)eg aruosur ztrl - z1 uorlenbenueJtleg eql segql"s fpressacau 1nq'alqerlueresrp ue reprsuor o? lu"^r eM lou $ qtrqa 3[ ursrqdrouroauoqSur,,rraserd-uorlelueuo

We want to consider an orientation-preserving homeomorphism I which is not necessarily differentiable, but satisfies the Beltrami equation Iz = J.tlz in some suitable sense. Because of this requirement, I should be, at least, partially differentiable a.e. (almost everywhere) in the considered domain. This is guaranteed, for example, by assuming that I is ACL: here, a function I(z) = I(x,y) on a planar domain D is said to be ACL (absolutely continuous on lines) if the following condition holds for every rectangle R = [a, b] x [c, d] in D whose sides are parallel either to the real axis or to the imaginary axis: As a function of x, I(x, y) is absolutely continuous on [a, b] for almost every y E [c, d], and as a function of y, I(x, y) is absolutely continuous on [c, d] for almost every x E [a, b].

4. Quasiconformal Quasiconformal Mappings Mappings 4.

78 ?8

Remark .1. 1. An An absolutely absolutely continuous continuous function function .F(t) F(t) on on an an interval interval 1I is is differendifferen(z) on a domain D tiable at at almost every ttEl. Hence, when a function function fI(z) D is tiable € /. Hence, ACL, the the partial partial derivatives derivatives /,Iz and fz !z of of fI are well-defined well-defined and finite finite at at almost almost ACL, every zzED. It is not not difficult difficult to to show that that they they are measurable. € D. It

Now, as a natural natural generalization generalization of of the the notion notion of of conformal conformal mappings, mappings, we Now, make the the following following definition: definition: Analytic definition definition A A of of quasiconformal quasiconformal mappings. mappings. Let Let f(z) I(z) be an Analytic orientation-preserving homeomorphism homeomorphism of of a planar planar domain domain D D into into the the complex complex orientation-preserving quasiconlormal (qc) on D D ifif /I satisfies satisfies the following (z) is quasiconformal plane. We say that that fI(z) conditions: ACL on D. D. (i) /I is ACL < & (ii) There exists exists a constant & k with with 00::; k< < I1 such such that that (ii)

I/zl kl/z I a.e. a.e. on on D. D. S klf"l lfd ::; on D. Setting K = = (l+k)/(I-k), we say that If is is K-qc K-qcon D. We We call call the the infimum infimum saythat (l+k)/(L-/c), we SettingX it by I(y and denote dilatation of of 1((> K(~ 1) such that I is K-qc the maximal of I, denote it Kf marimal K-qc such that / /, or K(f). or quasiconformal on D. Since Since Example -1.A conformal mapping of a domain D is quasiconformal Example 1. is can take 0 as k in the above condition (ii), a conformal mapping I is l-qc we above as I we / and 1. and Kf K1 = 1.

=

is C , l b l < lal) (a,b c E c (a, az + Example b,,c m a p p i n g I(z) € C,lbl A n affine a f f i n e mapping * c 2 . An * b zbi + E x a m p l e 2. l o l ) is f ( r ) = az = (al+ hence Ks as k, &, and and hence K f = (Ial + Ibl)/(Ialquasiconformal.We tal<eIbllial quasiconformal. can take We can lbl). l0l)/(ldl- Ibl). lDl/lol as

Example given k, we set set lc, we Emrnple 3. 3. For aa given ( z, z.

zE ze H

f ( t )== { xr ++il'K I(z) H ,K K > !1.. c -- H, * i iy y eE C K vy,, zz = xx + \ and K Ktf = K. I{. Then I/ is quasiconformal mapping of C, and is aa quasiconformal

Remark Set Remark 2. 2. Set

z

I ( , )==l-lzI T : E2p' , I(z)

z €..1. A.

z E

disk ..1 .4 onto C, the unit disk Then I/ is diffeomorphism of the orientation-preserving diffeomorphism is an an orientation-preserving quasiconformal mappings 4 mappings of ..1 but not quasiconformal. there are are no no quasiconformal quasiconformal. Actually, there onto C. See Proposition 4.32 later in this chapter. chapter. 4.32 See domain D', D', another domain domain D D onto onto another Remark -qc mapping of aa domain mapping of be aa K K-qc Let If be 3. Let Remark 3. is K K-qc. mapping go and -qc. the composite composite mapping Then the of D'. Then mapping of conformal mapping and 9g be be aa conformal I oI/ is (ii), and and (i) and and (ii), satisfies(i) that 9g 0o I/ satisfies In definition that the definition to see seefrom from the it is is easy easy to In fact, fact, it • that = K I{y. KsoT that K f gof

sarlrador4 ,(reluauralg pu" suorlrug:aq 'I't

4.1. Definitions and Elementary Properties

79

'peurtslqo uaeq seq olqarl pue Sur.rqag o? enp llnser elqe:lreurar Suro,olloJ eqt 'stusrqdrouroeuroq Jo ese) aql ur '.rale.tro11'uorle3rlselur JaqlrnJ o1 alqecrldde ;l';o sarl.radordpoo3 aalue.ren3 o1 qEnoua 1ou sr a/ pue ,t salrlelrrep lerlred ar{} Jo ef,uelsrxe 'leraue3 u1

In general, existence of the partial derivatives fz and fz is not enough to guarantee good properties of f applicable to further investigation. However, in the case of homeomorphisms, the following remarkable result due to Gehring and Lehto has been obtained.

'O uo 'e'D alqn4uaufitp fi11o7o7 st t uaqT'g uo 'a'o ut pu, 'l saatToau,ap eql sDq C ory! e urDtuopDIo I tustyiltotuoauoq Dfi .tV uorlrsodo.r4 1ory.r,od

Proposition 4.1. If a homeomorphism f of a domain D into C has the partial derivatives fx and fy a.e. on D, then f is totally differentiable a.e. on D.

'u 'f1tre.r1rqr€, 1as draae .rog 1u€lsuoc e,r111sod e xld 'eer" alrug e s€q O ueq^\ uorlJass€eql a,rord o1 sargns ?;''/ oo.t4

Proof. It suffices to prove the assertion when D has a finite area. Fix a positive constant t arbitrarily. For every n, set

It't't Q)l - ? t+,)l

I

l u ) q ' u l r > l q l >- o dns (r)"0 I _ Un () Z -

sup

O
If(Z+h)-f(Z)_f()1 x Z h

•

slesqns aq1 go slurod ,(lrsuep a.re 0f pue 0r 'g ) ont * 0e dre,ra lsorule '1s.rrg ro; 'pq1 dldun uraroaql s.lurqng pue rualoeql ,,{lrsuap s,an3saqel 'f.rerlrqre sr ro;'g uo r 'e'e elqetlueragrp 'uorlresse ,t1e1o1 ^roqs sl ol secgns aq1 a,rord o5 leqf 1r / 'snonurluoc sr .rog'g uo ;l' snonurluoc erc nt pue t/ 1eq1sarldur ecueS.ranuoc ruroJrun aql leql aloN .Ar ue q)ns xU em Joord srqlJo 1sa.ratll uI'0 - q se fl uo.,(1uro;run (r)uI o+ se3ra,ruoc VI (?) I - @! + z) l) teql etunsseosle {eu aar'1uarun3.rer€lrrurs€ /tg'0 1- q e g uo flurrogrun (r)'l ol sa3raruoc,t/(@)l -(rt+t)l) teq? pue r u€q? ssel$ rr-O Jo €are er{} }?ql qrns o Jo ar lasqns elqerns"eeue puu uer e,r,r't}{"6} eeuenbas eql ol ueroeql s,go.ro3g Surflddy 'oo + u se (I uo 'o'e 0 * uf ?sq? aloN

Note that Un - 0 a.e. on D as n - 00. Applying Egoroff's theorem to the sequence {Un} ;;:'=1' we can find a measurable subset E of D such that the area of D - E is less than t and that (f(z +h) - f(z»jh converges to fx(z) uniformly on E as h - O. By a similar argument, we may also assume that (f(z+ih)- f(z»jh converges to fy(z) uniformly on E as h - O. In the rest of this proof, we fix such an E. Note that the uniform convergence implies that fx and fy are continuous on E, for f is continuous. To prove the assertion, it suffices to show that f is totally differentiable a.e. on E, for t is arbitrary. First, Lebesgue's density theorem and Fubini's theorem imply that, for almost every Xo + iyo E E, Xo and Yo are density points of the subsets

E xa = {y E R

I Xo + iy E E}, J\

0n!+o u ) n} = ong {s ) I

pue

and

> fi!+0r I u ) fr) =o"g

I x + iyo E E}

'{g

E Ya = {x E R

'flerrrlcedse.r Jo lurod flrsuap e sr 0r 1eq1 ,{es ealr,e.reg)

respectively. (Here we say that Xo is a density point of E Ya if

l

ong

'*#,i" 'r - *p(r)oor, ,:*"""'[T

1 -h h>O,h-.O 2

lim

xa

h

+ IE (x)dx = 1,

Xa-

h

.

Yo

1eg'dperlrqre I > L I 6 q1r,ul,l xrg '0- 0z ?eq1eunss€ ar*'flrcqdursJo a{es eq} rod'02 }e elqelluareJrp,tlelol sl / ?eql ^roqs Iprls e \ pue 'lfi! + oa = 0z lurod " qrns xlJ ('"? ;o slurod dlrsuep eugap e^\ 'fgelnurg 'ofrg lo uorlcunJ orlsrralr€req, eql fl onal araq,u

where 1E ya is the characteristic function of E ya . Similarly, we define density points of E xa .) Fix such a point Zo = Xo + iyo, and we shall show that f is totally differentiable at zo0 For the sake of simplicity, we assume that Zo = O. Fix TJ with 0 < TJ < 1 arbitrarily. Set

I(z) = If(z) - f(O) - xfx(O) - yfy(O)I·

'l ( o) o /n -ft),l a -(o ) / - ( r ) { l= ( z) t

ef,urs

Since

' l(( o)u-/ ( *)u t)n+l l (s)'tx- ( o ) /- ( ") / l - @)l - @p+ n)tl ) Q)1 l@)utn I(z) ::; If(x

+ iy) -

f(x) - yfy(x)1

+ If(x) - f(O) - xfx(O) I + Iy(fy(x) - fy(O» I,

'g n! pue 9 > lrl qlp z fra,rerc1lzlh > (t)t ) g rell€urse 3ur1e1'luarun3re.repturse,,(g leql osl€arunss deurerrr',,(lessacauJr 'g ) n pue > to1lzll"t> (z)t leql q)ns 9 e,rrlrsode sr araql 9 lrl qtt^ z .,l.ra,re

there is a positive 6 such that I(z) ::; TJlzl for every z with Izi < 6 and x E E. By a similar argument, taking a smaller 6 if necessary, we may assume also that I(z) ::; TJlzl for every z with Izi < 6 and iy E E.

4. 4. Quasiconformal Quasiconformal Mappings Mappings

80 80

Further, taking taking a smaller smaller 68 ifif necessary, necessary, we may assume assume that, that, for every z = = Further, x1t X2, r 2,iy1 x *+ iy iy with with c, x, y )> 0 and lzl Izi << 6/2, 8/2, there there exist exist points points Xl, iY1,, and iy2 iY2 of E E a that such that - rt), < (1-1J)x < nXll << eX << xX22 << ((11 ++41J)x, )c, Q r < ((1L -1J)Y - q ) a (<e Y1 < aY l
=

=

ys = 0 a.re (Note that that oo Xo = 0 and Yo are density points of Euo E yO and.8ro, and E xo ' respectively.) respectively.) = lx1,x2)x qlz* I| for every z* on the boundary of the rectangle ,R = I(z*) <~ 1Jlz* rectangle R [Xl, X2] X Then, 1(z*) principle holds for since other hand, the maximal [Y1' Y2]. maximal I, since the On /, /I is a tyr,yz). homeomorphism, and hence hence is an open mapping. Therefore, Therefore, for some some suitable z* homeomorphism, boundary of of R, .R, we have on the boundary

(r.) -- 1(0) I(r) I(z) < ~ lf I/(z*) xlx(O) -- y/y(0)l yly(O)1 /(0) -- 0f,(0) < r.l (l/,(0)l ~ I(r') I(z*) + z*1 (1Ix(O)1 + I/y (O)I).. + Iz + l/y(0)l) l, everyzz with Since z*1 ~ 1Jlzl, we conclude conclude that, that, for for every with c,y x, Y > 0 and and Izi < 8/2, ) 0 SinceIz lzl<6/2, lz -- z*l
L ~ A(E) l"t,p1a,a,s Jj(z)dxdy

A(E)

(4.1) (4.1)

for every measurable subset E of D. differentiable we see see that I/ is totally totally differentiable On the other hand, 4.1 we hand, by Proposition 4.1 point z we we can can show that that at almost every zED, such a point z € D, and at such

JyQ)= l f" (r)l ' - lf,( ' ) l' . Since implies that Since the condition (ii) on I/ implies 2 1 a.e. on D, I/zl liz I2 ~s 1lfA' ~slil' *rt pJj a.e. on D,

(a.1). the followsby by (4.1). assertionfollows the assertion

otr

y'. Actually, continis absolutely absolutelycontinRemark A in the the above aboveproof is function.A the set set function Actually, the Remark4. 4'12in in §1.3. uous, (4.1).See Lemma4.12 in (4.1). equalityholds holdsin SeeLemma and hence hencethe the equality uous,and $1.3.

el€q a^r uaql '(f)zd(t),d *toJ qlr^1,uollrunJ e a{€}'((oo)tl)"$CJo ue sv'r o1 qg,rtr/Jo elrleArrep 1er1.red }uetuala leadsar 'W'")xlor'of = (or)U f"r pue ,[g,o] uo 0c Fuorlnqrrtsrp eql ["/] ,(q aloueq 'fgrerlrqJe e xg ,esod.rndsrqt roJ lurod e xrg O ur W'rlx [g'r] = g, e13ue1ca.r '1CY

Proof. We have already shown as Propositions 4.2 and 4.3 that a quasiconformal mapping f in the sense of the definition A satisfies the condition (i),. Hence, f is quasiconformal in the sense of the definition A'. Let f be quasiconformal on D in the sense of the definition A'. To show that f is quasiconformal in the sense of the definition A, it suffices to show that f is ACL. For this purpose, fix a rectangle R = [a, b) X [c, d) in D arbitrarily. Fix a point Xo on [a, b], and set R(xo) [a, xo] X [c, d). Denote by [fx] the distributional partial derivative of f with respect to x. As an element of ego (R(xo)), take a function with form
=

sl ./ l€rl? /roqs ol secgns t! 'y uorlrugep eql Jo esues aql ur leuJoJuocrsenb sr 3l' leql ^\oqs oL' ,V uorlrusap eql Jo esuaseql ul O uo leuroJuocrsenb aq / la1 '/H uorlrugap aql Jo esueseql ur I€ruroJuocrsenbsr 'eaua11'r(r) uorlrpuoc arll sess-rpsy uorlrugep aql Jo esuesaq? ur / Surddeur / .loo.t4 IeruroJuo?rsenbe 1eq1 t'7 pue 6'p suorlrsodord ss uiroqs fpee.qe a^"q a \

'Tuapatnba fr11on7nu.t a.to s0utddou loru.totuoctsonblo rV puo y suotytu{ag .7.? uraroaql

equivalent.

Theorem 4.4. Definitions A and A' of quasiconformal mappings are mutually

'O uo'e'€ l'tlq)lttl

Ifzl ~ klfz 1 a.e. on

D.

(i)' The distributional partial derivatives of f with respect to z and z can be represented by locally integrable functions fz and fz, respectively, on D. (ii) There exists a constant k with 0 ~ k < 1 such that

? s q l q r n s I > { t 0 q ? l i a{ } u e l s u o c s s } s f f e a r a q l ( r r ) '6r uo'f1a,rr1cadsa.r'rt pnt "/ suorlcun; alqer3alur r(1eco1fq paluasa.rda.r aq u€? Z pue z o1 laadser qll,lr / Jo salrlelrrep 1er1.redlsuorlnqrrlsrp aql /(l)

Analytic definition A' of quasiconformal mappings. Let f be a homeomorphism of a domain D into C which preserves orientation. We say that f is quasiconformalon D if f satisfies the following two conditions:

:suorllpuoc oall Eur,rnollo;eql segsrlesI y O uo lout"loluocrconb sl / leql ,(es e711'uorl"luarJo se,r.resardq?!q^a C olq O uretuop e yo ursrqdrour .sturddBru -oeuoq s eq / 1a1 [BturoJuocrsenb Jo ,v uol+rugop c1+rtpuy

:s^{,olloJqrrqa auo a{t se qcns 'sEurdderu leruroJuocrsenb go y 'g'? pue uoltlugep eqt Jo uorlecglpour e raprsuocu?c a/r4, 6'7 suorlrsodo.r43ur1o11

Noting Propositions 4.2 and 4.3, we can consider a modification of the definition A of quasiconformal mappings, such as the one which follows: .2.T.?

4.1.2. Analytic Definition A' of Quasiconformal Mappings

s8urddel4l lBruroJuocrsen$ go /I/ uolllugaq

c11,tpuv

's1rcd ,tq uorler3alur z(q .no11o3 O suollrasse aqt 'TCV st ;f aaurg 'sp.r3elur paleeder e{l se ue}lrr!\er eq uec (g'p) pue (Z'7)Jo saprsprrerl Ual eql(ureroaql s,lulqnJ pue Z,'Vuorlrsodor4 Ag.loo.r,4

Proof. By Proposition 4.2 and Fubini's theorem, the left hand sides of (4.2) and (4.3) can be rewritten as the repeated integrals. Since f is ACL, the assertions 0 follow by integration by parts.

JL JLfz .

(z'v)

'fi,pxpz61 'ttf ttil - - fi,papdt

(e'r)

I

and

JL JL

f·

f .
puD

'frprpzdt t"il -=ftpapdt,t"lI fz .
(4.3)

(4.2)

'sq.toililns Toqysnollol 7t Tcndu.tocypm .tot 'frputo7J 0 uo suotl?unt qToorus11oto 7aseW'(O)JC to dt Tuauala fi.r,aaa 'uorryqpprp zt saatToauappty.toil to asuaseUIur ?soql Vpn lu?pN?uroJ?JD,l puo urDurop o lo I Futrldoutlouttotuoctsonbfuaaa JoI .g.V uorlrsoilo.r6

with compact supports, it follows that

Proposition 4.3. For every quasiconformal mapping f of a domain D, the parti~l derivatives fz and fz are coincident with those in the sense of distribution. Namely, for every element

?t#'O

4.1. Definitions and Elementary Properties

81

sarlrador6 freluauralg pu" suorlrugtag'I'?

T8

4. Quasiconformal Quasiconformal Mappings Mappings 4.

8822

Jr(

Jr{

(p r)' (c))
1/ (Xo

fq

11/ ro

lso

lxo

fn

[fx](x, y)
ro (r, v)y)(
"'

| ""

for almost every y on [c, [c, dl. dJ. Next, for every sufficiently large integer n, take as as oo' 00, then by the above above equality, equality, we have

=

(xo,il JaIro lLl@,y)dxdy [fx](x, y)dxdy = ff(xo, y) -fto

y) almost almost every every y y e1",4. E [c, dJ· ff(a, @,v)

(4.4) e.4)

Ja

dependence, Here, xo. To get rid of this dependence, depends on 00. set of y depends Here, the exceptional set countable, Since,E consider b]. Since E is countable, numbers in [a, say E, -8, of all rational numbers set, say consider the set, [a,b]. continuous (a.a) are sides (4.4) dJ for every Xo in E. Since both sides of (4.4) are continuous o0 since d] every (4.4) holds a.e. on [c, holds a.e. [c, a.e. (a.a) holds holds a.e. we conclude conclude that (4.4) with D],we denseon [a, .E is is dense since E rs, and since with respect respect to Xo, [o, b], every holds where (4.4) on [c, dJ for every XQ in [a, b]. Note that for every y where (4.4) holds for every every b]. cs every d] [o, [c, coincident o, and that [fx] Xo, f(x, y) is absolutely with respect respect to x, continuous with absolutely continuous xo, f(x,y) [/'] is coincident with fx a.e. b]. pa,rtial derivative a.e. on [a, derivative f, usual partial with the usual [o,6]. y, we we can can confirm aa similar As for the partial respect to y, partial derivative derivative of f/ with respect distributional and that the distributional assertion. ACL on D and is ACL we conclude conclude that f/ is assertion. Thus we derivatives 0tr usual ones. ones. are coincident coincident with the usual derivatives are Dt , another D', onto another domain D onto Corollary mapping of aa domain confonnal mapping be aa conformal Let 9g be L. Let corollary 1. mapping of D. and -qc mapping -qc mapping is aa K l{'qc of D'. Dt. Then Then fog mapping of K-qc be aa K and ff be f og is the there exist exist the a.ssumption,there Proof. = urtr+ By the assumption, variable on on D'. D' . By the variable lu be be the u) = Let w Proof. Let * iv on D' . integrable distributional and fw, and they are locally integrable on D'. are locally and and derivatives fw distributional derivatives /,5, /. seenthat that easily seen it is is easily Since (D') for every

Corollary is conformal. conformal. mapping is l-qc mapping 2. AA l-qc Corollary 2.

'0 - npnp dl"Ut) - "H)lt [ [ T;lli J J

'g uo t"ql (g'?) 3ur1ou ,tq ,uoqs u?c eilr ecurs pue

on F, and since we can show by noting (4.6) that

zt = z(lb) pu€ ,I = "(tb) (TJf)z

= fz

and

(TJfh

= fz

acurs 'oo ts u se g uo fprloJrun 3f o1 saSrerr '.reloe.rotr41 -uoc u/ pue 'u a3rel rflluarcgns fraaa lof (O)JC o1 sEuolaq $

Moreover, fn belongs to C[f(D) for every sufficiently large n, and fn converges to f uniformly on F as n ---.. 00. Since (4.6) (g'f)

.t]h)

* "dt = z("1)

pu€

"(ttt) * "d = "("t) eleq eilr 'u fleaa roJ uerlJ

Then for every n, we have 'npxp(z)!(z)t(, -*)"d,"[[

Jl


wE C. n

J J

=
'))

=@)(!u)*udt

- (m)u!

fn(w)

and

Pu€

zEC C)z

'(zu)6.ru-(z)"dt


'u .re3alure,ulrsod fre,re ro; 'reqlrng

Further, for every positive integer n, set 1es

vf f '1=npxp(z)dt | | JJ '{srp 'e.rag lu"lsuo, e esooqC }run aq1 sr y

JL


Here, ..:1 is the unit disk. Choose a constant C so that leql os ,

'v-c)z

'0)

z E C-..:1.

0,

(-1-l I2)' ,("+-)

{c.ex

Proof. Fix an element TJ E C[f(D) which is identically equal to 1 in some neighborhood of F. Then TJf has a compact support in D. Further, (TJf)z and (TJfh exist and belong to LP on D. We set p z E..:1 z
v>z

d x ae

]=?)d)

les eA\ 'O vo o1 3uo1eq pue lsrxa a7 'd z(lL) pue z(/lr) 'raq1.rnd 'O ul l.roddns lceduroc e seq ll-t uaql Jo poor{roq -q3reu euros ur 1 o1 lenba {pecrluapr sr qctq^r (O)"$.e ) lr luaurela ue ng'too.t4

's= nprpolz! -,("!)l"lf *tf (4.5)

(q'')

'o - npap - ,('!)1"[[ *tf al"!

'oo + u s0 tDq?puD f uo fiyl^tolmn t "{ p1t o7 safi.taauoe u? r-;j{"!} acuanbasD s, areql ,O lo I lasqns Vcns (q)}g tuana.tot ueUJ'CI uo a1qo$ayur fi,11oco7 ato fils1,7os Taodu.toc alzllpuD alzllll?t? 'fr,pu.rou'e uo "t fr11oco1 zt ^to puD senqDtuuap pqlod puorlnquqtp esoqm aI 'uaafi eq .g,V Bururarl O uxprilopn uo uotTcunlsnonuNlu@D ?q t 7a1 l
whose distributional partial derivatives fz and /z are locally LP on D, namely, satisfy that Ifz IP and 1/zIP are locally integrable on D. Then for every compact subset F of D, there is a sequence {In};;''=l in C[f(D) such that fn converges to f uniformly on F as n ---.. 00, and that

Lemma 4.5. Let p ;::: 1 be given. Let f be a continuous function on a domain D

'etutual uorleurrxordde Suurrolloyeql esn e^rd eterqut'outtual s,fiap1 ;o;oord e elrS e.,n'acuarualuo? Jo e{€s aq? ro; ,ere11

Here, for the sake of convenience, we give a proof of Weyl's lemma, where we use the following approximation lemma.

'1fa6 'ecua11 'uor]nqrJlsrp go €rmuel eql ,iq crqdrouoloq sr D l€crssrlc 3[ - z/ sagsrles .too"t4 Jo esuas eql q O uo 0 O ur€ruop e Jo I Surddeur cb-1 y

Proof. A l-qc mapping f of a domain D satisfies /z = 0 on D in the sense of D distribution. Hence, f is holomorphic by the classical lemma of Weyl. sarlrador4 freluauralg pu" suorlruyae 'I't

83

4.1. Definitions and Elementary Properties

84

4. 4. Quasiconformal Quasiconforrnal Mappings Mappings

and and

ardy= o, lim I I l$"), - Qtf)rlo J JD

N-6

obtain (4.5). we obtain

tr D

sequence {/.} {fn} *as in in Lemma Lemma 4.5 an LP-srnoothing LP -smoothing Hereafter, we call such a sequence sequence for /f with with respect respect to to F. F. seqaence

Lemma 4.6. 4.6. (Weyl's (Weyl's lemma) lemma) Let ff be be a continuous continuous function function on D D whose whose Lemma distributional derivative f2 fz is locally locally integmble integrable on D. D. IfIf fz fz =0 0 in lhe the sense sense of of distribational ileriaatioe distributions on D, D, then then ff is holomorphic holomorphic on D. D. distributions

=

Fix a relatively relatively compact compact subdomain sub domain Dr D I of of D D arbitrarily, arbitrarily, and construct construct Proof. Fu LI-smoothing sequence sequence for /f with with respect respect to 4D I as as in the proof proof of of Lemma an .tl-smoothing From the construction there, we see see that (TJfh = 0 in some some neighborhood 4.5. Flom thaf (nf)z 4.5. = sufciently large . of D By (4.6), see that (fn)z = D for every sufficiently large n. Thus, Thus, on D1 for every (a.6), we see thar (f")7 0 of E[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[[. I By I converges to /f fn is holomorphic D sufficiently Since fn converges large n. every sufficiently Since on D1 for holomorphic I /' /,. --+ we D1 is arbitrary, uniformly D as -+ 00, f D Since D we on D1 . Since n oo, is holomorphic on as uniformly I I I I / obtain the assertion. assertion. D

=

4.1.3. Definition G of Mappings of Quasiconformal Geometric Definition 4.L.3. Geometric Quasiconformal Mappings we first quasiconformality. For this purpose, we We give a geometric definition definition of quasiconformality. introduce the notion of quadrilaterals. quadrilaterals. called a Jordan closed closed The closure curve is called closure of a domain bounded by a Jordan curve (Q;qt,9z,Qs,Qq) of a Jordan domain. A is, by definition, a pair (Q;ql,q2,q3,q4) quadrilateralis, A, quadrilateral q4 on the boundary aQ 0Q of Q which closed closed domain Q 8r, q2, Q2,q3, 93, q4 Q and four points ql, oriendistinct and located in this order with respect to the positive orienare mutually with respect located are mutually confusion, tation of aQ. We call each qj a verlexofthe quadrilateral. If there is no confusion, q1 If there aertexof the each a 0Q. we quadrilateral (Q;ql,q2,q3,q4) (Q; er,8z,Qe,qa)simply simply by Q. we denote denote aa quadrilateral Q. lhere is is aa homeohomeo' quadrilateral(Q;QI,Q2,Q3,q4), (Q;Cr,Q2,QB,qs),there Proposition euery quadrilateral Proposition 4.7. For every (a,b which is 0) morphism onto some rectangle R = [0, a] x [0, b] (a, b > 0) which is onto some reclangle R ) morphism hh of Q Q [0,4] [0,b] conformal Int Q of Q, and satisfies and satisfies the interior interiorlntQ conforntal in the Q, h(gr) : 0,

h ( q 2 )= a ,

h(c") = a*ib,

a n d h ( q a )= i b .

Moreover, o h. i s independent i n d e p e n d e nof t f h. a f b is M o r o u e r , alb quadrilateral (Q; and (Q;qt,!2,Q3,9+)' the quadrilateral We QI, Q2, Q3, Q4), and the module moduleof the value alb the call the the value We call denote b V M(Q). M(Q). o, r simply s i m p l y by y M(Q;QI,Q2,Q3,Q4), M ( Q ; 8 r , Q z , e e , Q e )or d e n o t eiti t bby conformal mapping mapping Proof. there exists exists aa conformal theorem, there mapping theorem, by Riemann's Proof. First, First, by Riemann's mapping is theorem, hI h1 is hI H. By By^Caratheodory's the upper upper half-plane half-plane H. Carath6odory's theorem, onto the h1 of of Int Int Q Q onto suitable by composing composing aa suitable extended fl U R. Here, Here, by of Q onto H U It. homeomorphism of extended to to aa homeomorphism Q onto Mobius we may assume that that may assume transformation, we Miibius transformation,

98

Pu" suorlruyeq'I'}

4.1. Definitions and Elementary Properties sarlrador4,{rcluauralg

= -1,

= 1,

85

= -h 1(q4) > 1.

Pue'(sr;tq7l { ?aS 't < (vb)rq- = (80)It1 pu€ '1 = (zb)tq 'I- = (ID)rtl

h1(qt}

h 1(q2)

and

h1(qa)

Set k = 1/h 1 (qa), and

z

P

dz

,Qt"rt--lG,

| =e)",

r z

v'(1- z2)(1- P z2)

-il

10

, z E H.

'H)z

h 2 (z) =

a13ue1cer auos Jo rolralul eql o?uo g ;o Sutddetu lsruroJuoc 3 sr zV ueqJ

Then h2 is a conformal mapping of H onto the interior of some rectangle [-K, K] x [0, K'] (K, K' > 0). Hence we see that

'0] x .(0 leql eesel'aaruaH < ,y,x) l,x [y'>l-]

z E Q,

'b)z

h(z) = h2 0 h1(z) + K,

'x*Q)tq"zr1-@)rt 'EU) Eutddeur P?rrsaPe sI

is a desired mapping (Fig. 4.1). '(t'f

I't

h

<_

~

q,

q,

x+'l ~

h,

"q ffi

I :

s

' T

!

T

a

Fig. 4.1.

.I'?'EIJ

'P '(O f"I ) z) P * Q)t1c = (z\rt eleq e,rr Pu" t sreqrunu xalduoc elqellns qlr^1' 'flpaleedar 'acue11'(c)pV to lueruela u€' o1 papue1xeaq uea , -V o V leqt ees a,u '2'7 uI se suolllPuoc aures eql uotltsodord aldrcur.rd uorlcegar (zJs^rq?SEutflddy 'lxap sessrles qcrqa'rEurddeu raqlou" aq [g'o] x [g'o] = v *@ : q le1

=

Next, let h : Q ~ R [0, a] x [0, b] be another mapping which satisfies the same conditions as in Proposition 4.7. Applying Schwarz' reflection principle repeatedly, we see that hoh- 1 can be extended to!\n element of Aut(C). Hence, with suitable complex numbers c and d, we have h(z) = ch(z) + d (z E Int Q). Since h(qd = h(ql) = 0, h(q2) > 0, and h(q2) > 0, A?UIS

'l'? uollls tr -odo.r4Jo uorlress" puoceserll se[du4 qclqa '0 = p pu€ g 4 a ]e{} aPnlf,uocert.r '0 < (b)q '0 = (ID)tl= (b)t! '0 < (zb)y Pu€

we conclude that c > 0 and d = 0, which implies the second assertion of Proposition 4.7. 0

'erutuel 3uuro11o;aql e^"q eir ueqtr, '(vb)!' ...'(ID)/ rllllrl Plel"Urpenbe t" (O)f lePlsuotaru'3 olut @;o seclllel ro; '.rary:earag e pu" (tD,ab'zbtrb1fl) le.Ialeyrpenb,(.ra,ra /' ursrqdrouroauoq

Hereafter, for every quadrilateral (Q; ql, q2, qa, q4) and a homeomorphism f of Q into C, we consider f(Q) as a quadrilateral with vertices f(qt},··· ,f(q4)' Then we have the following lemma. sa{stqosO uroulopo lo { tutililoru cb-51fr.nog'8'? BurtnaT

Lemma 4.8. Every K -qc mapping

f of a domain D

satisfies

686 0

4. Quasiconformal Quasiconformal Mappings Mappings 4.

~M(Q) ~ I

M(f(Q))

~ KM(Q)

KM(Q)<MU@))
F=hofohF = - h o f o h - r1 of the the interior interior of of R R onto the interior interior of of E R which is a quasiconformal mapping of n$ ( c f . maps 0,, aa,, iib, 0,, iii, and Remark §1.1 + i D , rrespectively e s p e c t i v e l(cf. y R e m a r k3 iin n d 6a+ib, b , aand n d aa+ib * i b t o to 0,A b , ib, a m a p s0 1.1 particular, hence is ACL on Ii, and Corollary 1 to Theorem 4.4). In particular, .F(z) F(z) ACL R, hence and Corollary every y €E [0,D], [0, b], we have have for almost every

iv) -- F(iy)l a< ~ lF(a IF(a + + iy) F(iy)1 = =

11

a

l

a

~~ (x + + iv)drl= iY)dxl ~ (lFzl + IFzl)dx. l.zt)dr 1,"F"t+ l,' #o

Since ffR Jpdrdy JFdxdy S ~ ,4(n) A(R) = = dD ab (as (as was was stated in the proof proof of Proposition 4.2), 4.2), Since /J* we obtain integrating both both sides above inequality over over [0, b], we sides of the above [0,b],

r

( [ [ (lFzl ~ (fL + l+Da"av) IFzl)dxdy "Ur,l+ / \"r.h

(ab)2 -< @b)'

f IFzl + IFzl

Jr f (I 12 1 12 )

dxdy . J , Fz'' - Fz dxdy - JJJ^, ~ JR' IFzl_lFzl JJo," R 1r1:77d'oa' r

fL,

fL,

t o o , o y !~KK(ab)(ab). (ab)@b). I ( d x d- y . [ [ JFdxdy ~-. J[J[ n ' Kdxdy· JJn' M(f@)) ~ haveM(f(Q)) Thus^we Here, R' = {w have 0}. Thus_we ,R11Fz(w)l:/; we set setl? Here,we eR < KM(Q). lIF"(*)l I O}. {tr,'E (or considering = h 0o ff 0o h(ih)-r (or considering Next, h-r1 by by (ih) (ih) 0o f 0o (ih)-l f' = replacingF Next, replacing that (Q;q2,q3,q4,qt)), the same sameargument argumentthat we can canshow showby by the (Q;qr,qs,q+,qr)),we 1I K M(f(Q)) ~ M(Q)' Ma) W@Ds tr D

Thus we we have assertion. have the assertion.

quasiconformal Now, we give give another another definition of quasiconformal 4.8, we Now, by noting Lemma 4.8, mappings derivatives. without using using partial derivatives. mappings without be aa homeohomeoquasiconformal mappings. Geometric mappings. Let f/ be of quasiconformal definition G of Geometric definition that preserves say orientation. We We say morphism of aa domain f/ isis which preserves orientation. C which domain D into C quasiconformalon condition: satisfiesthe following following condition: quasiconfonnal on D if f/ satisfies

(iii) (iii) There > 11 such such that that constant K K 2: is aa constant There is

IiM(Q) M(f(Q)) MU@)) ~S KM(Q) quadrilateral Q holds in D. D. every quadrilateral holds for for every Q in

'a.rolaqso ftpnp{ilt11

= @)V alaqm 'g lo g ?esqnselglrnsoelrz fri,aaa.ro!

for every measurable subset E of D, where A(E) = ffJ(E) dxdy as before.

(4.7)

Q'v)

sf f

- ,l"lD (s)v= nprpQl'll IJ sa{sr7os CI utvluop Dlo I |utddoru Totu.to{uocrsonb fuaag 'ZT'V r.urrrroa

Lemma 4.12. Every quasiconformal mapping

I

of a domain D satisfies

'oraz ear? seq ((A)/)r -t = g D '91'7 ureroaqJ les aqt wq+ zI'v eurtuerl {q ureSeees e/lt Jo (l) fq pu.ro;uocrsenb osle sI r-l eculs 'orez ear€ s€r{ (ar)/ }as eql '71'V eluuo.urc1 }xeu eql dq acuag 'A uo'e'e "t 0 = zt pu€ elqernseeu $ A ueqJ'{0 = I O ) z} = g pg'too"r4

Proof. Set E = {z E D I Iz = O}. Then E is measurable and !z = 0 a.e. on E. Hence by the next Lemma 4.12, the set I(E) has area zero. Since f- 1 is also quasiconformal by (i) of Theorem 4.10, we see again by Lemma 4.12 that the set E = 1-1 (f(E)) has area zero. 0

'o

Proposition 4.11. If f is quasiconformal on a domain D, then D.

uo 'a'o 0+ "l ueql'(I utDu.topo uo lout"toluoctsonb s! I lI 'II'7

Iz # 0

a.e. on

uolllsodo.r4

',) uollluuecl ruoq r€al) sI PJIqI eql tr '(2'7 uorlrsodo.r4go yoo.rd aq? aag) 'appour ar{} Jo e)u€rJelur leurJoJuocaql ,(q ueas sI puo?asaqtr'{reureg Eurpace.rdeq} urorJ s^{olloJuorlresse lsrg aqa /oo.l2,

Proof. The first assertion follows from the preceding Remark. The second is seen by the conformal invariance of the module. (See the proof of Proposition 4.7). The third is clear from Definition G. 0

'cb-eXrX '(O)l st t o 6 0utddout pasodruoceq? .tog (t$) {o 6 0utddout cb-zX fi.taaapun e urotilop D Io I |utildout ab-rX fr".taaa 'ab-y oslo s? ?q?'e oluo O to t |uzrldout t_\o I oU 0utddou.tpasodu.t,oc cb-y fr.tana.tol puo'fr1aai7tadsa.t'epuo q suwu,6p lo tq puo t1 |urildotu 'fi1atuo111 'Tuottiaut fi11ow.to{uoc st fiTtfoutoluoct,sonb-y Tout^totuocfi.taaa.tot 'cb-y oslo st Outildou cb-y o to as.taauraytr

(i) The inverse of a K -qc mapping is also K -qc. (ii) K -quasiconformality is conformally invariant. Namely, for every conformal mapping hand h of domains D and D, respectively, and for every K -qc mapping I of D onto D, the composed mapping hoi 0 h- 1 is also K-qc. (iii) For every K 1 -qc mapping I of a domain D and every K 2 -qc mapping g of I(D), the composed mapping go f is K 1 K 2 -qc.

(r) (r)

'0I'7

Theorem 4.10. urorooql

( '''t ureroaql o1 1 frego.roC pu€ I'I$ q I {reureg se perlord.{pearle ueaq s€q gl'f ureroeqtr Surmolo; erll Jo (11)wut II"csU) '6'? ureroaqJ prre 5l uorlrugao Jo sarrsllorot 'a.rag 'y'1$ ol Ierelas e,rr3 e.r,r. 6'? ruaroeql yo goord aq1 auodlsod geqs all

We shall postpone the proof of Theorem 4.9 to §1.4. Here, we give several corollaries of Definition G and Theorem 4.9. (Recall that (ii) of the following Theorem 4.10 has been already proved as Remark 3 in §1.1 and Corollary 1 to Theorem 4.4. )

'guapamba Q7on7 '6'V urarooql -nur, etD s0utddou.t loru.totuoctsonblo g puo V suotTru{ap onJ

Theorem 4.9. Two definitions A and G of quasiconformal mappings are mutually equivalent.

'O ul roJ sploq 0l€replrrpenb .,(.re,ra

holds for every quadrilateral Q in D.

~M(Q) :::; M(f(Q))

:::; KM(Q)

@ ) w > r > ( } i l w > @ wI ! (iii)' There is a constant K

~

1 such that

l€rll qcns I < >I lu€lsuoc e st a.req; ,(ttt)

Remark. Condition (iii) in the foregoing definition is equivalent to the following one:

:euo eql ol lualerlrnba fl uorlrugep Suro3e.ro;eql ul ($) uorltpuoC 'qrDuaq Surr*o11o; 't to t X uorlet"llp lerurxetu eql ol lenbe sr jyr qlns Jo turuugur aq? 1eql aasa^{'ra1e1pa,rord$ qrrqa gI'? €ruurarl pue 8'} eurural {g

By Lemma 4.8 and Lemma 4.15 which is proved later, we see that the infimum of such K is equal to the maximal dilatation K f of I. 4.1. Definitions and Elementary Properties

87

sarlrador4 dreluarualg pu€ suorlluu:aq'I't

L8

4. 4. Quasiconformal Qua.siconformal MaPPings Ma.ppings

88 88

Proof First, First, we we consider consider the the case case that that EE isis aa rectangle rectangle contained contained in in DD and and that that Prool. I/ isis absolutely absolutely continuous continuous on on the the boundary boundary 0E 8E of of E. E. In this this case, case, in in view view of of Proposition Proposition 4.2, 4.2, we we find find an an tr2-smoothing L 2-smoothing sequence sequence In = put un* ian.n . n' every For (cf. Lemma 4.5). {fn}~=l for I with respect to E (cf. Lemma 4.5). For every n, put In = Un +iv to E' respect with for /' / {/"}Lpr By Green's Green's formula, formula, we we have have By tt - (.,-)r('")"} I I {@^),(an)v J Jn

dxdv -

I u^d'an J| , E

for every every nnand m. Let Let m m *-+ oo. 00. Next, Next, to to the the right right hand hand side, side, apply apply the the formula formula and.m. for for integration integration by by parts parts for for the the Stieltjes Stieltjes integral. integral. Letting Letting 7r n *'+ -+ o9, 00, we we obtain obtain for t[ t [n

- uuo,)dtdy[ uda. Uf"P- lf1\ara,- = [ [ (u,oc Jan JJn'

=

Here, we write write /I = u+ia. u+iv. The The right right hand side of of the above above equality is interpreted interpreted Here, as the the line line integral integral of of uda udv along the Jordan curve df(E) 81(E) on the w(w( = u*ia)-plane. u+iv)-plane. as By the the assumption, assumption, Af@) 81(E) is rectifiable. Hence, Hence, we can show that that By t

(

tl

{ udv = J~ dudv = A(E)' A(E). JJ,r"rdudo J aE JJ(E) Ju""o'

=

=

Thus we we have have the assertion in in the case case stated stated at the beginning beginning of of the proof. Thus such Since contained in D can be approximated by such every rectangle recta,nglecontained ACL, every Since I/ is ACL, D. By a routine rectangles. contained in D. rectangle contained every rectangle holds for every Hence, (4.7) holds rectangles. Hence, tr subset E of D. 0 measurable subset argument, every measurable (a.7) holds holds for every that (4.7) it is proved that argument, it quasiconformal mapping If of a domain Now, by Propiosition 4.11, every quasiconformal 4.11, for every D, we can consider a quantity quantity a consider we can D,

Iz f, Iz

J1.J r t t= =T and satisfies satisfies a.e. bounded measurable measurablefunction on D, and pt is b arbounded D. This J1.J a.e. on on D.

< ~J ess. sup IJ1.J (z)1 ::; - ~ ess.sup lptQ)l 9+ t fJy + *L zED zED

<< 1.r.

We of Ion D. dilot'ationof prythe complerdilatation the complex call J1.J We call f on D. D, proposition 4.13. domainD, of ao domain andgg 01 Proposition mapping qaasiconfortnal moppingIf and For every eoeryquasiconlormal 4.13. For 1 giaenby by g o1mappinggo the mapping composed pso!-t 01 of the the composed dilatationJ1.goJ-t I-r isis given complexdilatation thecomplex

Izf, J1.g J1.J . Fo--!!_, J1.goJ-t l r s o l -0 r I" I== T=I _ - t r r_ l r n, Iz 1- J1.JJ1.g

D. a.e.on on D. a.e.

(4.8) (4.8)

Hence proof.By quasiconformalon on I(D). gof-L isis quasiconformal Proof By (i) 4.10,go/-l of Theorem Theorem 4.10, (iii) of /(D). Hence (i) and and (iii) 1 for a subset proposition except for a subset by on I(D) differentiable on g o1totally differentiable 4.1, go /(D) except by Proposition 4.1, f-r isis totally 1 quasiconformalmapping mapping 1E.Oof to the the quasiconformal 4.12to Lemma 4.12 /-1, , Applying Lemma zero. Applying of measure ,rr"""rrre zero. 1 both 4.1, by Proposition Proposition 4.1, both I/ we (E) isis also zero. Hence Henceby of measure measurezero. also of we see seethat that 1f-r(E)

'1purs dlluatr leqt atunss€ feur aaa 'U -gns eql uo snonurluoc ,(pr.ro;tun sr 3f acuts 'alo11 & 3ul{"t }as lceduroc ('e't 'St,f aeg) 'oo - ll uaqn tlt Aq .{lrlenbeur eq} Jo eprs pueq lq3rr aq1 acelda.re^\ pue 'llq'lr) = {J 1es ein 'era11

Here, we set I j = [aj, bj ], and we replace the right hand side of the inequality by 1/{ when fj = 00. (See Fig. 4.2.) Now, since f is uniformly continuous on the compact set R, taking 1] sufficiently small, we may assume that

k=1

I={

'Z/,-!,1?lr-q)-{)13

L

and

I(k - (k-d

2: fj -

{/2.

n

Pu3

, ( o f r ?!+e ) l = o )

+ iVa),

+ iVa)

(4+!q)I =2

(0 = f(aj

(n = f(bj

1€rl1q?ns({on} x ft)/ (0 <) r pue f xrg uo slurod r=l{{)} }esalruge sr areql ueq; 'f1.re.r}lqre Jo 'd1e.rr1cadse.r'({on} x ft)/ pue f7 1osqfual eqt eq !,1pu, ll t"j '{ f.rer.aro;

for every j. Let fj and fj be the lengths of I j and f(Ij x {Yo}), respectively. Fix j and { (> 0) arbitrarily. Then there is a finite set {(d~=1 of points on f(Ij x {Yo}) such that

(A)t = lO pue fft(ofrlx !7 - lg '[g 'r] r--,I{!il e a{el ,tlF tJ ellug les pu€ Jo sle^ralurqns 1u1ofs1pf11en1ntuto 'uo a.requrorg '6 < & leql erunss? (orf- /t e qcns xlJ'[p'af 'on - n - & ) ff frale lsotule l€ alq€ItuereJlp sI 1aspue e sr (f)g ecurg '[p'a] uo /t f.ra,re.roy([n'a] x [g'r])/ Jo g 'uor1cun3Sursea.rcap-uou exr.g'too.t4 eer€ eql (n).I fq aloua(I'fp,re.ryq.re0 qW '"1x [g'rf =A e18ue1cer

Proof. Fix a rectangle R = [a, b] x [c, d] in D arbitrarily. Denote by F(y) the area of f([a, b] x [c, V]) for every y on [c, d]. Since F(y) is a non-decreasing function, F is differentiable at almost every y E [c, d]. Fix such a y = Yo, and set 1] = y - Yo. From here on, assume that 1] > O. Take a finite family {Ij }j=1 of mutually disjoint subintervals of [a, b], and set

'O uo ICV sr g uorTtu{ap aq7 ''I'' Bruurarl to asuas?W u? CI utDtuop D lo t |ut,rldout Toutlotuoctsonbfinag

Lemma 4.14. Every quasiconformal mapping f of a domain D in the sense of the definition G is ACL on D.

'(g) p"" (r) suorlrpuoe aql segslles gr uollluuep eql Jo asuesaql ur Suldderu leuroJuocrsenb e lsql ^ oqs gI'? pue ?I't s€ururaT 3ur,uo11o;eq1 'dlesre,ruoC'($) uorlrpuoc eqt segsltes y uolllugap aql Jo esuas aql q Sutddeu leuroJuoclsenb ts 1eql 8'' eururerl uI u^roqs fpeerp e^eq el6

We have already shown in Lemma 4.8 that a quasiconformal mapping in the sense of the definition A satisfies the condition (iii). Conversely, the following Lemmas 4.14 and 4.15 show that a quasiconformal mapping in the sense of the definition G satisfies the conditions (i) and (ii). 'v'T'v

4.1.4. Proof of Theorem 4.9 6'7 tuaroat{I

0

tr

Jo Joord

'uorlresse aq1 fldurr sarlrpnba e^oqe eql snqtr, 'O J z ,'hala lsorul€ roJ

for almost every zED. Thus the above equalities imply the assertion.

O*1"^G-t"6) pu€ '0*'6 '0#'l Using a similar argument, we can show by Proposition 4.11 and Lemma 4.12 that

leql fq ,raoqs uet e^t 'luaurn3rerelltuls e 3ursl Zl'V evutej pup II'f uo11tsodo.r4 'g). -'0 lo t(r_/o f) *'l' I o'(,_1o 6) Pu€ and

^G-t o6)= z6 "!' "t(r-l" 6)+'t' I t o eAeIIa/rr'(r)I - ^ 'r{lEurproccy'p[e^ q elnr urcq?eql (z lulod e qcns ?V 'O uo z frale Eur1r.r,n 'flarrlcadsar'(z)rf ?s pus z le elqellueraslpf1p1o1a.re1-./ o d pue lsourle .ro; gz =(gor 1 )w o f·fz+(gor 1 )w o f·lz

and go f- 1 are totally differentiable at z and at f(z), respectively, for almost every z on D. At such a point z, the chain rule is valid. Accordingly, writing w = f(z), we have 89

4.1. Definitions and Elementary Properties sarlrador4 dreluauralg pu" suorlruyaq'I't

90 90

4. Quasiconformal Quasiconformal Mappings

}~

Ir

Ir r5 0 r> 1 Fig. F i 9 . 44.2. .2.

en -

i(yo+ If(xo f(xo + +€)) - /("0 +;vil S~ lf(ro+ i(yo .iYo)1 ~ *4n

e

e

with 0 ~ < € ~ D] and every { with Ri for every Xo cs on [a, < TJ. 4. Take any curve L in Rj [o, b] parallel to the y-axis. Then connecting two sides a"reparallel Then we we can see see sides of oRj )Ri which are that f( L) is less than that the length of ol f(L) is not less

I,=il(*-(*-,1i. &=1

=

rectangleRj On hj ] x f, be be aa homeomorphism homeomorphismof some somerectangle fr.1= [aj, the other other hand, hand, let h On the la1,6il quadrilateralQj interior of Rj and [Cj, dj ] onto which is is conformal conformalin the the interior Ei and onto the the quadrilateral Qi which l\i,dil respects verticessuitably. suitably. Then we have have respectsthe the vertices Then we /

"

,6;

Ii s | |

\/a;

\2

,6;

tlt'ta,)s tai- ai).t |'ai li,'t'a,. /

we obtain obtain Integrating dj], we sideswith respect respectto y on on [Cj, Integratingboth both sides lei,dil, ii2

-

u { o i ) .A4j .i . lj S ~s M(Qj). Here, area of Qj. Here, we we denote denote by A li j the area Qi. Now, suppose senseof the definition G. Since suppose that that f/ is K-qc K-qc in the sense Since

u(Q)
(l-)
I6

91

4.1. Definitions and Elementary Properties sarlrador6,(rcluauralg pup suorlluya('I'p

'O uo rIcY sr D '[g'r] > 0c .frerralsotup ro; d ;o uorlcunJ e se [p'a] uo / reqr epnl)uof, ar'l snqtr, snonurluoc {1e1n1osqe* (fi'oa)l }€q} ^\oqs uec ea\ 'luaurn3re eures aq1 fg 'o Jo uorlrunJ * * [g'o] uo snonurluoc * llt--,t'3se 0 - !,lt--,j31€rll apnlruo? flalnlosqe 4(fr'r)!'flluanbesuoC'0 '6 sr f7 r(.ra,ra e.!r', - lA? l€ql ees rr€c ea\'relncrlred u1 <- & se \e?urs'ellug anl€ etlug e ol spuet (ofr - q/((on)a - (n)A) '0n te alqeltuereJlP sI dr e)uts

Since F is differentiable at Yo, (F(y) - F(yo)) j(y - Yo) tends to a finite value as 'TJ -+ O. In particular, we can see that every i} is finite. Since Ij ~ i} - f, we conclude that Ej:l i} -+ 0 as Ej=l i j -+ O. Consequently, I(x, Yo) is absolutely continuous on [a, b] as a function of x. By the same argument, we can show that I(xo, y) is absolutely continuous on [c, d] as a function of y for almost every Xo E [a, b]. Thus we conclude that I is ACL on D. 0

uayl 'N uor?Dlolrplourroou eUl ypn g uotTru{apaqy 'St'V BtutuaT {o asuas ?qI u? CI urvulop o to 0utddoru Tou.totuottsonb p sr I lt

Lemma 4.15. II I is a quasiconformal mapping of a domain D in the sense of

the definition G with the maximal dilatation K, then

'O uo'e'o l'llq > Itll

I/EI :5 kl/zl

where k = (K - l)j(K

a.e. on

D,

+ 1).

'0+x)10-x)=qer?qn

'6'7 uorlrsodordJo;oord eq1 ur pelou e \ sy'anrleEau-uou a.re(g)z/ pu€ (0)"/ (20 pve Id sraqurnu leer alqelrns qq^a (z . "sre)t . ,6re l€ql eurnsspJeqlrnJ feur an Sutreptsuoc ,(g '0 = oz Ie-qI'flqelaua! Jo ssol lnoqtl^t 'eurnss€ feur er* ara11 'acua11 'oz = z e q?ns xld 'O ) z f.rcrlo lsorul€ le elqerluaragp f11e1o1sr 3f '1'7 uorlrsodord 'loo.r4 Jo uorldurnsse aql segsrles / teqt saqdur yl't €ruural

Proof. Lemma 4.14 implies that f satisfies the assumption of Proposition 4.1. Hence, I is totally differentiable at almost every zED. Fix such a z = zoo Here we may assume, without loss of generality, that Zo = O. By considering ei81 . f( ei82 . z) with suitable real numbers (}1 and (}2, we may further assume that Iz(O) and fz(O) are non-negative. As we noted in the proof of Proposition 4.2, fz(O) ~ fz(O) (~ 0). If Iz(O) = 0, then the assertion is clear. Hence, we assume that fz(O) > O. In this case, note that I is expanded as

o < (o)"/ teqt 'r?elc$ uorlrassB aurnsss arrr'acue11 eql ueql'o = (o)"/JI'(0 <) $)tt 7(il"t se papuedxa q ./ l€rll elou 'asec srql uI

(lrl)o + z'(il'l + z'(o)"1+(o)/= ?)t f(z)

= 1(0) + Iz(O) . z + IE(O) . Z + o(lzl)

elEuelce.raq1 flaleurxo.rdde sr ('g,)/ .t"ql 'a elrlrsod frarr.aro; [r'0] * [r'0] = 'g a13ue1rere raprsuoC 'ur3rro sqlJo pooqroqq3rau e ur

in a neighborhood of the origin. Consider a rectangle R f = [0, f] positive L Then I(R f ) is approximately the rectangle

X

[0, f] for every

' [, ((o)"/- (o),/)+ s's]x P((o)"/+ (o)"/)+ D'Df

[a, a + (/z (0) + fz(O)) f) x [b, b + (/z (0) - fz(O)) f] ,

where 1(0) = a + ib. Hence, by Rengel's inequality (cf., for instance, Lehto and Virtanen [A-69), I 4.3), we obtain

ul"tqo r '(g'' f '[Og-V]ueuelrr1 pue olqerl'ecuelsurro; '';a) flrpnbaur s,1a3uag fq'acua11'g?*e = (0)/ eraq!\

Iz (0) + fz(O) ~ M(/(R f ) ) ~ Iz(O) _ IE(O)

:x ? (dDw 7 ('a)wx = K M(R

f )

'(t)o +ffi

K

+ 0(1).

'6 'snq6 13ql epnlcuoc a,ll o+ puel r 3ur11a1

Thus, letting

f

tend to 0, we conclude that

,$),t - 6)'l < (oETl6)"/ )/

K> fz(O) + fz(O) - Iz (0) - IE(O) ,

'ft)"lq j 6)tl

or equivalently, IE(O) :5 klz(O).

o

'f11ua1e,rrnbe ro

92 92

4. Quasiconformal Quasiconformal Mappings

4.1.5. 4.L.5, Other Other Fundamental F\rndamental Properties Properties of of Quasiconformal Mappings Quasiconformal Mappings proofs, two of the fundamental and important We state here, here, without without proofs, properties important properties quasiconformal mappings. on continuity continuity of quasiconformal (Mori's theorem Theorem -qc mapping Theorem 4.16. 4.16. (Mori's theorem [157]) mapping of the K-qc the unit A ff is a K ll57D If (0) = 0, disk disk..1 A onto onto itself with with ff(O) then 0, then - (rz)l < t 6 lzz11-- Z2!I/K, ,rltlK , If(zd < 161 l f ( r t ) - ff(z2)1

2 1 , 2 2E z 1 i= f Z2. A , ZI 22. ZI,Z2 € ..1,

Theorem -qc mappings fixing 0 and Eaery sequence sequenceof K Theorem 4.17. Every K-qc mappingsof C onto onto itself firing and 1I contains subsequencewhich which converges conlains a subsequence conaergesuniformly unifonnly with respect respectto the the spherical spherical distance. d,istance. function of such is again -qc. Moreover, Moreoaer, the the limit limil function such a subsequence subsequence again K I{-qc.

For proofs of of these these theorems and further further information information on quasiconformal mappings, see, pings, see,for instance, instance, Ahlfors [A-2], [A-2], and Lehto and Virtanen [A-69]. [A-69].

4.2. 4.2. Existence Theorem Theorem on Quasiconformal Quasiconformal Mappings We have have seen seen that that aa quasiconformal mapping mapping f/ of of aa domain D induces aa pJ on D which satisfies measurable function Pion satisfies ess.suPzEDlpl(z)1 ess.sup362lttt!)l < L. bounded measurable < 1. p with In this section, we shall show with section, we show the converse. converse.Namely, Namely, for every every measurable measurableP quasiconformal mapping whose whose complex ess.supzED Ip(z)1 < 1, we we construct aa quasiconformal €ss.supz(DlpQ)l < 1, dilatation dilatation is equal equal to p.

4.2.1. Preliminary Considerations Preliminary Considerations We denote the complex Banach bounded measurable measurable denote by LOO(D) I-(D) Banach space space of all bounded given by functions on aa domain D. Here, Here, the norm is given

=

pE ess.supz€Dlp!)|, L*(D). Ilplioo P e LOO(D). llpll- = ess.suPzEDlp(z)l, of LOO(D), L*(D), and andcall call Let be the II 00 < I} Let B(Dh B(D)1be theunit unitopen openball ball{p L*(D) I| lip 1} of e LOO(D) {p E llpllany element element of B(Dh B(D)1 aa Beltrami Beltrami coefficient coeficient on D. prescribed complex quasiconformal mapping with First, we with the prescribed we note that that aa quasiconformal precisely, we dilatation we have essentially unique. unique. More precisely, have the following: following: dilatation is essentially Proposition that there pr be be an an arbitrary arbitrvry element elernentof B(Dh. B(D)1. Suppose Supposethatthere Proposition 4.18. Let p pJ = p. Then quasiconformal mapping Then for exists for mapping ff with wilh the the complex cornpler dilatation PI erists a quasiconformal (D), the every mapping hho0 ff has has the the same same confortnal mapping mapping h of ff(D), the composed cornposedmapping eaery conformal complex complexdilatation dilatotion p. = p, quasiconformal mapping Conversely, for every pg = mapping g with Fg the composed composed euery quasiconformal Conaersely,for H, the 1 (D). mapping g o fconformal mapping mapping of "f ff(D). mapping go f-L is a conformal

stu-rddery l?uroJuof,rsen$

4.2. Existence Theorem on Quasiconformal Mappings

93

uo uraroaql

eluetsD(g 'Z'?

= =

'rb-1 sr 't'? ureroeql o1 f.re1oro3 ,(q tr 6 leurro;uoc8I etueq pue '-Io61€rll s&olloJtl'gl't uotltsodor4fq (O)/ uo'e'e g ol lenbas.r'-lotrl acu-ts '1xatr1'rl = trl - {ottil e^eq e/{'tI'} uo11rsodor43o;oord aql uI s€'1s.rrg/ool2'

Proof. First, as in the proof of Proposition 4.13, we have J-lhoj J-lj J-l. Next, since J-lgoj-l is equal to 0 a.e. on f(D) by Proposition 4.13, it follows that gof-l is 1-qc, and hence is conformal by Corollary 2 to Theorem 4.4. 0

uoll€lar e e^sq aruue{} ',t1en1ry 'zJ' ue^€ eq? urorJ 'et)C / / uorleluaserdarelqellns e 1eBa,n3r elqelrnse Surpug''a'r 'tualqold-g..If Eunlos 1srgrePlsuocea.'asodlnduq1 rog '"lrt - zs uorlenba1er1ue -regrprtu€rlleg eql e^losol lr.roqreplsuoceaa'I(C)B 3 r/ ua,r€ {ue ro;',tlo1q

Now, for any given J-l E B(Ch, we consider how to solve the Beltrami differential equation fz = J-lfz. For this purpose, we consider first solving the a-problem, i.e., finding a suitable f from the given fz. Actually, if we get a suitable representation f = G(fz) , then we have a relation fz = G(fz)z = G(J-lfz)z

'(trt)C='el)g=zt

,(q uarrrEfl rurou aql eJeqA{'C uo alqerSelur a.re alll 1etll qcns c uo / suollcunJ elq€rns?au 1e 3o aceds qc€ueg xalduroc eq+ eq (C) n lel 'oo > d ; I qll^{ d f.ra,ra rog 'uolleruroJsuerl fqcne3 'pueq Jaqlo eql uo aql m ua\our1iflecrsselc u e/ urog / lanrlsuooar o1 fem e 'uorlenba aql p ((rl)grl)9 = t ua,u3 ruerlleg ''l pue r/ uaaallaq uorlnlos e ur?lqo a,r,r,'(r/)g = zt tjd'rol aql u-r 11 3ut1t.ra,rag

between J-l and fz. Rewriting it in the form fz = F(J-l) , we obtain a solution f = G(J-lF(J-l)) of the given Beltrami equation. On the other hand, a way to reconstruct f from fz is classically known as the Cauchy transformation. For every p with 1 ~ p < 00, let LP(C) be the complex Banach space of all measurable functions f on C such that Ifl P are integrable on C, where the norm is given by

(fL

= oil,ltl ot,(np,por:[f)

'(r)ar),t

Ilhllp =

IhIPdXdY) liP,

hE LP(C).

'uralqord-g aql e^los ol Frluasse x opru"tot s,ntadtuo4 letrsselc Euu,ro11o;eqa

The following classical Pompeiu's formula is essential to solve the a-problem. tt puo lt s7uau;1afr,qpeTuesaul^teJo searry)auap1otq.tod PuorlnquqrP esoqn C uo uotTeunt cnonutluoz o ?q I pI '@ > d > ?, qfn d qI 'At'V uollrsodorg

Propositiun 4.19. Fix p with 2 < p < 00. Let f be a continuous function on C whose distributional partial derivatives are represented by elements fz and fz of LP(C). Then f satisfies

sa{s4nst uaqJ '(dn

Io

a)),nP,pfi'ff +_#"'[,+=o)/ f(() = ~ [ 211"l

1

&B

.!. j~ [

f(z)dz _ Z -

(

1

fz(z) dxdy,

B Z -

11"

(E B

(

for every open disk B in C.

'C u! g tlnp uado tueaa .tot

se,rr3elnurro; s(uearp 'u fla,ra rog ',tpe.r1rqre g ) ) lutod e xg aauanbesSutqloours-o7 ue a4e1 '1xa1i pu"'g o1 lcadsar qgi'a / roJ I*{"/} 'flqenbaur s(raploH fq s.uo11o; uorlresse aq? 'g uo alqe.rEalulsI rl() - z)/11 acurg 'trsJ u1 'alqer3alur f1a1n1osq€sI apls 'I = b/I +dlI fq paugep aq (Z >)D lal '1sq1g pueq lq3rr eql uo turel puo?es arll Jo puer8alut eqt l"ql a?ou erlr /oor2r

Proof. First, we note that the integrand of the second term on the right hand side is absolutely integrable. In fact, let q « 2) be defined by l/p + l/q = 1. Since 11/(z-(W is integrable on B, the assertion follows by Holder's inequality. Next, take an LP-smoothing sequence {fn}~=l for f with respect to fJ, and fix a point ( E B arbitrarily. For every n, Green's formula gives

[ (fnh(z) di

,Pv,Pffi"il+-ffi"f +=e)"! =~ [ 2n 1&B

fn(z)dz _ ~ j~ Z -

(

211"l

'uotllesse eql aAeq e^r "("/) pt" 'g uo ttpuroyun / +- u/ aautg

Since fn -+ f uniformly on fJ, and (fnh we have the assertion.

1B

Z -

1\ dz.

(

tr '@me u se 'flartlcadsa.r '(g)a7 ul zt *

fn(()

-+

fz in LP(B), respectively, as n

-+ 00,

0

3ur11es,tq (C)aZ uo d roleredo reeutl € augap eA\

We define a linear operator P on LP(C) by setting

( o ' r ) ' c r ) ' ( c ) )atzt ' o p (, p 1 -+ ) e t u " f+l -=o ) q a Ph(()

= _.!. jr[ h(z) (_1__ .!.) dxdy, 11"

lc

z- (

z

hE LP(C), (

EC.

(4.9)

94 94

4. 4. Quasiconformal Mappings QuasiconformalMappings

Then we have have the following: following: q and Lemma L e m m a 4.20. 4 . 2 O . For F o r every e a e r y p with w i t h 22 < a n d ffor e u e r yh E L p ( C ) , Ph Ph < p < < 00 e LP(C), o r every is a unifonnly function on - 2/p), and uniformly Holder Hiilder continuous continuousfunction (7-2/p), on C, with wilh exponent erponenl (1 and satisfies satisfiesPh(O) Ph(0) = O. 0. Moreover, Moreouer, P Pff satisfies satisfies (Ph)a = h

on on C in in the lhe sense senseof distribution. distribution.

proof of Proposition Proof. we shall Prool. First, First, as as in the the proof Proposition4.19, 4.19,we shall show showthat the the integral integral (4.9) is purpose,define well-defined.For on on the the right right hand hand side sideof (4.9) is well-defined. For this this purpose, defineqq by the the equationl/p \/p+I/q equation + l/q = 1.1. Since Since 1

11

(C

---zz -( - C - -zr = z(z r 1 r- -()q belongs belongsto Lq(C), Holder's Hcilder'sinequality inequalityimplies impliesthat 1 (a h IPh(() : : ; );lI I P A (IC ; * n_ l()l cIlq ;ll
( 00. @. <

Further, the variable, variable,we we have have when ( :I 0, by by changing changingthe Further, when I 0, q . t tqQ trt 1 lo, 2 2, f l l axay=lCl2-z'II d,rda. l--;----l II l--:--l L ) l dxdy. J J c l z(z z ( z- () dxdy 1(1 - qJ J g lz(z z ( z- 1)

fL I

~ C I) l

fL I

=

~ I

Hence, depending only on p such such that Hence, there is aa constant constant K I(op depending

l P h ( ( )Sl K r l l h l l o ' l c l ' - ' t(o€, c , e + 0 .

=

(4.10) (4.10)

=

(4.10)is valid even evenwhen when ( = O. Since 0. Ph(0) - 00 by by the the definition, definition,(4.10) is valid SincePh(O) we obtain Next, Then we obtain Next, set set h hr(r) h(z + + (d. 1 (z) = h(z Ct).Then

phlCz(,)=-+il"h(z+c,)(4+_6- l) o,o, 1

f f

/

1

zrJJc "\z-(z = Ph(cz)- Ph((r).

1

\

z-Q/

(4.10),we we conclude Combining this with (4.10), concludethat Combiningthis 1 2P pn4)l ::::; 'lc,-- (21 c, Kollhllo - / , IPh((d ·1(1 (1,(2 crlt-'to, cr,czEQC, S Kpllhllp lph((r)-- Ph((2)1

(4.11) (4.11)

with exponent or equivalently, that Ph is aa uniformly uniformly Holder continuous continuous function with equivalently, that | - 2/p. 2/P' 1(C) such that sequence{h such that To show show the second second assertion, assertion, take take aa sequence Cf (C) n };:"=l in COO {h"}Lr ---+00. - h"llop -+ * 0 as example, as as in Ilh as n -+ oo. (Such (Such aa sequence sequenceis is constructed, constructed, for example, llh - hnll we have the proof of Lemma 4.5.) 4.5.) Then for every every h h., have n , we

96

s8urddel4l l"uroluof,rspn$

4.2. Existence Theorem on Quasiconformal Mappings

95

uo ruaroaqJ af,ualsrxg 'Z''

(^0.r6{;s"ll +)# =eFyta) =

(_~

8_

8(

71"

Jr f

JrJcf hn(zz+ () dXd y)

lzt'v)

(Phnk(()

(4.12)

cff v )-z 'np,pG)1",1 Jlt-=

= _~

(hn)z(z) dxdy.

Jc

71"

z-(

Hence Green's formula gives

se,rr3elnurro; s.uaalg ecueg

.()),q= rpY*{=tt-zt! = (illeltd) ' (z)"tl +t;ri t I 1 . 1 = lim -2 f-+O

7I"Z J{lz-(I=f}

hn(Z() dz = hn (().

z-

'(C)JC 1eBa,ra.

(Ph n )((()

'relnctlred u1 lt