The Uniformization Theorem
Author(s): William Abikoff
Source: The American Mathematical Monthly, Vol. 88, No. 8 (Oct., 1981), pp. 574-592
Published by: Mathematical Association of America
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574
WILLIAM ABIKOFF
i pis'ma, p. 352. Cited in G. J. Tee, "Sofya Vasil'yevnaKovalevskaya,"
22. Kovalevskaya,Vospominaniya
MathematicalChronicle,5 (1977) 132- 133.
i pis'ma, p. 532. Cited in Tee, p. 135.
23. Kovalevskaya,Vospominaniya
p. 171.
1891.Cited in Leffler,
Speech as Rectorof StockholmUniversity,
Commemorative
24. G. Mittag-Leffler,
25. L. Kronecker,Journ.furdie reineundangewandteMathematik,BD CVIII, March 1891, 1-18.
"Weierstrasset Sonja Kowalewsky,"Acta Mathematica,39 (1923) 170.
26. G. Mittag-Leffler,
Journalfur die reineund angewandte
27. S. Kovalevsky,"Zur Theorieder partiellenDifferentialgleichungen,"
Mathematik80 (1875) 1-32.
Klasse von Abel'scherIntegrale3-tenRanges auf elliptische
28. _
_, "Uber die Reductioneinerbestimmten
Integrale,"Acta Mathematica,4 (1884) 393-414.
zu Laplace's Untersuchunguber die Gestalt des Saturnringes,"
_,
"Zusatze und Bermerkungen
29. _
111 (1885) 37-48.
Nachrichten
Astronomische
Mitteln,"Acta Mathematica,6 (1883) 249-304.
30.
,"Uber die Brechungdes Lichtesin cristallinischen
_, "Sur la propagationde la lumieredans un milieu cristallise,"Comptesrendusdes seances de
31.
I 'academiedes sciences,98 (1884) 356-357.
kademiens
af Kongl. Vetenskaps-A
medium,"Ofversigt
uti ettkristalliniskt
32. _ _, "Om ljusetsfortplantning
41 (1884) 119-121.
Forhandlinger,
33. _
_, "Sur le problemede la rotationd'un corps solide autourd'un point fixe,"Acta Mathematica,12
(1889) 177-232.
, "Memoire sur un cas particulierdu problemede la rotationd'un corps pesant autourd'un point
34. _
par diverssavants
du temps,"Memoirespresentes
s'effectuea l'aide de fonctionsultraelliptiques
fixe,ou I'integration
a l'academie des sciencesde l'institutnationalde France,31 (1890) 1-62.
qui dfinit la rotationd'un corps solide
35. _
_, "Sur une proprietedu systemed'equationsdifferentielles
autourd'un point fixe,"Acta Mathematica,14 (1890) 81-93.
36.
__,
"Sur un theoreme de M. Bruns," Acta Mathematica, 15 (1891) 45-52.
AN SSSR,
37. S. V. Kovalevskaya,Nauchnyeraboty(ScientificWorks),ed. by P. Y. Polubarinova-Kochina,
Moscow, 1948.
38. G. Birkhoff,
ed., A Source Book in Classical Analysis,Harvard UniversityPress,Cambridge,Mass., 1973,
p. 318.
"On the ScientificWork of Sofya Kovalevskaya,"transl.by N. Koblitz, in
39. P. Y. Polubarinova-Kochina,
Kovalevskaya,A RussianChildhood,p. 233.
40. E. T. Whittaker,
AnalyticalDynamicsof Particlesand Rigid Bodies,CambridgeUniversityPress,London,
1965,p. 164.
du problemede la rotationd'un corpspesant... " (seeNote 34).
41. Kovalevsky,"Memoiresurun cas particulier
du systeme..." (see Note 35).
42. Kovalevsky,"Sur une proprinet
Dover Publications,New
and RotationalMotion,Theoryand Applications,
43. A. Gray,A Treatiseon Gyrostatics
York, 1959(firstappeared 1918),p. 369.
THE UNIFORMIZATION THEOREM
WILLIAM ABIKOFF
ofIllinois,Urbana,IL 61801
Department
ofMathematics,
University
Almostoile hundredyears have passed since Felix Klein discoveredthe uniformization
theorem.
Whilesuchtheorems
had earlierbeen provedin specificcases,no one had daredeven
by a variablewhose
conjecturethat everycompactRiemannsurfacecould be parametrized
WilliamAbikoffreceivedhis B.S. and M.S. in electricalengineeringand his Ph.D. in mathematicsfromthe
PolytechnicInstituteof Brooklyn.He workedat Bell TelephoneLaboratoriesand taughtat Columbia University
of Illinois
and theuniversities
of Perugia,Firenze,and Paris.He is currently
an associateprofessorat theUniversity
Instituteand the Institutdes Hautes
at Urbana-Champaign.He has been a researchfellowat the Mittag-Leffler
Etudes Scientifiques.His researchinterestshave centeredon the use of geometricfunctiontheoryin the studyof
Riemannsurfaces,Teichmillertheory,and Kleinian groups.Currentlyhe is studyingthe structureat infinityof
hyperbohcmanifolds.-Editors
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1981]
THE UNIFORMIZATION THEOREM
575
domainof definition
lay in theRiemannsphere,C = C U { oo}. For reasonsthatwillbecome
apparentin ? 3, Kleinassertedmuchmore,callinghisresultthelimitcircletheorem.
The theorem
occurredto himat 2:30 A.M. on March23, 1882,whilehe was in themidstof an asthmaattack.
His healthhad been poor and he was tryingto recoverin the North Sea coastal town of
Nordeney.The combination
of miserableweatherand a desireto announcethetheoremled him
to returnto hisnativecityof Dutsseldorf
almostimmediately.
Whenhe receivedthegalleyproofs
oftheannouncement
fromthepublisher,
he dispatched
copiesto Hurwitz,
Schwarz,and Poincare;
Hurwitzwas also sent an outlineof the proof.The responsesaccorded this resultby his
distinguished
colleaguesareclassicexamplesofthecamaraderie
ofscientists-at leastin myloose
of Klein'srecollections
interpretation
[14,p. 584].
HURWITZ: I acceptit without
reservation.
SCHWARZ: It's false.
POINCARt: It's true.I knewit and I have a betterwayof lookingat theproblem.
However,thestorydoes notend there.Schwarzrecantedshortly
thereafter.
He madetwogreat
contributions
to the theoryof uniformization.
The firstwas a methodforprovingthetheorem.
The secondwas to establishtherelationship
betweenuniformization
and thestudyof conformal
representation
of Riemannsurfaceson C. The linkis providedby thetheoryof coveringspaces
whichSchwarzdevelopedspecifically
to studythe uniformization
problem.In 1907,Poincare
provedthemostgeneralknownuniformization
theoremforthecase wheretheparameter
varies
overa simplyconnecteddomain.
The result,
whichis nowcalledtheuniformization
theorem,
is a problemin conformal
mapping
whichis solvedbypotentialtheory.
It is relatedto theoriginalproblemby algebraictopology.Of
themajormathematical
disciplines,
fewhavenotbeenenrichedby theuniformization
theorem
or
themethodsdevelopedforthestudyof theproblem.
Researchin uniformization
theory
has gonethrough
severaldormantperiodssincetheconcept
of a (global)uniformization
was introduced
by Kleinin 1882.Uniformization
theorems
of power
to theclassicalmastershavebeenprovedin thepast twenty
unimaginable
yearsand no end is in
sight.
My originalpurposein writingthispaper was to put into printa relatively
efficient
and
elementary
proofofhalfof theclassicaluniformization
theorem
whichI havecirculated
privately
forseveralyears.RalphBoas suggested
thatI writeit in a moreexpository
form.I havetakenthis
suggestionas an invitationto exercisemy personalfascination
withthe variousfacetsof the
uniformization
problem.To do historicaljustice to the problem,I was requiredto tracethe
development
ofthenotionof a Riemannsurface;foras thisnotionmatured,
so did thestatements
and proofsinvolvedin the solutionof the uniformization
problem.The discussionin the text
containsan outlineof thisproofwiththedetailsplacedin an appendix.For completeness
I have
also includeda briefsketchof Maskit'sworkon thegeneraluniformization
problem.
In translating
heuristic
arguments
intomathematics,
I shalloftenreferthereaderto Ahlfors's
Conformal
Invariants[2],whichwillbe abbreviated
as CI.
I wouAd
liketo expressmythanksto Lars Ahlfors,
LipmanBers,JozefDodziuk,and Bernard
Maskitfortheircomments,
manyof whichare incorporated
in the ensuingtext.The figures,
which,in a sense,are thesoul of Riemannsurfacetheory,
weredesignedand drawnby George
Francis.My debtto himis clear.
1. An Example.ConsiderthevarietyS in C2 definedby theequationX2 + y2 - 1
. Here
X and Y are complexand S is thesolutionset.Thereis an obviousmethodforparametrizing
S.
For z E C, set X(z) = cos z and Y(z) = sinz. S is completely
parametrized
or,in thelanguage
we willuse here,S is uniformized
by thevariablez.
Thisis usuallyconsidereda poorchoiceof uniformization
forthefollowing
reasons.X(z) and
Y(z) are transcendental
functions
withessentialsingularities
at oo. The structure
at oo is not
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576
[October
WILLIAM ABIKOFF
clearlydisplayedby theuniformization.
Anotherchoiceis to letz varyoverC = C U { oo}
withX(z) = 2z/ (I +
Z2 )
and Y(z) = (1
-Z2)/
(I + Z2).
has poles at ?+i, i.e.,
therationaluniformization
The behaviorat oo is notexceptional;however,
at oo. If we adjointo S theideal points(X(? i), Y(?i)), we obtaina
structure
thereis a definite
to C as maybe seenfromRiemann'smethodof
compactRiemannsurfaceS. S is homeomorphic
branchcuts.
To eliminatethespecialroleof thepointat oo, thevarietyS maybe viewedas lyingin the
i.e.,we considerthevarietyin
theoriginalpolynomial,
planebyhomogenizing
complexprojective
space definedby X2 + y2 W2= 0 (see, forexample,Kendig[12]).
projective
werefirstintroduced
oftheConceptofa RiemannSurface.Riemannsurfaces
2. The Evolution
variables.The
of
complex
functions
multivalued
by Riemannin an attemptto understand
ifX and Y are
of
Y.
However,
function
X
valued
0
as
a
single
does
not
define
y2=
X2_
equation
madeif Y is notpermitted
chosento haverealvalues,a choiceof signof Vfmaybe consistently
of P(X, Y) = X2- y2 vanishes,thatis,
to assumethevaluezero.At X = Y = 0, thedifferential
to
is not sufficient
of theequationP(X, Y) = 0. Avoidingthesingularities
(0,0) is a singularity
to havecomplexvalues.
permitsuchchoiceifX and Y arepermitted
complexvariablestext,Riemann
thatmaybe foundin any elementary
Usinga construction
sheetsabove thecomplexplane. On each sheet,a choice
resolvedtheproblemby constructing
forthe
to forma naturaldomainS, ofdefinition
maybe made.The sheetsarethengluedtogether
X
and
of
Fig. 2
Y;
for
values
solution
real
In
Riemann's
we
demonstrate
function X2.
Fig. 1,
is made on thedomainof the
givesthelocal picturein C near0. Noticethatthemodification
ofhissheetsas lyingoverthecomplexplane
thought
noton therange.Riemanninitially
function
of thenotionof Riemannsurfacewas
development
in Eucidean 3-space.Muchofthesubsequent
in R3. As Weyl
of Riemann'sembedding
and arbitrary
character
done to removetheartificiality
noted, S, is a naturalparameterspace for the varietyS. In this sense Riemannhad a
butitwas notthefirst.
theorem,
uniformization
nutLvodLuecL
ore cka.rt C|
funcl'oLr
functiorn
Y
Y
FIG. I
was provedbyPuiseuxin 1850.The theorem
theorem
As faras I know,thefirstuniformization
of plane algebraiccurves.It
of singularities
theoremforthe structure
is a local uniformization
statesthatif X and Y satisfya polynomialrelationP(X, Y) = 0, then,up to a linearchangeof
functionof y = (Y - 0)1/k forsome
X can be writtenlocallyas a holomorphic
coordinates,
>
forthevarietyP( X, Y) = 0. A proofmay
parameter
k 1; y thenbecomesa local uniformizing
be foundin Hille [11,vol. 1,p. 265 ff.].
Linear
functions.
Algebraicequationsin twovariablesare not theonlysourceof multivalued
havemultivaldifferential
equationswithregularsingularpoints,suchas XY' + Y = 0, generally
ued solutions.
in thecomplexplaneoftheformfR(z, w) dz whereR
arecontourintegrals
Algebraicintegrals
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1981]
THE UNIFORMIZATION
~~~1;
I
4 3~~~~~~~~Z1
577
THEOREM
11a
C:21.
v2
Construction
FIG. 2
is rationaland z and w satisfya polynomial
identity.
As thepathof integration
varies,we again
obtainmultivalued
functions
of thepathand upperendpointof integration.
Siegel[21,pp. 1ff.]
tracesback to Fagnanoin 1719thestudyofthealgebraicproperties
of theseintegrals.
In thefirst
halfof thenineteenth
theseproperties
century,
werestudiedbyAbel and Jacobi;theintegrals
are
now calledabelianintegrals.
Riemannlaterstudiedtheseintegrals
themas definedon thevarietyS rather
by considering
in thecomplexplane.
thanas integrals
The methoddiscoveredby Weierstrass
forstudying
multivalued
functions
is quite different
fromRiemann's.It is probablythefirststepin thedevelopment
ofthemodemnotionsofabstract
manifoldand sheaf.Weierstrass's
originalpaper on the subjectseemsto have been writtenin
1842,althoughit did not appearin printuntil1894.The basic idea is to construct
a Riemann
surface,in thiscontextcalled an analyticconfiguration,
by startingwitha power seriesf(z)
centeredat some z0 of positiveradius of convergence.
One thenconsidersall meromorphic
continuations
off in C. Each continuation
off to a pointz is givenby a chainof discsin which
thecontinuation
is possible.
The set of pointsto whichf may be continuedis subjectto an equivalencerelation,and a
naturaltopologymaybe givento theequivalenceclasses(see,e.g.,Ahlfors[1] or Hille [11,vol. 2]
fordetails).Singularpointsadmitting
uniformizations
by Puiseuxseriesare usuallyadded to give
complete
analytic
configurations,
whichareRiemannsurfaces
in thesenseofWeyl(see below).The
analyticconfigurations
are Riemannsurfaceswhichlie in thesheafof germsof holomorphic
or
meromorphic
functions
on C. Overlapping
discsdefinethetopology,a conceptwhichunderlies
themodemnotionof manifold.
In his 1907paperon uniformization,
Poincareremovedtherelianceof Weierstrass's
construction on theinitialpowerseriesf(z). He provedthe uniformization
theoremforcollectionsof
overlapping
discson whichthereexistssomemeromorphic
function
in thesenseof Weierstrass.
The geometric
entitybecameparamount;theroleof thefunction
was secondary.
The modemera in the studyof Riemannsurfaces,and, indeed,of all manifolds,
openedin
1913withthepublicationof HermannWeyl'sbook The Conceptofa RiemannSurface.Here,for
the firsttime,no definingfunctionis assumed.The definition
is purelygeometricand also
independent
of anyembedding
in Eucidean 3-space.
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578
WILLIAM ABIKOFF
[October
On the justificationfor abandoningRiemann'sembeddingof the Riemann surfacesof
Weylwrote:
functions,
multivalued
space has nothingto do with analytic forms[Riemann surfaces of
In essence, three-dimensional
grounds,but because it is closely
and one appeals to it not on logical-mathematical
multi-valuedfunctions],
associated with our sense perception.To satisfyour desire for picturesand analogies in this fashionby
on objects instead of taking them as they are could be called an
forcinginessentialrepresentations
contraryto scientificprinciples.However,thesereproachesof the pure logician are no
anthropomorphism
longer pertinentif we pursue the other approach ... in which the analytic formis a two-dimensional
not to use thisapproachis to overlookone of themostessentialaspectsof the
manifold... To thecontrary,
topic.
follows.Let S be a
hypotheses,
unnecessary
in modemlanguageand without
Weyl'sdefinition,
space and U = { DI Da C S} be an open coverof S. Assumethatforall a
connectedHausdorff
thereis a homeomorphismz -: Da - C which satisfies:z1l z< is holomorphicwhere defined.
The set A -{(za, Da) is called a holomorphicor conformalatlas and the pair (S, A) is called a
surface.Usually,by abuseoflanguage,we speakof theRiemannsurfaceS. za is calleda
Riemann
variable at the points of D.. A functionf on S is holomorphic,
local coordinateor uniformizing
for
each a, f o za has that propertywhereverit is defined. A map
if,
or harmonic
meromorphic
->
whenever
if z-1 -f-z is holomorphic
two
surfaces
between
f: SI S2
S, and S2 is holomorphic
on S, and S2, respectively.
defined.Herez1 and z2 arelocal coordinates
butpossible,to provethateveryRiemannsurfacein thesenseofWeylcarriesa
It is nottrivial,
and maybe embeddedin R 3. It
functions
meromorphic
linearlyindependent
pairofnonconstant
equivalence,thenotionsof Riemannsurfacesdevelopedby
thenfollowsthat,up to holomorphic
and Weylare equivalent.
Weierstrass,
Riemann(suitablygeneralized),
problem,the questionchangedas the notionof Riemann
To returnto the uniformization
surfaceschanged.Klein's originalclaim is to have provedthe theoremforcompactRiemann
possiblymissinga finitenumberof points,in thesenseof Riemann.All proofsuntil,I
surfaces,
was made
werefirstcountable;thislatterassumption
believe,the1920'sassumedthatthesurfaces
was removedbyRad6,butnowmaybe derivedas a simpleconsequence
byWeyl.The assumption
theorem.
of theuniformization
problem.Let S be a Riemannsurface.Find all
We maynow statethegeneraluniformization
t: D -> S so thatat each pointp E S, t is a local
functions
domainsD c C and holomorphic
variableatp.
uniformizing
of t to each
thereis a topologicaldiscB c S withcenterp so thattherestriction
Equivalently,
The readershouldnoticethattherequiredconditions
of t- l ( B) is a homeomorphism.
component
on t and D say preciselythatthetriple(D,S,t) is a (smooth)coveringspace withbase S, total
projectiont.
spaceD, and holomorphic
butwhereD is
It was simplyto findone uniformization,
Theearlyproblemwas lessambitious.
in 1907.
Poincare
and
Koebe
by
independently
proved
was
This
theorem
connected.
simply
here.Until
Poincare'ssolutionwas somewhatmoregeneralbutwe willignorethatgeneralization
recentlythe most commonproofwas thatofferedby Hilbertin 1909. We give the modem
thattieit
thepreliminaries
Theoremin thenextsectionfollowing
statement
of th,eUniformization
problem.
to theuniformization
Surfacesand ClassicalPlane Geometries.Let S, and S2 be twoRiemannsurfaces
3. Covering
-* S2 be a local homeomorphism
so thateach point on S2 has an evenlycovered
and Tr:
S,
thatS, is a
surfaceor,morecommonly,
neighborhood.We then say that (S, ,S2 ,5) is a covering
coveringsurface of S2 with cover map or projection 7T. The number of points in 7-I (p) is
independentof p C S2 and is called the numberof sheetsof the covering.
For
Coveringsurfacesoccurclassicallyin thestudyof algebraicequationswithsymmetries.
this
=
In
w.
w
by
0
by
replacing
obtained
w2
the
z
symmetry
admits
the
example, equation
over
The RiemannsurfaceS of w2 - z
is called sheetinterchange.
contextthe symmetry
cannotbe definedto
C \ {0} is a coveringsurfaceof C \ {0}. Noticethatthesheetinterchange
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1981]
THE UNIFORMIZATION
THEOREM
579
looks like z _ zn are called
0 Pointswherethemap topologically
be a coveringsurfaceat z O.
points.
ramification
all of whichare ramification
points,are called
Coveringsurfaceswithisolatedsingularities,
covers.
ramified
surfaceis to taketwocopiesS' and S" ofa givensurface
a covering
A methodforconstructing
surfaceis a
curves,and glueas indicatedin Fig. 3. The resulting
S, slicethemalongcorresponding
coverof S withtheprojectiontakingz' and z" to z. Notice thatthereis a simple
two-sheeted
closed curvethatcovers/3twice.In a sense thatmay be made quite precise,passingto the
coveringsurfacehas replaced/Bby a curveof twicethelengthof /3.If we repeatthisprocess
willno longer
long,simplecurve;i.e.,/3
often,/3willhavebeenreplacedby an infinitely
infinitely
manycurves
We mayrepeatthisprocessforsufficiently
to simpleconnectivity.
be an obstruction
trivialcurves.S is called the
coveringsurfaceS has onlyhomotopically
so thatthe resulting
surface.Thisapproachis themostclassicalone; themodemapproach(see,e.g.,
covering
universal
of thebasic idea of openingup all homotopically
Greenberg[8]) is an abstractreformulation
aboveis due to Schwarz.
closedcurves.Kleinstatesthattheconstruction
nontrivial
FIG. 3
As we have notedin theprevioussection,if S is theRiemannsurfaceof some multivalued
of D if D variesoverthepointsof S. If S, is a
functionY of X, thenbothX and Y are functions
thevarietyin C2
coveringsurfaceof S, and , variesoverS,, thenX v and Y v parametrize
betweenX and Y. ThusifS, liesin C we obtaina solution
relationship
definedby thefunctional
our
problem.We willreturnto thisquestionin ? 6 butnowrestrict
to thegeneraluniformization
attention
to S, = S.
on S. S is a simply
structure
S inherits
of a Riemannsurfacefromtheconformal
thestructure
nontrivial
closedcurvehas been opened.
connectedRiemannsurfacebecauseanyhomotopically
problemby showing
thatwe mayobtaina solutionto theuniformization
It followsimmediately
equivalentto a subdomainof
thateverysimplyconnectedRiemannsurfaceis biholomorphically
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580
WILLIAM ABIKOFF
[October
of theRiemannmappingtheorem,
(see ? 4). So a solutionto the
C; thisis merelya rephrasing
nowknownas:
uniformization
problemfollowsfromthestatement
connected
Riemannsurfaceis biholomorphically
THE UNIFORMIZATION THEOREM. Everysimply
equivalent
to eitherC, C, or theunitdiscA.
is an immediate
consequenceof Liouville'stheorem.
That thethreepossibilities
are distinct
The universalcoverhas severalverystrongproperties.Since it is simplyconnected,every
on S; herewe are
function
on S liftsto a meromorphic
function
multivalued
locallymeromorphic
restatingthe monodromy
theorem.The universalcoveris the only coveringsurfacewiththis
property.
Anotherimportant
propertyis the following.Suppose t E S and N is an evenlycovered
neighborhood
oft. The covermap ST:S -* S definesa mapy fromone component
N, of ST - l(N)
to another,
sayN2, by theruleD, + 2 wheret2 iS theuniquepointin N2 forwhichT(t,) = TG2)
of N,, one showsthaty extendsto a biholomor(see Fig. 4). Usingtheevenlycoveredcharacter
phicself-map
of S. Thesemapsforma groupG calledthegroupofthecovering
(5,S,, ) or cover
Clearly,G actswithoutfixedpoints.G also has the
groupor groupofdeck(cover)transformations.
whichmeans that each point f E S has a
stronger
propertyof beingproperlydiscontinuous,
N so thaty(N) n N 7i 0 if and onlyify = identity.
Sincethesheetinterchange
neighborhood
withthefreehomotopy
classesof closedcurveson
mapsy E G arein one-to-one
correspondence
to thefundamental
group Ti,Sof S based at anypoint.The orbit
S, thegroupG is isomorphic
space S/G is the set of equivalenceclassesin S whereDl - t2 if thereexistsy E G so that
a conformal
structure
fromS and thatS is
to showthatS/G inherits
(2 = y(;,). It is notdifficult
to
biholomorphically
equiyalent SIG.
c~~N,N
cON2
c
,
}S
NpJ
NC
CV4
FIG. 4
accuratewhenS is C or C
The lineof reasoningdevelopedaboveis, moreor less,historically
betweenbiholomorphically
equivalentsurfacesexcept when
(we shall no longerdistinguish
It showsimmediately
necessary).
thatS = C onlyif S = C. If S is compactand S = C, 7Tis the
inverseto an ellipticintegral
inversefunctions
(see Siegel[21,Chapter1]).The problemof finding
to algebraicintegrals,
problem,therichhistoryof
i.e.,covermaps,is calledtheJacobiinversion
which leads directlyto Riemann surfacesand uniformization(cf. Weyl [23, p. 144]).
WhenS = C, G mustbe a properly
discontinuous
subgroupof thegroupAutC ofbiholomorof C. Thereare preciselythreetypes.The firstis whenG is trivial.The Riemann
phicself-maps
surfaceis thenC. The secondare thecyclicgroups{z + z + nz0 n E Z}. All are conjugatein
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1981]
THE UNIFORMIZATION
THEOREM
581
AutC to G= (zF-z + n n C Z} and conjugategroupsdetermine
biholomorphically
equivalent
surfaces.
The surfaceis C \ (0) andST: C -> S is themap exp(27Tiz).The lastis a classof groups,
the lattices,whichare, up to conjugation,
generatedby Yl: z Fz + 1 and Y2: z - z + T where
Im T> 0. In thiscase 7T,Sis freeabelianon twogenerators
and S is topologically
a torus.The two
generators
maybe chosenas in Fig. 5.
t~~~~~~
FIG. 5
We havethis"found"all Riemannsurfaces
withS C or S =. The uniformization
theorem
tellsus thateveryotherRiemannsurfaceS has S = A. Herethehistory
proceededin theopposite
direction.
Kleinknewin 1882thatif G is a properly
discontinuous
subgroupof Aut A thenA/G
lookslikea Riemannsurface.Howeverin 1882,a Riemannsurfacerequireda defining
multivalued function.
Whileit seemsthatKlein had a methodforobtainingsuch a function,
Poincare
claimedtwo methods,one of whichremainsthe standardmethodforobtainingautomorphic
functions
fordiscretegroupsactingon boundeddomains.The existenceof thisfunctionshows
thatall thedefinitions
of Riemannsurfacescoincide.
The Riemannspherehas a metricof constantpositivecurvatureinduced by the usual
in R3. C, hencebyprojection
embedding
C \ {O} and tori,haveEucidean or flatmetrics,
i.e.,of
zero curvature.
Usingtheuniformization
theorem,
we see thatall otherRiemannsurfacesinherit
anyAutlxinvariant
metricfroml. Up to scalefactor,thereis onlyone; it is thePoincaremetric
on thehyperbolic
plane. It is a metricof constantnegativecurvature.
The universalcoverof a
surfaceof constantnegativecurvaturemay be conformally
and injectivelydevelopedon a
hyperbolicplane. This is one of many successfulapproachesto provingthe uniformization
theorem.
The relevantdifferential
equationis Au = e2u. Studiesof thisequationwereamongthe
earliestin nonlinearpartialdifferential
equations.
We also recallthatthehyperbolic
plane,in Poincaresmodel,is theunitdisc i\.Whenwe map
S to A\,
we givenaturalboundaryto S, namelyMA.To Kleintheedgewas a limiting
circleand he
calledhis uniformization
theorem,
thelimitcircletheorem.
To comment
on thepre-1907"proofs"of theuniformization
is to treadon themostdangerous
ground.One gathersfromHilbert's1900lectureat theInternational
Congress[10] thathe did not
acceptKlein'sproof,forit is notmentioned.
Poincare'suniformization
is spokenof,but Hilbert
comments
thatit does notparametrize
thewholevariety;so it is nota uniformization
in thesense
consideredhere(see also theintroduction
to Poincare[20]).
One is first
Fromthevantagepointof 1980,it is notquiteso easyto dismissKlein'sargument.
To thoseof us trainedin the
presentedwitha majorobstacle,namely,to findthe argument.
Satz-Beweisschoolof mathematical
expositionand discourse,readingKlein is oftena mystical
In fact,manya mathematician
has provedand publisheda deep and elegantresult,
experience.
laterto discoverwithchagrinthatthereis a casual and vaguereference
to theresultin Klein.
Most of Klein's writingon areas relatedto uniformization
are collectedin two books written
jointlywith Fricke; these compriseover 2,000 pages withoutan index, and oftenwithout
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582
WILLIAM ABIKOFF
[October
or theoremstatements.
The mathematical
definitions
insightcontainedtherein
is astounding,
but
it oftenseemsthatone can onlyappreciatea partofit afterhavingindependently
rediscovered
the
results.To themodemobserver
Kleinonlyclaimedtheproofoftheuniformization
theorem
when
S is conformally
finite,thatis, whenS is a compactsurfacemissinga finitenumberof points.A
is thatit is an excellentoutline,but farfroma proof;it
tersemodernappraisalof theargument
appearson pages698-705 ofvolume3 ofhis collectedworks[14].
A briefdescription
follows.Firstconstruct
one algebraicequationdefining
P(z, w) = 0 as a
multivalued
function
ofz. Let S be theRiemannsurfaceof P in thesenseof Riemann.On S we
finda finitesetofpiecewisecircularclosedcurvesa, so thatS \ U a, is a polygonII c S. On S, z
H in C, and a z( H) consistsof circulararcs.P maybe
is a well-defined
function,
z 1I7 immerses
Then forsome choice of P and the a,,
chosenin any genusg, so thatit is verysymmetric.
z(HI) C A and az(IH) are circulararcs lyingon circlesorthogonalto Ma. z-l'z(H) may be
butin a multivalued
analytically,
fashion,continuedalongall pathsin S. If theimagedomainis
is theinverseof theuniversalcovermap. He thenconsidersthe
theunitdisc,thecontinuation
and he notesthatthespace of such(normalized)
corresponding
groupof covertransformations,
groupsG and the space of dissectedRiemannsurfacesin genusg both have real dimension
betweenthemhe assumesis a local real analyticdiffeomor6g - 6. The local correspondence
phism.He, moreor less, showsthatthemappingis injectiveand proper,hencebijective.This
Evenassuming
Thistechniqueofproofis calledthecontinuity
method.
completesKlein'sattempt.
thelasttwoproperties
are truebutnoteasilyproved.The fundamentheuniformization
theorem,
tal difficulty
withthisproofwas recognizedquite earlyand goes as follows.Forgetthatthe
is a local diffeomorphism
correspondence
(or wait some 40 to 80 yearsuntilthe theoremis
proved).You only haye a proper,continuousinjectionf of R' into R'. Brouweressentially
the
and therebyresurrected
developeddimensiontheoryto provethatf is a homeomorphism
method.WithBrouwer'sproofadded, Klein's techniquebecomesviable; however,
continuity
theoremhad alreadybeen proved in
beforeBrouwer'sproof appeared,the uniformization
completegenerality.
Uniformization
dormantfrom1883to 1900.In 1900,Hilbertdelivereda
theorywas relatively
in whichhe stated23 problemswhich
lectureto the International
Congressof Mathematicians
havehad a profound
on thecourseofmathematics
in thetwentieth
effect
century.
Uniformization
was Problem22. This renewedinterest
in thequestionled to thesolutionin 1907.Bothsolutions
givenin 1907and theargument
givenin 1909by Hilbertcomefrompotentialtheoryand it is to
thatstreamof ideas thatwe nowturn.
4. Some PotentialTheory.Riemann'sthesisis contemporary
withmanyof the greatdiscoveriesofnineteenth-century
physicalscience.Klein[13,p. x] wrote,"Riemannas we knowused
of whichKlein speaks are those
Dirichlet'sPrinciplein theirplace." The physicalarguments
associatedto a conservative
vectorfieldE and its associatedpotentialfunctionV. The classic
fluid.I shallnotdwell
examplesof thesefieldsareelectricfieldsand theflowof an incompressible
?
on theseconceptssave to say thatthefollowing
equationshold: divE
0, E - gradV, and
AV= 0. In the last equation, = a 2/ax2+ a2/ay2 is the Laplacian, and thereforeV is a
harmonicfunction.
Amongtheearliestnontrivial
fieldsto be studiedis thatinducedby a pointcharge.In theunit
disc, the potentialdefinedby a (positive)pointchargeat z = 0 is V(z) = log(l/Iz I) if aA is
grounded,
i.e., VIaA = 0. Riemannconsideredthequestionof whether
a pointchargecouldlive
in an arbitrary
simplyconnectedplanedomainD whoseboundaryis a groundedconductor.
The latterconditionis thatthepotentialVIaD = 0. Assumewe have a pointchargein D. The
levelcurvesL v(c) of V for0 < c < oo areanalyticJordancurvesseparating
thepointchargeat z0
fromaD. The integralcurvesC of thegradientfieldof V, thepathsof electionsin thisfield,are
againanalyticcurvesfromaD to z0. Theseintegralcurvesmaybe parametrized
by theanglea at
whichtheyenterzo. We maytherefore
writeC C(O). Each pointz E D lies on a uniquelevel
curveLv(c) and a uniqueintegralcurveC(O). We forma map
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1981]
THE UNIFORMIZATION
THEOREM
583
f : D -->A
-c a0)
ZF-(
in polar coordinates.(See Fig. 6.) This map is conformaland provesthe Riemannmapping
Riemanninvoked
once we knowthata pointchargecan livein D. To proveexistence,
theorem,
minimizethe energyin an
the Dirichletprinciple.The principlestatesthatharmonicfunctions
was used by Riemannin
C. This argument
existshereif D C
electricfieldand sucha minimum
The latternoted that even elementary
severalcontexts,but was questionedby Weierstrass.
butwas later
problemsneedhaveno solution.The Dirichletprinciplefellintodisrepute
extremum
theory.
Hilbert'sproof
byHilbert.It is thekeyto his 1909proofoftheuniformization
resurrected
associatedwithelectricdipoles.
usesmappingproperties
FIG. 6
fluidflow.(See Fig. 7.) If
we nextconsiderincompressible
discussion,
To continueourintuitive
we havea pointsourceof waterin theplane,forexamplea faucet,theremustbe an edgeacross
thefluidwillcompress.If thedomainhas
whichthewatermayflowoutof thedomain.Otherwise
then
a
"thick
we
call
a
pointsourcecan existat anypointin the
which
boundary,"
suchan edge,
FIG. 7
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584
[October
WILLIAM ABIKOFF
domain.It is intuitively
obviousthatany edge save a point forthe domainwill enable it to
supporta pointsource.
Thereare nowmanywaysto provetheexistence
of a pointchargeor source.Perhapsthemost
elegantis due to Perron.His methodmaybe foundin mostcomplexanalysisbooks.Basicallythe
idea is thata harmonicfunction
is the2 (or n) variableanalogueof a linearfunction
f(x) of one
variable.f(x) has the propertythatf(x) = supg(x) whereg(x) is convexand the boundary
valuesof g are less thanor equal to thoseoff (see Fig. 8). Locallyone replacesg by a linear
function
and g is "bootstrapped"
up tof bytakingsuprema.
I
_S~~~~~~~~~~~~~~~~~~
S
FIG. 8
On a Riemann surface,subharmonicfunctionsassume the role of convex functions.A
subharmonic
function
is a continuous
real-valuedfunction
whosevalue at a point;0 E S is less
thanor equal to theaverageofthevaluesof thefunction
on theboundaryofanysmallsymmetric
oft0. Specifically,
lets: S -- R be continuous.
s is subharmonic
neighborhood
if,foreach o0E S
and each smallneighborhood
N and local coordinate
z = z(') in N withz('0) = 0,
S(W?) < 2g fs(relO)dO
forr z(t)j sufficiently
smalland 0 = argz( ').
A Perronclass'F is a nonvoidsetofsubharmonic
functions
s: S -* R whichis closedunderthe
following
operations:
(i) takingthemaximum
of twofunctions
(ii) local harmonicmajorization.
Property
(ii) generalizes
thelocal replacement
of a convexfunction
To obtain
bya linearfunction.
theprecisedefinition,
letD be a closeddiscon S. On D, let3 be theharmonicfunction
suchthat
3IaD = s IaD. Extend3 to S by setting
3 = s in S \ D. 3 is calledthelocalharmonic
ofs
majorant
in D. Local harmonic
majorization
is theprocessof replacings by3sforsomedisc D.
Perronshowedthats(') = sups,s( ') is eitherharmonicor identically
infinite.
We nowgivea precisedefinition
of theGreen'sfunction,
or potentialof a pointcharge,using
Perron'smethod.The potentialshouldbe thesmallestpotentialwhichis positiveand growsas
does - loglz nearthepointcharge.
Let t0 E S and 'J be the set of subharmonic
functions
s in S \ {0} havingthe following
properties:
(i) SMt 2_0.
(ii) {' E S \ {O)} Is(; ) #70} is compact.
(iii) For any local coordinatez on a neighborhood
N of t0 withz('0) = 0, s(z) + loglz is
bounded above near t0.
As a consequenceof Perron'sTheorem(see, e.g.,Ahlfors[1, p. 240]),s-(') = supsaqs(') is
eitheridentically
infiniteor harmonicin S \ { 0}. In thelattercase we call g(', '0) = s-(') the
of S withsingularity
Green'sfunction
at t0. In theformer
case we say thatS does not admita
Green'sfunctionwithsingularity
at t0. It is knownthat,if z is a local coordinateat t0 with
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1981]
585
THE UNIFORMIZATION THEOREM
of t0 (see CI, p.
z(tO) = 0, g(', '0) + loglz I has a harmonicextensionto anysmallneighborhood
141).
AssumethatS has a Green'sfunction
g(',tO). Then,in thesenseof fluidflow,it musthave a
thickboundary-a notionwe must also formalize.Assertingthat a domain S has a thick
boundaryis equivalentto statingthat the "edge" of S may serveas a source or sink for
fluidflow(withboundedpotential).Equivalentlythe associatedpotentialmay
incompressible
bythefollowing:
oftheboundaryis formalized
alongtheedge.The nonthickness
havea minimum
DEFINITION. Let S be a Riemannsurfaceand K be a compactsubsetof S. We say thatS
f: S \ K -*> R
relativeto K if everyboundedharmonicfunction
principle
satisfiesthemaximum
that
has theproperty
sup f(A)0
tE=S\K
lim f(0).
-->aK
withsingularity
NoticethatifS has a Green'sfunction
t0 E IntK, then- g I(S \ K) showsthatS
does not satisfythe maximumprinciplerelativeto K. The converseis also true,namely,the
of themaximumprinciplerelativeto K impliestheexistenceof theGreen'sfunction
invalidity
pointt0 E IntK (see CI, p. 139).
at an arbitrary
withsingularity
principle
Givena Riemannsurfaceor anyplanedomain,thevalidityof therelativemaximum
a standardtechniqueused to showthatit is notvalid.
Thereis,however,
verifiable.
is notdirectly
on K. If the
h on S. Thenh has a maximum
harmonic
function
Supposethereexistsa nonconstant
contradicting
wouldbe a globalmaximum
principlewerevalid,thatmaximum
relativemaximum
We haveproved:
principle.
theusualmaximum
harmonic
f: S -*1 R
function
surface.If thereis a bounded
PROPOSITION 4.1. Let S be a Riemann
themaximum
principle
then,forall compactK C S withnonvoidinterior,
whichis notconstant,
at anypointt0 E S.
withsingularity
relativetoK is notvalidand S has a Green'sfunction
on Riemannsurfaces.
boundedharmonicfunctions
to producenonconstant
It is nota triviality
For example,the usual maximumprincipleimpliesthat a harmonicfunctionon a compact
Rez, but,by applying
harmonicfunction
Riemannsurfaceis constant.C admitsthenonconstant
C admitsno boundednonconof infinity,
theoremto a neighborhood
theremovablesingularity
stantharmonicfunctions.
on a Riemannsurfacebringsus to the
boundedharmonicfunctions
Producingnonconstant
venerableDirichletproblem.The problemis easilystated.Let S C S, be Riemannsurfacesand,
forsimplicity,
assumeaS is a finitecollectionof piecewiseanalyticcurves.Let f: aS -* R be
continuous.The Dirichletproblemis to finda continuousfunctionh: S -* R so that h S is
harmonicand hIaS = f. Notice thata S correspondsto our intuitivepictureof a thickedge.
f, then,by the
Further,if we may solve the Dirichletproblemfor nonconstantfunctions
principlerelativeto a compactsubsetand our
above,S willnotsatisfythemaximum
proposition
willhavebeenproved.
of thickness
analyticcharacterization
The solutionof the Dirichletproblemis an applicationof Perron'smethod.Assumef is
bounded or, more simply,assume aS is compact.We forma Perronclass 'Y of continuous
s Ias < f and s < supf. If m = inff,
in S and satisfy
s: S -1 R whichare subharmonic
functions
then the constantfunctionm E CY.Also, the maximumprinciple,applied to subharmonic
functions,implies thatforall E S and all s E CY,s(') < supf. It followsthath(') = supa,s( ')
< oo and henceis harmonicin S. It remainsto showthath extendsto aS and hIaS = f. The
thenotionoflocal thickness
usualtechniquefordoingso is due to Poincare(1899) and formalizes
at a point C aS, and we
fineresolution
witharbitrarily
of theboundary.One aimsa microscope
see whether
it is possibleto pushwateracrossaS near '. The potentialsof theseflowsare called
and proofsmaybe foundin Bers[4,p. 139ff.]and Conway[7,p.
barriers.
The formaldefinitions
are statedforplane domainsbut are equallyvalid on Riemann
265 ff.].Conway'sarguments
surfaces.The preciseresultthatwe needis
'
'
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[October
WILLIAM ABIKOFF
586
and S C SI. If a S is a finiteunionofclosed
surfaces
PROPOSITION 4.2. Let SI and S be Riemann
bounded
f: a S -* R thereexistsh: S - 11Rso that
analyticarcs,then,forall continuous,
(i) Ih IsuplfI
(ii) h S is harmonic
(iii) hadS=
f.
4.1 and 4.2, we obtainthefirstconclusionof
As an immediate
consequenceof Propositions
surfaces
and S C SI. If as is a finiteunionofclosed
PROPOSITION 4.3. Let SI and S be Riemann
analyticarcsthen
g(', '0)
(i) S has a Green'sfunction
and
(ii) limv asg(G, W0 = ?.
Proof:We provethesecondconclusion.Let C be a smallcirclearoundt0. UsingProposition
4.2, on S we may solve the DirichletproblemoutsideC withboundarydata f C = g and
h is
functions,
principleforsubharmonic
f asS = 0. Call theresulting
solutionh. By themaximum
an upperboundforall functions
s lyingin thePerronclass IFdefining
g(', t0). It followsthat
lim g(',W<)
-->as
0.
Sincethefunction
s(') = 0 lies in IF,
lim g(t, t0) 2- 0,
-->as
whichprovestheProposition.
eventoday,commandsa nonTheorem.The uniformization
theorem,
5. The Uniformization
trivialproof.Herewe willsketchone styleof proofwithsomedetailsomitted.For theinterested
or to theappendixto thispaper.
eitherto theaccessibleliterature
reader,we givereferences
Let S be a simplyconnectedRiemannsurface.We firstassumethat,forfixedt0 E S, S admits
in
at t0. The completeargument
a pointchargeor Green'sfunction
g = g(','0) withsingularity
this case may be foundin CI, p. 136 ff.We must definea conformalmap f:S -* C. Let
S' = S \ {(0} and choosea simplyconnectedchartNDin S' neareachD E S'. Sinceg is harmonic
near '.
in ND,it has a harmonicconjugatehD thereand fi = exp[- (g + ih)] is holomorphic
by a complexnumberof modulusone. Let No be a
Further,
fDis uniqueup to multiplication
z('0) = 0. g(',q'0) + loglz() I is
simplyconnectedchartneart0withlocal coordinatez satisfying
in
and
harmonic No
hencehas a harmonicconjugateho. Set
=
(g + log| | +
ih0)].
ofmodulusone,fi
fois holomorphic
Noand vanishesto firstorderat t0. By adjustingconstants
aa Z(a) in-iexpN[N-theoremimplies
is an analyticcontinuation
offo.SinceS is simplyconnected,themonodromy
function
definesa homomorphic
thattheanalyticcontinuation
f: S -* C. Also If(t) = e -g < I
sinceg(','0) > 0. It is possibleto showdirectlythatf is a bijectionof S withA; however,an
We omitthedetailssave to notethatthe
due to Heins [9] is farmoreefficient.
elegantargument
with
a Green'sfunction
proofmakesdecisiveuse of thefactthata Riemannsurfaceadmitting
at any prescribedpoint. This
at t0 also has a Green's functionwith singularity
singularity
completesour sketchof theproofof
PROPOSITION 5.1. If S is a simply
Riemannsurfacewhichadmitsa Green'sfunction,
connected
i.e., conformal,
thenthereis a biholomorphic,
mapf: S -* A.
Henceforth
we assumethatS is simplyconnectedand does notadmita Green'sfunction.
DEFINITION. A divergent
curveon S is a piecewise analytic simple arc 0: [0,oo)
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-*
S so that,
1981]
587
THE UNIFORMIZATION THEOREM
foranycompactK
C
S, 'V'(K) is compact.
Now assumeS admitsa divergent
curveand set St = S \f([t,oo)). It shouldbe intuitively
of S impliesthatSt sharesthatproperty.
clear,but requiresproof,thatthe simpleconnectivity
UsingProposition
4.3,we thenobtain
LEMMA5.1. For all t 2 0, St is simplyconnected
and for any t0 E So, St admitsa Green's
withsingularity
function
at t0. Further,
lim g(',;O) = 0.
?as,
Proof:See theappendix.
We shallneedthefollowing
standardresultin function
theory.
z < r} and Sr be theset ofholomorphic
LEMMA 5.2. Let lA(r)
injections
f: A(r) -? C and
satisfying
(i) f(O) = 0
(ii) f'(O) = 1.
on compactsubsets.
ThenSr is (sequentially)
compactin thetopology
ofuniform
convergence
Proof.The map
UC:
Sr
f(z)
5I
F(z) = r-'f(rz)
is obviouslya homeomorphism.
It therefore
sufficesto show that 5 is compact.Montel's
Theorem(see anygraduate-level
complexanalysistext)statesthatone mustshowonlythatSI is
closedand bounded.Hurwitz'sTheoremstatesthata limitof holomorphic
injectionsis holomorphic and injectiveor constant.Condition(ii) rulesout a constantlimit.Thus 5 is closed.The
estimates
to showthat5 , is boundedaregivenby Koebe's distortion
necessary
theorem
(see CI, p.
84, or Conway[7,p. 351 ff.].
SinceStis a simplyconnectedRiemannsurfacewitha Green'sfunction,
thereis a holomorphic
we mayassumethatft(O) 0Oforsomefixed;0 E S. Now choosea
bijectionft: St -- A\.Further
sequence(t,) increasingto infinity
and denoteSt by S, and ft by f. Fix the local coordinate
z = fo(') near t0. We may then compute cl = ft'(z(?))KI
and let F,(t)
c-lf ().
Fj: S,-1- A = A(c7-) is a holomorphic
bijection.
Recursively
we definesubsequencesNi of Z ? as follows:
(i) N1= Z +
(ii) If N, is definedandj E NAandj > i, F3is definedand injectiveon S, and FI'(z(o)) = 1.
C is injective,
FJo F- 1: Amaps0 to 0, and has derivative
equal to 1. Thus,by Lemma
O
5.2, we findthattheremustbe a subsequenceNA+IC N, so that,forj c N+ 1, FJFo-'
-0
O
=
to
an
have
C.
and
1.
We
defined
Ib
converges
injectivemap HI:
HJ(O)
HI(0)
Choose ni to be thejth entryin the sequenceNJ.For k > i, on S,
injectionand
Hk oFk=
-
(limF
?oFk'
(limE
oFJ')oFi=HoF.
)Fk=
(limFn
oF
0)
Hk o Fk
is a holomorphic
oFloIFk
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[October
WILLIAMABIKOFF
588
Thus HIF, is therestriction
to S, of a globallydefinedholomorphic
mapf: S -? C. f is injective
sincef IS, is injectiveforall i.
f(S) is
f(S) is simplyconnected.If f(S) #&C, then,by the Riemannmappingtheorem,
conformallyequivalent to A and there is a conformal map h: S -1 A. Re h is a bounded
4.1, S mustthenhavea Green'sfunction
nonconstant
harmonicfunction
on S. As in Proposition
whichcontradicts
We havetherefore
proved
ouroriginalassumption.
PROPOSITION5.2. If S is a simply connectedRiemann surface with a divergentcurve and
equivalentto C.
admittingno Green's function,thenS is conformally
We shallneedthefollowing
PROPOSITION5.3. If S is a simplyconnectedRiemannsurfacewithno divergentcurves,then,for
all t, C S, S = S \ { ,} is simplyconnected.
Proof.Here the readeris offereda choiceof two proofs.A proofvia potentialtheoryand
of
coveringspacesis givenin theappendix.The deepestbutquickestproofuses theclassification
simplyconnectedtopologicalsurfaces.A simplyconnectedRiemannsurfaceS is homeomorphic
curvek.
to C or to ? (see Ahlforsand Sario [3,pp. 90-104]). In C it is easyto finda divergent
S is
curve.Otherwise
closelyapproximated
by a divergent
The imageof 0 in S maybe arbitrarily
to C and henceis simplyconnected.
homeomorphic
to ?, S is homeomorphic
THE UNIFORMIZATIONTHEOREM.If S is a simply connectedRiemann surface, then S is
conformally
equivalentto Av,C or C.
thenProposition5.1 showsthatS is equivalentto A\.If S
Proof.If S has a Green'sfunction
curve,thenS is equivalentto C. In anyothercase,
has no Green'sfunction
but has a divergent
curve.It followsthatS A or
S S \ {t0} is simplyconnectedand obviouslyhas a divergent
S
C.
If f: S A is a conformalequivalence,thenf is a boundedholomorpicfunctionon S, in
on removablesingularities,
it is boundedneart0. By Riemann'stheorem
f extendsto a
particular
A, thereis
holomorphic
map of S intoA. By themaximumprinciple,If(to)I < 1. Sincef(S)
thereare points 0, t neart0
somet, E S so thatf(t0) = f(t'). By theopen mappingtheorem,
and t,, respectively,
so thatf(D) f(?4).But this contradictsthe fact thatf 1S is injective.
S C and S ?C whichcompletestheproof.
Therefore
To illustrate
theuse of theuniformization
we note
theorem,
COROLLARY1. Every Riemannsurfaceis second countableand separable.
Proof.Theseproperties
projectfromtheuniversalcover.A, C, and ? have theseproperties.
COROLLARY2 (Picard's Theorem). Let f be a meromorphic
functionin C. If C \ f(C) containsat
least threepoints,thenf is constant.
coverD ofD is conformally
theorem,
theuniversal
Proof.Let D = f(C). By theuniformization
equivalentto A. Let ST denotetheuniversalcovermap,ST: A- D. Then,locally,ST -I existsand
7 -f maybe continued
alongall pathsin C to definea map F: C -1 A. SinceF is holomorphic,
Liouville'stheorem
saysF is constant;henceso is f.
6. Maskit'sWorkon the GeneralUniformization
seriesof papers,
Problem.In a fifteen-year
Maskitresolvedthegeneraluniformization
type.Weyl
problemforsurfacesof finiteconformal
notedthattheproblemhas twoaspects.One startswitha RiemannsurfaceS. The firstpartof the
of S. Maskit's planarity
problemis topological-namely,find all coveringsurfacesD C
way.On S we finda setof simpleclosedloops {al}
theorem
[12]classifiesthemin thefollowing
whichare homotopically
By thiswe mean thatthea, are disjointand not freely
independent.
e
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1981]
THE UNIFORMIZATION THEOREM
589
homotopicto each otheror to theideal boundaryof S. To eacha, we assigna positiveintegern,.
a planarcoveringsurfaceSp whichis definedas the
The set of pairsP ={(a, n,)} determine
gives
largestcoveringsurfaceon whichax' is a simpleloop buta k is notfork < nI. This theorem
a completesolutionto thetopologicalpartof theproblem.
mappingproblem.Here we tryto findall
The secondpartof theproblemis theconformal
of sucha mapwas provedby Koebe; it is
mapsof all surfacesSpinto?. The existence
conformal
(see,forexample,Tsuji [18]).Letfbe themap and D = f(Sp). The groupof
theorem
hisplanarity
decktransformations
G forthecoveringD -* S liesin theconformal
groupAutD
automorphism
mapsof Sp into?
to knowall conformal
it is notimportant
of D. For purposesof classification,
mapsof domainsD in C. Maskit[13]
butjust one goodone.All othersareobtainedbyconformal
chosenso thatforall y E AutD, y is
may
be
D
that
he proved
founda verygoodone. Specifically
acting
The groupG thenbecomesa groupof Mobiustransformations
a Mobiustransformation.
on a domainD C ?. Such objectshad firstbeen studiedby Schottky
properlydiscontinuously
and later by Fricke and Klein. They are called functiongroups.Now assume S has finite
showed [14] D may be chosen so that each componentof
conformaltype.Maskit further
Int(C \ D) is a Eucidean disc. Such groups,he called Koebegroups.In [15],he classifiedthe
type.
forsurfacesof finiteconformal
Koebe groups.This solvesthegeneraluniformization
Appendix
5.3. The proofof theformer
Thisappendixcontainstheproofsof Lemma5.1 and Proposition
is rathershortand we giveit first.
Proofof Lemma 5.1. St C S and as, is a piecewise analyticarc. As we noted in ? 5, Proposition
forall t > 0. We claimthatSt,is simplyconnectedfor
4.3 impliesthatS, has a Green'sfunction
all to. Let a be a closedcurvein Stobased at t0. Let
A(a) ={t
E [to, oo)la
is nullhomotopicin
St.
Since a is null homotopicin S, and the homotopytakes place in a compactsubsetof S,
A(a) + 0 and is open. To see thatA(a) is closed,observethatif t1E A(a) thenso is t forall
F1 of a
t > t . Let t2 = inf{tltE A(a)}. Choosea smalldisc D about k(t2).Thereis a homotopy
h ofD so thathIa D = id and h moves
to theconstantmap t0in S?2+e. Choosea homeomorphism
to a homeomorphism
by theidentity
0(t2 + e) to 0(t2 - e) as shownin Fig. 9. h maybe extended
of [to, cc), it
of So. h o F1(12) C St2;hencet2 C A(a) andA(a) is closed.Fromtheconnectedness
followsthatt2 toand a is nullhomotopicin Sto.
FIG. 9
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590
[October
WILLIAM ABIKOFF
is modeledaftertheideasofRiemann5.3. The proofof thisproposition
ProofofProposition
of S.
by Klein [13]. We shall solve a potentialproblemon a subsurface
at least as interpreted,
we shall
curvesofthegradientfieldof thatpotential,
thelevelsetsand integral
Then,by studying
readerfor
of theknowledgeable
obtaina globaltopologicalresult.The authorbegstheindulgence
and harmonicfunctions.
severalbasic resultson holomorphic
presenting
of 0 E C withf(O) 7#0. Then
in a neighborhood
holomorphic
Letf be a function
f(z) =
Eanz1=
ao+
akzk+
o(lzlk)
so that
whereo( z k) meansa function
lim o(Zk)
z-O
k is the order of f(z)
-
zk
ao at 0.
arg(f(z) - ao) = argak+ kargz + o(1 Z I k). It followsthat{z larg(f(z) - ao) = 0} consists
of
of k analyticarcswhichmeetat 0 and whosetangentsat 0 are equallyspaced as a function
argz. If k = 1, then,near0, {z Iarg(f(z) - a0) = 0} is a simpleanalyticarc. The pointswhere
of 0,
in a simplyconnectedneighborhood
f'(z) = 0 are discrete.If u(z) is a harmonicfunction
thenit has a harmonicconjugatev(z) there.
pointofu iff'(0) = 0. { z Iv(z) = v(0)}
0 is calleda critical
f(z) = exp(u + iv) is holomorphic.
0.
consistsof k analyticarcsthrough
The discussionabove remainsvalid forpointsotherthan0 and,indeed,is valid on Riemann
surfaces.
curves.Let dl E S and N
Now letS be a simplyconnectedRiemannsurfacewithno divergent
be a closeddisc about t, withanalyticboundary.Set S, = S \ N. The readermayeasilyverify
By Proposition
to S, and we mustonlyshowthatS, is simplyconnected.
thatS, is homeomorphic
4.3, S, has a Green's functiong(q, t0) forany t0 E S,.
5.1,we leth0 be a harmonicconjugateofg(q, t0) + loglz near
As in theproofof Proposition
=
continuedalong all
+
and
let
z(')exp[-(g
f0(')
t0
loglzI + ih0)].f0(') may be analytically
theanalyticcontinuation
pathsin S,. Sincewe do notknowa priorithatS, is simplyconnected,
maynotdefinea function.
Near t0, f(z(')) = a,z + o(Iz 1)and it followsthatf is locallyinjectiveneart0 and thereis a
neighborhoodof t0 containingno criticalpoints of g. If g has criticalpoints,let M=
sup{g(', 9'o) I 7#t0 and D a criticalpoint of g}. Otherwiselet M = 0. Let D ={
M} U {t0} and B C C be the disc of radius e-M about 0.
LEMMA
E
B. 1. Analytic
a holomorphic
continuation
f: D
offoin D defines
bijection
S I g(t', '0) >
-)
B.
Proof.Let T(6) be the curvein D startingat t0 along whichfo continueswithconstant
by e 9.
Since T(6) containsno criticalpoints,it is uniqueand maybe parametrized
argument.
curves,T(6) is a simplearc fromt0 to a D = {Ig(t o0) = M}.
SinceS has no divergent
of some analyticcontinuation
Let D t D \ {t0} and 00 be the argument
f, offo to t. Then
D withargf1= 00. ExtendA
uniquearc through
f f(D)I exp(-g(t,9'0). Let A be thenecessarily
so thatit is maximalwithrespectto being a curvealong whichf, continueswithconstant
in D. Since g is monotoneon A and D containsno criticalpoints,A is simpleand
argument
is
or containsan arc T( 6) fromD to t0. The firstpossibility
analytic.Thus it is eitherdivergent
ThusA = T(61) and we maywrite0(D) = 61. Thuseachpoint
ruledoutbyourinitialassumption.
in polar coordinatesby (e -g,0(')), witht0 beingtheorigin.The
; E D \ {t0} is parametrized
of D withB. It follows
henceis a homeomorphism
is continuousand injective,
parametrization
offo.
is preciselytheanalyticcontinuation
thatD is simplyconnectedand theparametrization
of D withA. If M > 0, we
If M = 0, we are done,sinceD = S, andf is a homeomorphism
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1981]
THE UNIFORMIZATION
591
THEOREM
have
LEMMA
B.2. If M > 0, thenthereis a critical
pointt'2
E aD
Proof.Firstsupposethatthereis a criticalpointt'2 so thatg(W2,W' = M. Then arbitrarily
close to t'2 arepointsin D and t'2 & aD. Otherwise
thereis a sequenceof criticalpointst'1 so that
10) -- M. Througheach tn thereis an arcAn alongwhichg increasesand anycontinuation
g(W,
offohas constantargument.
SinceS has no divergent
curves,A,nconnectst,nto t0. For some0,
=
D
6
On
a
and
subsequence,
T(6) musthave an endpointt. Fromthelocal
A,1
n
T(n).
f9n
of thecurvesargf = constantnearta, we see thatta is a limitof criticalpoints.Since
structure
criticalpointsofg are discretein S1, thisis impossibleand theLemmais proved.
LEMMA
B.3. If M > 0, thenthereexists_a
closedannulusA C S withpiecewiseanalytic
boundary
so thatS \A is connected.
HereA = IntA.
Proof.Usingthepreviouslemma,thereis a criticalpoint '2 E aD. Againby thelocal structure
near a criticalpoint,thereare at least twocurvesT(61) and T(62) emanatingfrom'2* Choose
closeddiscsBo and B2,withcenterst0 and '2, respectively,
whicharedefinedby Bo {= Ig(', to)
For
c'} and B2
small,theB, are disjointand
={-Ig(t,
t)-g(t2,
-o) < e}.
E sufficiently
containno criticalpoints.For 6 sufficiently
close to 6,, T(6) is a curvefromBo to B2. Let
E = U{T(6)1 I0-6.k < 8}. , is a stripfromBo to B2 fori= l,2. LetA = E1U E2 U Bo U B2 as
in Fig. 10. For nearlyall but finitely
many6,, T(6,) n B2 = 0 for E sufficiently
small. Let
=
\
A
E
S
where
A
A.
Any
analytic
Int
continuation
of
to
a
neighborhood
of
,q
f, fo
q maybe
continuedalong a curve with decreasingmodulusand constantargumentto aS1. Thus if
of S \ A.
g(), t0) : M, thecurvewillnotmeetA and q and as1lie in thesamepathcomponent
If q E D \ A, thenq lies in a complementary
sectorof 2 U 2 in D. Choose T(6) in thatsame
sectorso thatcontinuation
alongT(6) withdecreasing
modulusand constantargument
does not
lead us intoB2. We maythenfurther
continueto MB., T(6) and aB thuslie in thesamepath
ofS \ A, and S \ A is pathconnected.
component
PROPOSITION
B.1. If S is anyRiemannsurfaceand A is a closedannuluson S withpiecewise
analyticboundary
and suchthatS \ A is pathconnected,
thenS is notsimplyconnected.
T0)
T(O~~
~2
8B.
FIG. 10
Proof.A has twoboundarycomponents
F1 and F2. Further,
thereis a simpleclosedcurvea in
S so thata n IntA is a simplearc fromone boundarycomponentto theother.
SolvetheDirichletproblemin A withboundarydata p1 F, = I1 and 4 11F2 = 0. The Dirichlet
problemin S \ IntAmaybe solvedwithboundary
data021 F1 = 1 and021 F2 2. Set4 = expi7r+,
4 oa has windingnumber 1 aboutz = 0. If a
and noticethatp is definedon S and continuous.
is nullhomotopicin S then4 oa is nullhomotopicin aix. The latteris impossible;henceS is not
simplyconnected.
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592
PATRICIA C. KENSCHAFT
[October
The researchforthispaper was partiallysupportedby theNational ScienceFoundationand theAlfredP. Sloan
Foundation.
References
1. L. Ahlfors,ComplexAnalysis,2nd ed., McGraw-Hill,New York, 1966.
, ConformalInvariants,McGraw-Hill,New York, 1973.
2. __
3. L. Ahlforsand L. Sario,RiemannSurfaces,PrincetonUniversityPress,Princeton,N. J., 1960.
4. L. Bers,RiemannSurfaces,CourantInstituteLectureNotes, New York, 1958.
5.
, Uniformization,
moduliand Kleiniangroups,Bull. London Math. Soc., 4 (1972) 257-300.
, On Hilbert's22nd Problem,Proceedingsof Symposiain Pure Mathematics,
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Amer.Math. Soc., 28
(1976) 559-609.
7. J. Conway,Functionsof One ComplexVariable,Springer-Verlag,
New York, 1973.
8. M. Greenberg,Lectureson AlgebraicTopology,Benjamin,New York, 1966.
9. M. Heins, The conformalmappingof simplyconnectedRiemannsurfaces,Ann. Math.,50 (1949) 686-690.
10. D. Hilbert,MathematicalProblems,Bull. Amer.Math. Soc., 8 (1902) 437-479.
11. E. Hille, AnalyticFunctionTheory,Blaisdell,New York, 1959 and 1962.
12. K. Kendig,ElementaryAlgebraicGeometry,Springer-Verlag,
New York, 1977.
13. F. Klein, On Riemann'sTheoryof AlgebraicFunctionsand TheirIntegrals,Dover, New York, 1963.
14. F. Klein, CollectedWorks,vol. 3, Springer-Verlag,
Berlin,1923.
15' P. Koebe, Uber die Uniformisierung
beliebigeranalytischenKurven,GbttingerNachr. (1907) 191-210.
16. B. Maskit,A theoremon planarcoveringsurfaceswithapplicationsto 3-manifolds,
Ann. of Math.,81 (1965)
341- 355.
, The conformalgroupof a plane domain,Amer.J. of Math.,90 (1968) 718-722.
17.
, Uniformizations
18.
of Riemannsurfaces,Contributions
to Analysis,AcademicPress,New York, 1974,
pp. 293-312.
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19.
of Kleinian groups:I-Koebe groups,Acta Math. 135 (1975) 249-270.
20. H. Poincare,Sur l'uniformisation
des fonctionsanalytiques,Acta Math. 31 (1907) 1-63.
New York, 1969.
21. C. Siegel,Topics in ComplexFunctionTheory,vol. 1, Wiley-Interscience,
22. M. Tsuji, PotentialTheoryin Modem FunctionTheory,Maruzen,Tokyo, 1959.
23. H. Weyl,The Conceptof a RiemannSurface,3rd ed., Addison-Wesley,
Reading,Mass, 1955.
BLACK WOMEN IN MATHEMATICS IN THE UNITED STATES
PATRICIA C. KENSCHAFT
Department
ofMathematics
and Computer
Science,MontclairState College,UpperMontclair,
NJ 07043
Increasedattention
has been focusedon womenin mathematics
duringthepast decade,but
whenI was invitedto speakon Blackwomenin mathematics,
I couldfindonlytworeferences-a
talkby VivienneMalone Mayes[1] at theSummerMeetingin Kalamazooin 1975sponsoredby
theAssociationforWomenin Mathematics,
and theAWM panelI chairedin Atlantain January,
1978[2]. SincethenI havecollectedmuchinformation,
and thisarticletellsabouttheAmerican
Blackwomenholdingdoctoraldegreesin mathematics,
all but twoofwhomI havetalkedwithin
thepast threeyears.
The 1970decennialcensusrevealedmorethan1,100Blackwomenwhoreportedthemselves
as
mathematicians.
In thatcensus244 said thattheywerecollegeor university
teachers,and, of
Adapted froman invitedaddressgivenat the annual meetingof the Associationof MathematicsTeachers of
New Englandin Springfield,
Massachusetts,on November2, 1979.
PatriciaKenschaftreceivedherPh.D. witha specialtyin functionalanalysisfromtheUniversity
of Pennsylvania
in 1973 underthedirectionof Edward Effros.Since thenshe has taughtat MontclairState College in New Jersey.
She is theauthoror co-authorof threetextbooksfornontechnicalmajorspublishedby WorthPublishers,Inc., and is
currently
preparinga paper on thelifeof CharlotteScott,vice presidentof theAMS in 1906.-Editors
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