や
ON SOME PROPERTIES OF A TEICHMULLER MAPPING
BYYAsusHI MIYAHARA
A TeiclmU皿er mapping has an important property in the theory of quasiconformal homeomorphisms of． a compact Riemann surface． It gives the extremal quasicon− formal mapping which、 minimizes the max㎞al dilatation in the farnily of a皿quasi− conformal homeomorphisms homotopic to it． On the other．hanq， a TeichrnUller mapping is considered to be a hamonic mappi皿g． But， the property of a Teich− mU皿er mapping as a harmonic mapping seems not to be㎞own． It is the．purpose． of the present paper to show an extremal propepty of a TeictmU皿er mapping in a ロ の family of ha頂onlc mappmgs・1．Let R andぷbe Compact Riemann surfaces which are homeomorphic andη＝
ρ佃）1吻12be a collfbmal metτic on 8， whereρ佃）is a positive and continuoUs func− tion of a local parameter w on 51． Now we consider an orientation−preserving homeo− morphism∫of R onto S which is L2・derivable， that is， w＝f（z） has genera五zed partial derivatives which are square integrable， where w＝f（z）denotes a 10cal representation of∫fbr local parameters z and w on R and S res】Figctively． The metricηon 5 is trans佃med by the mapP垣9∫as f（）皿ows； ・一・（・）1醐・一・㈹裟此＋髪凌2 −・㈹（霧12＋1裟12）1此1・＋・R・［・㈹）鵠此・］・The fbrm
Of∂〆 ρ（f（z）） 改2 ∂z∂乏 in the bracket of the second term defines a quadratic qifferentials on R． If it is an analytic quadratic di丘brential，∫is called a』r〃lonic〃lapPing relative toη（cf・［3］， ［6］）．We denote it by∫fn and set ・・ω一㈱）裟裟・ We call the analytic quadratic di丘’erential qv（z）∼た20n R to be attached to the har− monic mapPing fv． A conformal metric V＝ρ（w）ldw 12 iS said to be nor〃alized if it satiSfies the condition ∬，ρ（・）・・dv−1（・＋⑳・When a nomlaUzed confbrmal metricηo皿5and a homotopy dassαof orienta一
［36］ON SOME PROPERTIES OF A TEICHMU】［LER MAPPING
37 tion−preserving homeomorphisms of R onto S are arbitrarily given， there alwaysexists a ha㎜onic mapping五relative toηwhich圃ongs toα． In血ct， Shibata’s
paper［6］has shown that a harmollic mapping fv is obtained aS a homcomorphis血 which min㎞izes the 1）ouglaぷ一Diric〃6t fuitctional 瑚一∬。ρ㈹）（3212＋霧2）勅 in the family 8γ， M of all orientation−preserving homeomorphisms f of R ontoぷliatisfy− ing the fbllowing conditions； （i）∫belongs to the homotopy classα， （ii） ∫andノ「−1 are L2−derivable， （・・）ll。ρ㈹）（裟2＋裟2）d・dy≦K＋K−・・ where K is the maximal dilatation of an arbitrarily fixed quasiconfbrmal mapping belonging to α， （・・）∬、・（f−・⑩））（筈2＋荒2）͡M（K±・K−・）・ ．whereγ＝λ（訓此12 is a normakzed collfbrmal metτic on R， andハイis a positive co皿一 stant． In this paper， by a hamonic mapping fv we shall mean the hamonic map． ping which minimizes Iv［∫］in the血mily．審r， M． If∫is． an orientation−preserving and L2−derivable homeomorphism of」R ontoぷ，then
裏2一墓2≧・ almost everywhere in each parametric disk on R（cf．［4］）． It is also known that f is a measurable mapPi皿g and m蛎E）一∬。（砦2−322＞d・dy fbr．every measurable set E contained in a parametric disk on R（cf．［4］）． When anormaUzed confbr血al metricη＝ρ（w）ldw l20n S is given， we have ・・［∫コ＋・∬。ρ㈹）裏2血⑳， Slnce ’ ∬。ρ㈹）（3212一裟2）d・dy−1・ Hence，賜［∫］≧1 fbr any mapping∫ifηis normalized． Furttiermore the equality holds if and only if f is ．conformal． We have obtained the fo皿owing lemma．LEMMA 1・Lθ’R繊8bθω〃rpact Rie％nηぷ〃rfaceぷ晒c乃are乃o〃leo〃iorph’c
αη砺加αη0〃nalized C・nf・〃nal〃2β〃ic on S．ザ∫輌ぷan・r∫θπ’a伽一preser吻9励 L2−derivab∼e乃0〃2．αη0／P乃’ぷ〃i of R oη’O S， then In［∫］≧1． Here， the equal妙加倣rぴand onlyぴ∫’ぷa eonfo〃na’〃2卿’カ9∼38
Y．MIYAHARA
The fb皿owing le㎜a is weU㎞own．
LEMMA 2．．Let S加aco〃rρac’Rie〃mnn surfacq．ぷ｛ゾgenuぷ9＞1・ lf f iぷαeo〃− formal〃laPlガngげ8ρnto匡’sell㍉vhich is乃0〃10topic’O the identitγ，’乃επ∫匡ぷthe ide〃一 ガリ7（cf」 ［2］）．2．Let R and S be compact Riemann s曲ces which are homeomorphic and∫
be an orientation−preserving and L2−derivable homeomorphism of R op． to S． We ca皿 ∫a Teich〃1ti〃6r〃mPIガ〃g of R Onto S， if there exist a positive number k＜1 and． a pa辻of analytic． quadratic di丘’erentialsψ（z）dz2 andψ（w）吻20n・R and S respectively， such that｝v＝ノてz）iS related by the equation （1） ンψω吻一∼／q（z）dz＋んMψ（z）dz， where z and w are local parameters on R and S respectively（c￡［1］）． In（1）it is understood that∼／爾「denotes the complex conjugate of∼x；q：7；（z）， and the sign． of ViOplih（z） wM detern血e the sign of Vψ（w）． For a TeicimU皿er mapping the◎onstant k is uniquely deterinined， while q（z）dz2 and ￠（w）dw2 are determined up to a common positive血ctor． There exists a unique Teichm皿er mapping in eaCh homotopy dass ．of orientation−preserving homeomorphisms of R onto S（cf．［2］）． From（1）it fbllows that Mψ㈹）32−v。ω， （2） Mψ㈹）票一砺（・，⇒consequently we have
Iψ㈹）1霊輌（・）・ Hence the TeichlnOller mapp垣g∫is hatmonic relative to the confbrmal metric lψ（w）ll吻120n S， if we extend the de丘hition of a harlno皿ic mappi皿g by admitt桓g of confbrmal metrics which have isolated zeros． The正efbre we have the fbHo亘ng theorem． THEOREM 1． Lε’Rand S be co〃ψαc’・Rie〃mnnぷ〃η勉cθぷwhich are ho〃leo〃iorphie and f be伽Teichmti脆r mappingげ・R on’o S・which iぷa∬oeiated・with aク0ぷitive constan’k〈1and a pair of quadratic晒’επ’輌α1∫9（z）此2 a〃4ψ（w）吻20ηRaud 3 respectively． Then f iぷα加朋・〃ic m卿ing．rela伽θ’・’he C・ψ・励1〃・etric lψωl l酬2，a〃d kq（Z）dz2 isど乃θα”ached quadraガC雌rential’O it・ If f is the TeiclmUller mapping assoc輌ated with k（0＜k＜1）and W2，ψ吻2）， we obtain from（2）・ 裟一・1鎧1裟・
that is，∫satis丘es the Beltrami equation with the coe伍cient kq（z）／1ψ（z）1（c￡［2］）． We notice that the inverse mapping f’i is also a TeiclmUller． mapping and．．that itON SOME PROPERTIES OF A TEICHMUI LER MAPPING
39 satis丘es the Beltrami equation with the coeMcient−kψ（ng）／1ψ（w）1． we have s㏄n that a Teichmti皿er・mapping is a harmonic mapping with a constant dilatation． The cnov6rse is also伽e， since we find the following lemma． The proof is㎞ediate．LEMMA 3・lf f iぷa． harmonic〃仰吻9げRonto S whoぷθdilataガon匡ぷco斑砺，
’乃θ〃fiぷαTeichmti〃er mapping． Leげbe a harmonic．mapping of R onto S relativ6 to I li（w）ll姻2， Whereψ㊥）吻2 is an analytic quadratic d．ifferential onぷ・We setρ（w）＝1ψωl and（・）’．‘
@・（・）一・㈹）霊・
By assumptiop q（z）is analytic， so qi≡…0． Since．ρ⑩）is twice d舐｝rentiable，∫is so by Morrey，s result（c￡［5］）． We obtain丘om（3）by a simple computation ∂2f ∂f・∂f （4） ρ（∫（z）） ＋ρ旬（1てz）） ＝0． ∂z∂乞 ∂z∂zSince
・・ω一ψ’（w）1ψ（w2ψ（w））1・ we have丘01n（4）（・） ・ψ㈹）器＋ψ’㈹）裟髪一・・
If we set ・・ω一ψ㈹）（砺）2…（・）一ψ㈹）（裟）2， then（5）implies that ipi（z） and q2（z）are analytic． Sin㏄ψ1（z）and Op2（z） define quad− ratic differentials o11．R，ψ2（z）／q1（z）determines a meromorphic fUnction on．R． Then it is constant， for 綴卦裟2／袈≦1・nR・ Hence it fbllows that 砦／32≡・・n…， consequently∫has a constant dilatation． Therefbre， by LEMMA 3，∫is a TeichmU皿er mapping． We have thus proved the following thcorem． THEOREM 2・ Let R and S be co〃ψαc’Rie〃lannぷurfaceぷw乃輌c乃are〃o〃leo〃lorp乃匡c and f be a’ harm・nic〃2卿輌η9・ザR碗・S・rela’ive’・1ψ（W川姻2， where di（w）吻2 is a〃卿砂’ie q〃α砺ガcぴ’επがal oη＆η∼θπ∫∫ぷαTeic加2∬∼已〃2卿晦 3．Let R and“S be compact Riemann surfaces of genus g＞1 which are not con− formally equivalent， and letαbe an arbitrary homotopy class of orientation−preserv− ing homeomorphisms of R ontoぷ． If fo is the TeichmU皿er mapping which belongs toαand is associated with a constant k and an analytic quadratic di丘brentialψo（z）dz2 0n R， we have40
Y．MIYAHARA
（・） 普一・1‘i81普
and
（・） 筈一暢
for local parameters z and w on R and S respectively． Iffv is a harmonic mapping belonging to a for an arl）itrary normalized conformal metricη＝ρ（w）［ dw ［2， then（・） ・・（z）一・㈹砦
de丘nes an analytic quadratic differential on・R・ an L2−derivable homeomorphi諏of S onto itself． functional of this mapping； Ofv ∂7− The composite mapPing fn・fo−1 is We consider the Douglas−Dirichlet ・V［赫・コー∬，ρ（鯛・））［ρ（雛λ2＋亘籠一1）2］d・dv・ If we change the variable by w＝、fe（z）， then班航・コーll。ρ㈱）［璽讐一112＋醇㍑己一2］（」芸2−」諺2）拗・
By use of the relations 翫 OfeOf。一㌧ ∂z 処一1＿＿ ∂z
∂W−∂ん2 ∂ん2’ ∂W ∂f・2＿Of・2’
∂z ∂z ∂z ∂z we obtain ∂（fn・f。−1∂w）2＋∂（漂一1）2一普答1＋誓答12＋砦答1＋裟酷12
（讐12＋團2）（普2＋普2）一・R・（普普砦裟）
罐2暢2）2 ’ Hence， it fbUows by（6），（7）and（8）that ∂fe 2＋・∂f・2（・）班赫・コー∬。ρ㈲）（裟2＋誓2腸・一畿・d・dy
∂z ∂z Of。2 −・k∬。R・｛・㈹裟裟一1膓1｛薪｝。f。・警ぴ・d・d・ ∂z ∂z −1圭21∬。ρ㈹（裟2＋髪2）d・dy 1一毒R・｛∬。・・（・）膿｝IW｝・ Now we denote by（∼（R）the linear space consisting of aU anaIytic quadratic d距 ‘ferentials on R． For brevity， we shaU simply denote byψan elementψfz）dz20f ＠（R）． Let us define a linear fUnctio皿a1
（1・） ・［・コー∬．・・（z）1畿｝1卿（・∈ρ㈹）
On（∼（R）． By reWriting（9）We haVe（11） ・n［f・・f・一・コー｝彗；刷一1竺、，R・｛・［・・D・
We know by LEMMA l that lv［fn・fo−1］≧1． Since fv minimiZes In［∫］in a㏄rta㎞ 飽mily審r， M containing fo， we find that ・v［fnコ≦刷一∬。ρ㈱（讐2＋讐2）d・dy −∼！，，（。）藁il；＋1藁i： ∂w ∂万 1十kZ 1−k2’ Hence， it fbllows from（11）that 1≦G彗；）2−rk−kk、 R・｛・［・・コ｝・consequently
（12） R・｛・［・・コ｝≦ft−k、・
If the equality holds in（12）， then it follows from（11）that In［f，・fo−1］＝1． Hence f》・fo−1 is confbmal by LEMMA 1． Therefore， by LEMMA 2 we see that fv・fo−1 is the identity， namely fv ＝fe， for fv・fo−1 is homotopic to the identity． It is dear that the equality holds in（12）if∫fv ＝fo． Thus we have proved the following thcorem． THEOREM 3． Lθ’R‘z，id 5 beω〃lpact Rie〃m〃〃surfaceぷθゾge〃us 9＞1 w乃fc乃are π・τωψ朋・∼ly equivalen’，απ〃εオf・ be’乃ε蹴励∼ler〃卿加9〆R・〃・ぷwhich ∫ぷa∬ocia’e4 w輌th oクoぷ∫’∫昭ω〃5励’k〈1 andαπα朋砂’∫c quadra’∫c diVeren’ial q・oπR．Then we加りθ
R・｛・［・・コ｝≦1ξ、， ノb’θαC乃〃0’〃ialized eonfor％1 metricηon S， where・L・iぷ’he・line〃ルπC’∫0〃al on e（R） defined bγ（10）， and 9n∫ぷ’he at’ached quadra’∫cぴ’επ’ia’to a加rmonic〃卿∫〃9 五乃0〃10toρ’C’o f。． The equali’γ加ldsぴand onlyヴ五一fe． This theorem shows a certain extremal property of a Teictm皿er mapping， that is， a TeichmO皿er mapping fo maximizes the value Re｛L［￠v］｝in the飽mily of har− monic mappingS∫fv homotopic to fo． We sha皿state another expression of T朋oREM 3． 4． Now we de丘11e an inner product in the space e（R）as fbllows： （・・…）一∬。・・（・）・3（・）・ω一・dUdy42
Y．MIYAHARA
forψ1，ψ2∈ρ㈹， whereλ（z）1dz 12 is the hyperbo血c metric on」R． we set ‖・ll一ン（・，・）一［∬。1・（・）1・λω一id・d・］’・ Sin㏄ρ（R）is血並te d㎞ensiona1， it is complete with respect to this norm． For every rp∈αR） 1・［・コ1≦∬。1・（・）1・・d・ ＜［∬。・（拗］‘［∬。IPtz）1・・ω一・d・d・］’ 一σ㈹illqll． whereσ（R）denotes the hyperbolic measure of」R． Hence也e lillear∩mctional L on Q（R）is bounded． By Riesz’theorem thele exists a unique elementψ＊∈0（R）suchthat
L［ψ］＝（ψ，ψ＊） f（）ra11ψ∈（2（R） We denote by N（R） the set of all q∈Q（R） such as L［ψ］＝0． N（R）is a closed sub− space ofρ（R）． Let∧r（R）⊥be its orthogonal complement． Then we have ρ㈹＝ハr（R）㊥ハr（R）⊥． The space N（R）consists of a皿ψ∈（≧（R）orthogonal toψ＊， consequently its dimen・ sion is 3g−4， fbT the dimension of（］（R）is 3g−3． Hence the d㎞ension of N（R）⊥ is 1． Let fo be the Teichm皿er mapping associated with a constant k and an analytic quadratic differential￠o on R． Since qo is detennined uniquely up to a posiUve constant factor， we may nonna血ze it with ∬。1・・ωlw−1・ We can decomposeψo uniquely in the f（）皿 （13） q。−q。＊＋q。＊＊ （ψ・＊∈N㈹⊥，ψ・＊＊∈N㈹）・ For every normalized co皿formal metricηon S we d㏄compose the attached quadratic ditferentialψηto a hamonic mapPi 19 fv homotopic to fo as fo皿ows； ψη＝ψη＊十ψη＊＊ （ψη＊∈N（R）⊥，ψη＊＊∈N（R））． If we set qn＊＝avgo＊， where aηis a complex number， then L［OPe］＝L［qη＊］＝＝aηL［q。＊］−avL［￠・］ 一・a・ 撃戟Di…（・）1・・d・一… Therefbre， by THEoREM 3 we get k Rea・≦1＿ん・・ Thus we have obta㎞ed the fbllowing theorem． THEoREM 4． Under the hypotゐeぷ’ぷρゾTHEoREM 3， le’ηbεαπαrbξ〃ary nor〃1alizedconformal〃letric O〃Sandノ）be a har〃lonic〃mpping乃0〃lotopie’0／6． 」rf Wθdeeom− pose〃吻uely’舵attached quadratic differentia’￠v to fn in the／b㎜ ψη＝aηePO＊十ψη＊＊ （qv＊＊∈N（R））， whereψo＊ is the ele〃lent q〆ハ1（」R）⊥ deゾ『n昭d by（13）and aηiぷαco〃ψ∼ex nu〃必ε’， there k Re・・≦1＿k・・ The eguality ho倣rぴand ontyぴノ）±『ん．
