ALEMMA ON A FAMILY OF HARMONIC MAPPINGS
BY
YAsusHI M】「YAHARA
■ ’
Let R and∫be compact Riemann surfaces of genus g， and letη＝ρ（w）1dw 12 be
aconfbrmal Inetric on S， whereρ（w）is positive and continuous with respect to each
local parameter w＝u十iv onぷ． We ca11ηanomlalized confbrmal metric， if it
satisfies
∬、・（・）・・d・・一・1・
Let ’
?@be an orientation−preserving homeomorphism of R ontoぷ． We assume that
∫is L2−derivable， that is， w＝f（z）has genera1セed partial derivatives which are square
integrable， where w＝∫（z）denotes a local representation of f f（）r local parameters z
and w on R and S resp㏄tively． We set
・・［∫コー∬。ρ㈹）（裟2＋裟2）w・『
For a normakzed confbrmal metricη＝ρ⑩）ldw l 20n S， an orientation−preservi lg
and L2−derivable homeomorphism∫of、R onto S is ca皿ed a harmonic mapping rela−
tive toη， if the quadratic di旺brential
Of’ Of
ρ（f（z））
此2
∂z∂z
is analytic． When a nomla丘zed conforma1 metric Op on S and a homotopy classα
of orientatio皿一preServing homeomorphisms of R ontoぷare arbitrarily given， there
exists a harmonic mapping relative toηwhich belongs toα（c￡［1］）．． We denote
it byノ）， and we set
・・（・）一・働讐讐・
The quadratic d血fferentialψη（z）dz2 is said to be attached toノ）． In the paper ［11
it is proved that a harmonic mapping fv is obtained as a homeomorphism which
mi舳es班∫］hl the family｛Yr，」if of all homθomorphisms∫satisfyi皿g the fbllow−
ing conditions：
（i）∫belongs to the homotopy classα，
（il）∫and∫㌔1 are L2−derivable，
価）班∫］≦K＋．K−1，
where K is the max口nal dilatation of a fixed quasiconf（）㎝al Inapping belonging to
α，alld
（・・） ∬，・（∫一・（・））（筈：12＋1答12）d・dv＜M（K＋K−・）・
’
●
［5］
6
．Y． MIYAHARA
’
whereγ＝Z（z）ldz I 2 is a confbmal metric on R， and M is a positive constant． In
this paper， by a hamonic mapping we shall mean the homeomorphism which min−
imizes、班∫］in． a certain family魯γ， M． Therefore we have
（1） ち［f？］≦K＋K’“1． ・
If g≧2， then the universal covering surfaces of R and S are con丘）㎝any equiv−
alent to unit disksσ＝｛lzl＜1｝’1 and V＝＝｛l wl＜1｝resp．ectively． From now on， we
iden卵σandγwith the universal cove血g surfaees of R and S respectively． We
denote by G and H the groups of cover transfbrmations ofσand V over・R andぶ，
respectively． G．and H are properly discont口luous grOups of血1ear transfbrmations．
When ・a notmaliZe 1 confo頂al Metric’ s・ p（w）1・dw l 2 is ’given on S， we can define a
conthluous functio皿ρ（w）on V such that
ρ（B（w））1B’（w）12＝ρ（り f（）ra皿B∈H．
Ifwe set
Mn＝inf”∈vρ（w），
then mv is positive． For a positive number c， we denote 1）y 2， the family of．all
no頂aHzed Confor al metricsηon S sudl as
（2） 〃ln≧c．
Ahomeomorphism f or R onto S can be extended to a homeomorphism｝w・＝f（z）’
ofσonto V． The extension w＝∫ωis llot uniquely deternimed．
We shall show the fbnowing result．
Lε’C be a p・㊨θη醐bθ7励導加α吻吻㎡乃・施・〃卿hisms Wr五ω・fσ
0〃’0γノbr a”η∈2』， where eαC乃W「fn（Z）」ぷa〃α崩τ〃のノextensionρゾaha〃nonic
mapping fv加αぬε4乃0〃20鋤ッclasぷ． Then S∫ぷa no〃妬1允〃吻0πひ
、
It is su伍cient to show『．that蔓）is equic皿tinuous on lzl＜アo fbrアo with O＜γo＜1．
Suppose that導．is not equicontinu．ous o皿lzl＜ro． Then there exist a constantδ＞0，
aseqロ㎝㏄｛ηfi｝in 2c， and two・sequences｛a＃｝，｛●。｝il l z l＜r。 such that．．
（3） ［an−b。1→0 （n→。。），
（’4） 1∫（q）國一∫（bヵ）1≧δ （n＝1〔2， … ）◆
By（3）， it函sts a point zo in l zo l≦ro whiCh is an accumulation point of｛偽｝and
｛b”｝．We may． assume that a。 a皿d b外converge to zo．・
Now we丘X art．ll’ゴwith O＜r1＜1一γo’and setε外→aカLb．1． Then we may assume
that O≦ε蕗くrl fbr allπ． Each annulus 二． ・・
A。：ε。≦lz−a。1．≦r、 ．．・ ’一．
is contained j皿σ． The length Lヵ＠）of the image ri of the・circle l Z・−a外Iir under
the mapping w＝＝fv．（z） is finite for almost all r∈［ε。， r1］． It’is evident by（4）that
・ ’．．’，． L6ω≧1允（傷）−f，。（b。）1≧δ ・・：1
for almost a皿r∈［ε。， rl］． Consequently， it飴Uows知m（2）that． ．二．．．．∴
蹴ω・と（い1）2≦÷（1砺禰剛2’
A巳MMA ON A FAMILY OF HARMONIC MAPPINGS 7
≦÷［∼、＿＿＿＿1＿＿∼／禰一（莞一 ＋ 誓一Dl4zl］2，
where we setηヵ＝ρヵ⑩）1吻12． By use of Schwarz’inequaUty
一δ・≦÷11。．。。1。夙．。。1．．ρ・（fn・ω）（筆・＋誓・）2同
≦4芸711。．。．］．．P・㈱）（’af・・2＋Of・・2∂z． ∂z）ld21・
w・di・id・th・b・th・id・・by・and血t・9r・te ab・ut・丘・m・。 t。。、， th。n
・・1・g㌃≦÷∬。。・・㈱）（警12＋警2）卿・’
Since．4。 are contahled in a oompact subset ofσ， thとy’are covered by a finite num−
ber of f㎞皿damenlal polygons P1，1㌧，…， Pi qf〈｝l Therefore， 二
・・1・g㌃≦4：π∬。ρ・（f・・（・））障2＋誓・2）妙一4！klil・−Zin。［五β・
By（1）， we have
δ・1。9⊥≦4kπ（K＋K−・） 、．
ε”
c
オbr a皿η． Sin㏄εヵ→0， we have a contradiction．
Next， we剛s伍te another proof of t顕emma． The attach斑q皿dratic d醗ren−
tial・ψv（z）此弩toノ）can be extended to an analytic functionψv（z）On σsUch that
． ψη（A（z））A’（z）2＝ψη（z）・ ］bra皿A∈G．
Now we 6x an arbitrary r with O＜×1 and choose an rl with rぐ1＜1． Let k be
the number of血ndamental polygons of G which intersect l z l Sri． Then
∬同．．、1・・（・）1・・dy−＝∬，。、．．、ρ蹴讐芸卿
’ ．≦Tk∬。ρ㈱）（劉2＋髪2）w
consequently， we obtahl by（1）
（・） ∬同．。、1・・ω1勧≦Tk（K＋r・）
tor every normalized conformal met亘cη． We setδ＝rl−r． Sinceψη（z）is analytic，
we have
…（・｝一。5・∬、、．。1．、・・（㈱・（ζ一ξ＋・・）
for l z l≦r． Accordingly， it fbllows from（5）that
l・・ω1≦。∼、−i−k（K＋K−・）・
lfbr l z l．≦r and fbr allη∈、Ω￠． Namely，｛opr、（z）｝is unifbm皿y bounded on l z l≦r fbr
・au rt∈2e・1s血ce fv is o亘entation−preserving， we㎞ow that l afn／∂zl≧1砺／∂z l．
・Consequently， by use of（2）w6 have
1・・ω1≧・讐2・
lHence｛Ofv／∂z一｝is uniformly bounded on l z l≦r for all Op∈2c． We denote by M（r）
1
8 Y．MIYAHARA
an upper bound of lafn／∂Zl on lzl≦r fbr all T∈32c．
By mean・・f gC…aliZ・d・G・㏄・’・f・・mUl・th・f・ll・wi・g・e1・ti・n i・ea・ily・d・「i”ed；
（・） f”（・）一、｝／∼j、、．r、幾・ζ一÷∬。1．．、ZIEtS？（9dξdn
f・・ lzl≦・（・f，［2コ）． H・・e， th・i・t・9・al・f th・・㏄・・d t・㎜i・ab・・lut・ly・・nve「gent・
f（）rfv，ξis bounded on lζ1＜r1． By use of（6），
fv（・・）−f・（・・）一箒・！i、、。．、⊇9−。，）・ζ一Z1二Z2111、1．．、（ζ三ll｛摯。，）・ξ吻
fbr any two Points zl， z2 in l zl≦r・ since
lfn（ζ）1≦1， lf。，i（ζ）1≦Mω
fOr lζ1＜r・， we have 、
lf，（・、）−f”（・，）1≦1・・一・・1｛（r、≒），＋M￡1）∬；。．．、1ζ一需一。，1｝・
By means・of ihe est㎞ate
（・）’
@∬，、1．．、1ζ一：1書一z、1≦・・11・g1・・一・・“＋・・n・…
wρcan deduce
（8） lfn（z、）−f，（z・）1≦c（・）lz・−z・111・glzrz・1L
wh。，e Cωi・a・・n・tant・d・p・nd・nt・nly・f・・（（7）h・・been p・・v・d in［2コ・）The
・est㎞。t。（8）h・ld・u曲・面ly・f・・a皿η∈2c・Thu・w・hav・p・・v・d th・口i・equi−
continuouS on lzl≦r fbr any r with O＜r＜1．
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