Annales Academire Scientiarum Fennicre Series A. I. Mathematica
Volumen 15, 1990, 293-306
ON CONFORMAL WELDING HOMEOMORPHISMS
ASSOCIATED TO JORDAN CURVES
Y. Katznelson, Subhashis Nag, and Dennis P. Sullivan
Abstract
tr'or any Jordan curve
C
on the Riemann sphere, the conformal welding home- omorphism, Weld(C), of the circle or real line onto itself is obtained by comparing the bLundary values of the Riemann mappings of theunit
disk or upper half-plane onto the two complementa,ry Jordan regions separatedby C.
In
PartI
we showthat
a family ofinfinitely
spiralling curves lie intermediate between smooth curves and curveswith
corners,in
the sensethat
the welding forthe
spirals has a non-differentiablebut
Lipschitz singularity. We also show how the behaviour of Riemann mapping germs under analytic continuation impliesthat
spirals were the appropriate curvesto
produce such weldings.A
formula relatingexplicitly
therate of
spirallingto
the extent of non-differentiablityis
proved by two methods, and relatedto
known theoremsfor
chord-arc curves.Part II
studies familiesof
Jordan curves possessingthe
same welding. We utilise curveswith
positive area and the Ahlfors-Bers generalised Riemann map-ping
theoremto build
aninfinite
dimensional "Teichmiiller space"of
curves all having the same welding homeomorphism.A
point of view on welding as a prob- Iem of extension of conformal structure is also derivedin
this part.Introduction
In this paper we address two questions regarding the conformal welding home- omorphism, Wetd(C), associated
to
any Jordan curveC
on the Riemann sphereÖ.
W"ta1C) is the homeomorphism of the real line on itself obtained by compar- ing the boundary values of the two Riemann mappings of the upper half-plane U ontothe two
complementary Jordan regionsDr and D2
separatedby C.
(W"assume
that C
is an oriented Jordan curve, sothat
D1 is chosen to be the regionto
theleft of C.)
Precisely,
Iet fi: IJ
--.D;, i:Lr2,
denote any two Riemann mappings ontothe
relevantdomains. By
caratheodory's theorem one knowsthat fi *d f,
extend continuously
to
the boundaries.Let 0f;:0U : RU {oo} 'C, i:7,2,
classification: 30c20, 30c35.-Keywords: conformal mapping, boundary values, Jordan curves.
doi:10.5186/aasfm.1990.1517
294 Y. Katznelson, S. Nug, and D.P. Sullivan
denote these two boundary homeomorphisms.
Then Weld(C) is
the orientation reversing self-homeomorphism of the circle(R
U{*})
given byNote
that
reversal of orientationon C
replaces the welding byits
inverse.Since
fi
ar,df2
are ambiguous up to pre-composition byarbitrary
elementsof Aut(U) : PSLz(R),
we notethat Weld(C)
as definedis
ambiguousup
toarbitrary
pre and post compositionsby
elements ofPSL2(R).
Moreover,if
thecurve C is
replacedby its
image underany
Möbius automorphismof Ö,
the corresponding welding homeomorphism class remains preciselythe
same. Thus, one shouldthink
of the welding as a map:(1)
(2)
Weld(C)
-0f;'o0fr.
Weld: {Möbius equivalence classes of Jordan curves on
C}
--+ PSLz(R)
\
Ho*eo(^9t)I
YSLz(R)."(1) -
{Jordan curvesthat
are quasicircles}/
UoU(C)f G) -
{Quasisymmetric homeomorphisms ofR fixing
0, 1,m}.
By normalising the choice
of fi
andfz *"
may assumethat Weld(C)
fixes eachof the points 0, 1, oo.
Nofurther
normalisations are then possible usingpre
andpost
compositionsby PSL2(R) transformations. It is
also sometimes convenient to make Weld(C) orientation preserving by takingf2 tobe
a Riemannmapping
from
the lower half-plane.L
orrloD2.
Wewill
assume,without
lossof
generality,that C
passes through oo, andthat fi(m): oo, ,fz(oo):
oo.The conformal welding arises rather ubiquitously
in
various contextsin
ge- ometricfunction theory. As is
well-known,two
standard waysof
viewing the universal Teichmäller space7(1)
are:(3) and (4)
The passage between these two descriptions of the complex Banach manifold
7(1) is
preciselyvia
conformalwelding. In particular note that
Weld(C) for C
aquasicircle is quasisymmetric, and
this
correspondence is a nice bijective one.One may a,lso note
that
the homeomorphismWeld(C),
(andits
barycentric extension), was utilised critically for arbitrary Jordan curvesC
in the paper [E-N].With
the foregoing background we are ready to explain our workin
this arti- cle. Inside the universal Teichmiiller space7(1)
one may conveniently distinguish certain special subfamilies of quasicircles.First
of all there are the smooth(C-)
Jordan curves-whose corresponding welding homeomorphisms are
C-
diffeomor-phisms. Because,
by
classical results of S. Warshawski, the Riemann mapping to a Jordan domainwith C-
boundary extendsC- to
the bounclary.On conformal welding homeomorphis.rns 295
In
contrast, suppose the curveC
hasa
"corner" of positive angle oz-,(0 <
q < 1), at
somepoint. Weld(C) will
then havea
"powerlaw"
behaviour(t
r-,7Q-a)la,
neart : 0) at the
correspondingpoint (t : 0 say). Thus
Weld(C)will
have vanishingor infinite
derivatethere. A natural inquiry
thereforeis
to seek Jordan curves which arein
a sense intermediate betweenthe C-
ones andthe
oneswith
cornersby
requiringthe
welding homeomorphismsto
have non- differentiablebut
Lipschitz behaviour at some point(s).(5)
Table.Curve Welding
c@ C@ diffeomorphism
? Lipschitz
Corner Power law
In Part I of this
paper wewill ffll in the ? slot in the
above tableby
ex-hibiting
a largefamily
of. infrnitely spiralling Jordan cu.rve,s for which the welding homeomorphism, at the point correspondingto
the eye of the spirals, has the req- uisite behaviour. The derivatives from the left and right of Wetd(C)will
exist, befinite
and non-zero,but
are unequal. Moreover, we find an explicit formula (equa-tion
(8)) relating the rate of spiralling parameter to theratio
of theleft
and right derivatives. We deducethat
the discrepancy between the left andright
derivativeswill
decrease downto
zero as the rate of spiralling increasesto
infinity.This last assertion ties in intimately
with
certain general results of David et aI.regarding the welding homeomorphisms
of
"chord-arc" curveswith
small chord- arc constants. The "chord-arc" curves are a well-known subfamily of quasicircles, and our spiralsfall within
this subfamily.In
SectionI.2
we developa
criterionfor Weld(C) to
possessa
non-differ- entiablebut
Lipschitzpoint in terms
of. analytic continuation propertiesof
the Riemann mappings involved. This leadsto
a rather surprising explanation, from a very different point of view, of why the spirals were theright
Jordan curves for producing the type of Weld(C) desired. The explicit formula (8) mentioned above is reproved by the new method.In marked contrast to the previous part, Part
II
of this paper looks at Jordan curves quite outside the realmof
quasicircles. We considerthe
question of non-injectivity
of the welding correspondence (2).We reinterpret welding as a problem of extension of conformal structure. This gives us a criterion for a homeomorphism to be a welding, and also for there to be only one curve from which
it
arises as welding homeomorphism.One
then
observesthat
there exist Jordan curvesC with
positive two-di- mensional Lebesgue measure.The next
ideais to
usethe
,,p,-trick,(i.".
the Ahlfors-Bers generalized Riemann mapping theorem),to
construct a largefu*i|y
of Jordan curves possessing the same welding homeomorphism as C.
296 Y. Katznelson, S. Nug, and D.P. Sullivan
It is natural to
consider thena
"Teichmiiller spaceof C'-comprising
tJrevarious quasiconformal deformations
of C with dilatation
supportedon C.
Weconclude by conjecturing
that
the Jordan curves sharing the given welding home- omorphism are precisely the members of this Teichmiiller space.Ad<nowledgements.
T.
Wolff and S.C. Bagchi are thankedfor
useful conver- sations. The second author would liketo heartily
thank theInstitut
des Hautes Etudes Scientifiquesfor their kind
hospitality during Spring 1989, when much ofthis
work was done.Part I. Welding homeomorphisms for spiralling
curves1. A rnain result
We start
with
a simple but crucial observation on the"locality"
of the welding correspondence. Suppose we wishto
concentrateat
apoint
zs anthe
curve C.The
natureof Weld(C) at the
correspondingpoint to:?ftl(zs) is then
at issue. The point of the following Lemma isthat
although the Riemann mappings are global constructs, the smoothness/non-smoothness of Weld(C)at t6
are only dependent on the local natureof C
around zs.Lemma I.L,
LocaJIy the welding homeomorpåismis
determined, up to pte a.ndpost
compositionby real analytic
diffeomorphismsof real
intervaJs,by
an a,rbitra,rily smaJl neighbourhood of the cltrve.Proof.
Consider two different conformal mappings,"f *rd g, of
a. standardhalf-disk D onto a
topologicalhalf-disk B
boundedby a portion of the
curve around the relevantpoint
zs€
C .C
Figure I.1.
On
conformal welding homeomorphis"ms 297 The self-mapg-'o /
onD will
extend conformally, by the Schwarz reflection principle, throughout afull
disk containing the real interval AD. It
follows that the weldingti
determined by looking at any small portionof
C arounf zs is obtainedfrom the
actual welding ur, (determinedby
the global Riemann mappings), via the relation:(6)
where
$
andnate changes)
?i)-
QzowoQtQz are two real-analytic diffeomorphisms (i."., real-analytic coordi- of the relevant segments on the real axis. s
Figure I.2.
where
it
polar coordinates:given
by
Co- 7t
U{0}
U ^yz,0
-> *oo
(7) ^lt ', f : €-o0 , ^tZiT:-€-o0,
so each C o is a disjoint union of two logarithmic spirals that
join
up at the vortexpoint z : 0
(FigureI.2).
Thispoint
is our focus ofinterest-we will
callit
the"eye"
of
these spiralling curves.The quantity a, 0 ( o < oo, is the "rate
of spiralling" parameter.Theorem T.2,
For eacha,
0<
a< x,
the homeomorpåism Weld(Co),in
a neighbourhoodof
the point (sayts)
correspondingto
the eyeof
Co, is Lipschitz butnot
differentiableat ts.
The left derivative (,\1,
say) and theright
derivative (),2 , say)for Weld(C") both
exist, are finite and non-vanishing at ts, but
do notmatch.
In fact, the
rate explicitly:(8)
lml'
of
spiralling pararneter can be relatedto the ratio \t
I\z
Thus, the discrepancy between the derivatives
from
the opposite sides decreasesmonotonically to zero as the rate
of
spiralling blows up toa.
\
\ r,
298 Y. Katznelson, S. Nug, and
D.P.
SullivanRemark. Weld(Cr) is, of
course, diffeomorphic everywhere exceptat
f6since Co is
a C-
curve away from the eye. The last statement assertsthat
as the rate of spiralling becomes larger and larger the curve, or more precisely its welding, looks more and more smooth evenat
the eye.This
assertionwill
be ramified in twofurther
waysin
following sections.Proof. The proof is by a direct
computation keepingin mind the
locality lemma. Consider the entire functionG(z)
-
e-(a*i) zG
mapsthe positive r-axis
and certain parallel horizontal half-Iinesonto
the spirals.ft
and 12.In fact, let R
denotethe half-strip
region below and abovethe
horizontal half-lines U: 0
and A:
alr/(! * o'),
and bounded to the teft by the line segmentjoining
the origin to\-- 7f
-; ar
Lla2'"L+oz'
Then the maps G1
: G and
Gz: -G
restrictedto
.B are conformal mappings onto releva.nt portions ofthe
complementary Jordan domains D1and D2.
The boundarypoint
"oo!iy" of
.E correspondsto
the eye of. Co.Now, the welding
for
Co, looked at on the boundaryof
.8, becomes- lz*\,
ono-axis,'*- t r-),, orry: atr/(t*o').
Making a conformal change of variables, (using a holomorphic logarithm), we can replace the domain .R by the upper half-plane and get the
critical
boundarypoint
" oo+ iy" to
correspondto
the real numberlo :
1.
Normalising sothat
Weld(C") mapsts
to itself, we get finally the following expression for the welding:(e)
(10) * werd(cs)
f
((*,fi - il l@'fi - o))' for t >
1,t( v
t(("'fi+il|@rfi+o))'fort<1
in
a neighbourhood of t6:
1 on the real line. Herc B>
1> o )
0 are determinedfrom the spiralling parameter by
therelations 0:L+e-n/o, a:L- e-r/a.
[This expression (10)
for
WeId(Co) isup to
real-analytic changes of coordi- nates, as explainedin
Lemma I.1.]Now, calculating the derivatives of (10) from the
left
andright
oft :
1, one obtains(" - §)/(" +
B): -e-tla
from theleft,
andits
reciprocal,-eulo,
fromthe
right. But
equation (6)in theproof
of Lemma I.1 showsthat the
ratio )qf ),2does
not
depend on ma^king real-analytic coordinate changes. Consequently one gets Å1/)2- u-2r/a,-md
equation (8) follows. trOn
conformal welding homeomorphismsPart I.2. Lipschitz welding and analytic continuation of Riemann mappings
There is an interesting and distinctly different way to explain why the spirals were the right coordinates to produce Lipschitz welding homeomorphisms, utilising an idea about analytic continuation
of
Riemann map germs. Even formula (8) emerges naturally. We explain this method below.Assume
that
Weld(C):R
--»R
has a graphwith
a cornerat zero-say it
isr
++\1x for
small negative valuesof r
andr r+
),2afor
small positive values ofr.
(Namely, .\1 and .\2 are the derivatives of Weld(C) form the left and the rightat 0.
Rememberthat
we are interested onlyin
the local considerations.)Let f1: U -+ D1
andf2: L
--+Dz
be Riemann mappings ontothe
comple- mentary regions separatedby C, with fi(O): å(0):
thepoint
of interest(p)
on C.
We assumethat
the arc of the curveC
on each sideof p
is real-analytic up to,but
not including, thepoint p.
Thus the curveC in
a neighbourhood of p looks liker u {p}
U 72 where "[1 and1z
aJe two open real-a,nalytic arcs meetingat
p.Y-/
\---l
Figure L3.
Consider the restriction of the conformal map
f2
or.a small half-disk E1 (inthe lower half-ptane) located infinitesimally to the left of the origin (see Figure I.3).
The idea
is to
usethe
stipulated natureof the
welding homeomorphism to obtain a relation between/2
restrictedto E1
andits
analytic continuationin
acircuit
around theorigin. In
whatfollows'r4.'will
be generic notationto
denote analytic continuation along some path.Since
/2
mapsthe
real segmenton the
boundaryof E1 to a
pieceof
thereal-analyti
c
arc J1, one knowsthat f2
must extend conformallyto
thefull
diskDzU?Et UE1
(see,for
example, Nehari [N,p.
186]). Comparingthis
extended 299B
\ /
300 Y. Katznelson, S.
Nrg,
and D.P. Sullivanf2,
callit Af2 (on
D2 )with
the Riemann mapå ot
E2 we seethat
(11) fr'
oAfr: Kt, on
E2where
K1
is some conformal map on82.
We assumethat I(r
is analyticin
a ballB
around0.
Then the power seriesfor K1
must be( 12)
because (11), together
with
the stipulated natureof Weld(C)
on the left of zero,imply that
the derivativeof K1 at 0 is
,\1 1.Thus
the
analytic continuationof /215, up to the
regionE3 will
producethe function f1
o K1.
Now, againby an
argument as above,h
oKt
extendsanalytically to the disk
Es UEl,
and wemay
comparethis
extension,call it A(1,
".I(1) : Afi
oK1 with /2
itselfon Ea.
We get(13) fl'
oAh
o K1: !{r, on
»4 whereK2
has a power series expansionKr(r): {r, + O(r'), around
z:
0tKz(z\- 'J,
3\ ')
Ät+ O(r'), around.
z-
o.Afr(r): fz(fr)
This follows because Weld(C)
: 0f ;'o Ofi
has slope)2
to the right of the origin.Thus, frl»r,
analytically continuedin a
clockwise circle around the origin, ends up as f2oKz.
Wewill
neglect termsO(r')
in the Riemann mappings, since we are interested onlyin
local structure around0.
We therefore see thatif
the welding åas sJopes)1
and ),2 from the left and right of zero respectively, then the germof the
Riemann mapf2
restrictedto a
region infinitesima.llyto
the leftof
zerowill
satisfy the funtional equation:(14)
( 15)
( 17)
where
A
denotes analytic continuation alonga
clocl<wisecircuit
encircling zero.Notice
that
the analytic functionsin
.D satisfying (15) clearly form an algebra.The countable family of functions:
(16) px(z):
exp(§xlogz),with
0* -
1"s(\rl 2kri
)z )- 2ir'
k-
0,*L,*2,...
On conformal welding
homeomorpåisms
301 wherelogz
is any holomorphic login
the lower half-plane, aJI satisfy (15), (easyto
verify).Moreover, these
gx
linearly span(This
is becausegi?t -
gj+x,
sincec.)
To see therefore what
kind
of curveC will
produce the type of welding pre- scribed, we only needto
calculate the images of smaJl portions of the positive rcaJaxis
(R+)
andof
the negative real axis(R-)
underany
non-trivial(i.e.
k+ 0)
one
of
these functionsgx.
Indeed one obtains twologarithmic
spiraJswith
the samerate of
spirallingfrom
thetwo
sidesjoining up at their
common eye (thepoint 'p'
on C).
We indicate the calculation below.Setting
\l\z:
Å, one gets from (17)a subalgebra of the algebra satisfying (15).
the 0x
constitute an additive subgroup of,x:(ffi)+i(ffi)
,so that
( 18)
R.0*
r*
0*is independent
of k.
Now theportion 1of. C is
g7,(l)for t < 0, t -» 0-
andthe portion y of C is
g1,(f)for t ) 0, t - 0*.
Theseturn out to
be spirals,and the sign
of
fr hasto
be controlled sothat
the eye of these spirals ast
-+0
isnot at m.
(This depends on the signof Imgx.) In
polar coordinates one obtains the curves J1 and 72 to be of the form (assuming ,\1> lz
)t( 1e)
Here (20)
Ttir:Ate-o', jz:r:Aze-ao, 0++oo.
comes out independent -lfote
that (19)
andRemark. If
Å1<
.\2 one wouldfind
the spirals goingin
clockwiseto
their common eye,i.e.
as0
---+-m.
The equationsfor
the spirals are essentially the same as (19)(with a
replacedby -r).
By reflecting the curveC,
and changing the orientation onC if
necessary, one can get the spirals to match the form (19) or (7) exactly again. (ReflectingC
replaces the weldingä(t) by -h(-t);
reversingorientation replaces
h by h-r
.)--m 2r
a:1. ?"
Illos(
\l )r)
Iof the choice
of lc. At
andAz
are unequal real constants.(20) tally exactly
with
(7) and (8)in
the previous section.302 Y. Katznelson, S.
Nrg,
and D.P. SullivanSomewhat more generally
it
is easyto
showthat
evenif
one workswith
alinear
combinationg of
variousg*'s,
then onestill
obtainsinfinitely
spiralling curves as the image under underg of R+
andR-.
This is because the term ing with
oneparticul* gr will
eventually dominate ast
--+0,
and consequently the image curvewill
look asymptotically exactly like the logarithmic spirats (19) obtainedfrom that
dominatingcpr. (To be
precise, wecall a
curve infinitely spiralling around an "eye" p on it, if the argument function on the curve (measuredwith p
as origin) becomes unboundedin
every neighbourhood ofp.)
Our resuläs of the preceding section are thus exactly corroborated, and some-
what
generalised,by
the present analysis. Note that the rateof
spiralling always satisrles(20), (i.e. (8) of Theoreml.2),
and depends only on theratio of
slopesof Weld(C)
from thetwo
sides of the 'singular' point.Remark.
The functional equation (15)for
the approximate Riemann map- ping makes contactwith
J. Ecalle's theoryof
"resurgent functions".Part I.3. Chord-arc spirals and their weldings
The spiral
family
we have been deaJingwith
are "chord-arc" curves, and we want to indicate a relationship between our results above and known facts about the welding homeomorphism for chord-arc curyes.Recall
that
a Jordan curve (normalisedto
go through oo)
is called a chord- arccurveif itisarectifiablecurve z(s) (s isanarc-lengthparameter),suchthat
thereexistsIl>0 sothat
(21)
for all
s1 and s2 real; see Semmes [S].Condition (21) implies Ahlfors' well-known condition for being a quasi-circle.
In fact, chord-arc curves are the images of
R
under bi-Lipschitz homeomorphisms of the plane, where quasicircles are the imagesof R
under the more general qua- siconformal homeomorphisms of the plane.Our double-spiralling curves Co of Theorem I.2 are chord-arc curves. This is best seen by remembering (Semmes, loc.
cit.) that
chord-arc curves arein
one-one correpondencewith a
certain open regionO in fåe
real Banach spaceBMO(R)
(real valued functionson R
of bounded mean oscillation). The connection isthat
given 6€ O
one associates the chord-arc curve suchthat
(22)
For the choice (22)
On conformal welding homeomorphisms 303 one obtains precisely the double log-spiral
C"
(apto
a globalrotation
and mag-nification).
Thus, the subfarnily"f
quasicircles comprising the chord-arc curyes is a real Banach manifolddl
containing the one-parameterfr*ily of
curves Co.The welding homeomorphism
ä for
any chord-arc curve has derivative a.e.and
log(h')
is known to lie againin BMO(R).
(This is implied by the factthat
ä' itself is known to lie in the class Aoo of weights of Muckenhouptl see [S] and UK].) One thus has, from the welding correspondence (2), a well-defined map(24)
W: f)-* BMO(R)
by associating
to
any ö € Othe BMO
function log(Wetd'(26)).It
is knownthat W
maps a small neighbourhoold of the origininto
a small neighbourhood of the origin. Indeed, the curves z6 arising from small BMO func- tions ä, (which are precisely the chord-arc curves with small chord-arc constantK
of (21)), have been shown (David [D])
to
be characterized by having small BMO normfor log(ä').
See [S], [D] and work of Coifman and Meyer.This
tiesin
exactlywith what
we establishedin the
preceding two sectionsabout the
spiralswith
very largerate of spiralling a.
Indeedthe BMO
normof
åin
equation (23) andthe
correspondingK for
the spiral,is
small precisely whena
is very large.But
thenlog(h')
has ajump
discontinuityat
0 of jump sizel), - )zl,
where)1 and
Å2 are as usual the derivativesfrom
theleft
and rightof 0 of Weld(C,). The BMO
normof
a functionwith
ajump
discontinuityis easily estimated (locally around thejump)
as approximatelyhalf
thejump
size.ConsequentlS David's result
that log(ä')
should have smallBMo
norm for largerate of
spirallingfits
preciselywith the last
assertionof
TheoremI.2-which
is what equations (8) and (20) implied.Remark. If
one knewthat the map W in (24)
was continuous(in
theBMO
norms) then one could generalise the result of Theorem 1.2to
an infinite- dimensionalfamily of
chord-arcspirals. In fact then all
chord-arc curves closeto
thefamily C,
would possess weldingsthat
are(in BMo norm)
closeto
the weldingsfor the Cr.
SuchBMO
functions, which are closeto
functionswith
ajump
discontinuity, have also a characteristic kind of discontinuity. Unfortunately themap I7
is not known to be continuous (except at the origin, as \Me noted).Remark. In
concludingPart I
we would like to recallthat
spiralling curves figured prominentlyin
the well-known paper of Gehring [G].Part II. Jordan curves sharing the same welding
We
study
herethe non-injectivity of the
welding correspondence(2).
To handle the inverse mapto
(2) we consider the following point of view.Given any (orientation preserving) homeomorphism ä: ,91
- sl
we createa
2-sphere S3 bVjoining
two copies of the closedunit
disk along the boundary304
circle-the
attachit g usidentify the
circle identification space (25)Y. Katznelson, S. Nug, and D.P. Sullivan
map used
on the
boundary beingh. More
precisely, let5t
asR
U{*} _ R, on the
sphereÖ. Then Sl is
theS7: (u
un)
Un (2, un)
where
U
and.L are the upper and lower half-planes respectively. Noticethat
^9f isa topological^2-sphere
with
a well-defined conformal structure on the complement of the circleR.
The following proposition can be seen straight from the definitions.Proposition II.1. The
homeomorphism ä:R -- fr. o""rt"
asa
welding homeomorphismfor
some Jordan curveC if
a.nd onlyif Sf;
canies a conformal structurethat
extendsthe
sta,ndard conformal structure already existing on the open regionsIJ
andL.
Forany
sueh conformal structure onSf;
(supposing at least one exists) the Riemann surface 521, becomes conformdly equivalentto
Ö.The
imageof n(c
521) underany
such biholomorphic equivalenceis a
Jorda.n curveC
whose weldingis
h.Corollary I1,2,
The Jordan curveC o, Ö i"
the unique one (upto
Möbius transformations)that will
producethe
weldingh :
Weld(C)if
and onlyif
any homeomorphism ofÖ
whichis
conformalon
the complement ofC is
necessarily conformalon
all ofÖ,
i.e., isa
Möbius transformation.Proof. Suppose C1 and C2 are two Jordan curves producing the same welding
h. Let p1 and gz
be two conformal equivalencesof
Sfr(with
possibly different conformal structures)onto Ö-as in the
above proposition; rp; throwsfr.
ontoC;, i: L,2.
Then o:
gz og, I
is a self-homeomorphism of the Riemann sphereÖ
which mapsC1
orfto C2 and whichis
conformal onthe
complementof Ct.
The corollary follows. o
Remark. If C is
asin the
above corollary wecall C a
removable set for conformalmapping. In the
analogous sense one can a,lsotalk of
Jordan curvesthat
are removable setsfor
quasiconformal mappings. This concept is relevant to what follows, and is importantin
somefurther
work.Having noted
the
criterionfor injectivity of (2)
we now show howto
applythe
Ahlfors-Bers generalised Riemann mapping theoremin a
simplebut fruitful
fashion
to
construct many inequivalent curves sharing the same welding. Since on"(1)
the welding correspondence is bijective, we are working outside the realm of quasicircles.A
classicalfact.
There exists Jordan curves possessing positive two-dimen- sional Lebesgue measure. Indeed, there exist Jordan curveson R2 that
contain the setAx A
whereA
is a Cantor setin
[0,L] of linear measurearbitrarily
closeto
1.On
conformal welding homeomorphisms 305 References. see Gelbaum andolmsted [Go], pp.
135-138. For the original paper see Osgood [O].The idea is now to take a Jordan curve
7
of positive area as above and consider Beltrami coefficients supported on7.
Namely, look at the complex Banach spaceL*(i
and considerit
isometrically embeddedin .L'"(C)
by extending functions to the whole plane by setting them identically zero off7. Let .L-(7)1
'--+tr-(C)1
denote the open
unit
balls.By
Ahlfors-Bers [AB], for everyp in .[-(C)1
there is a unique quasiconfor- mal homeomorphismof C
solving the Beltrami equation:(26)
6w-p.0w
and fixing the points
0, 1,
oo. Call the solution tD: ltv
.Theorem II.3.
For anyp e L*(l),
the Jordan curve1p
which is the imageof 1
underwP
possessesfåe
same welding homeomorpåism as1
itself.If t
haspositive areal measure
L*(1)t is infinite
dimensionaJ, a,nd consequently one has many Jordan cttrveswith
the same welding Weld(7).Remark. If 7
has zero two-dimensional measure then top is Möbius and the theorem istrivially
true.Proof. wp
is conformal on the complementof 7'
Soif fi and /2
were twoBiemann mapping-s
U
ontothe two
regionsD1 and D2
separatedby 7,
thenft :
wpofi
andfz:
wpo/2 will
be Riemann mappings ofU
onto the respective ,"giorrr separatedby ^,'.lt
followsthat 7ilt o0i1: Of;'"7fr.
oIn analogy
with
the usual definitions of Teichmiiller theory,it
appears natural to define a "Teichmriller space" for the curve 7 as.t-(7)1
modulo the equivalence relation-
, where p*
u if and onlyif
u)F:
1!u on7.
But the equivalence relation istrivial in this
situation:Proposition TI.4.
We haveL@(f),
Proof.
If. p,- v
considerF : (*")-1
otoq.
.F' is a quasiconformal homeo- morphismof C
which hasdilatation
zero awayfrom 7
sincep and z
are bothsupported on
7. But F
is given to be the identity on7 itself-hence
its dilatationon 7
is also zero. HenceF
is identitX maP, sout' : llv
orLC,
and so Lt:
u . EWe end
by
questioning whetherfor
any Jordan curve7, all the
curves-tbat sharethe
same welding homeomorphismwith 7
arejust the
membersof
this"Teichmiiller space"
L- (l)r. In particular this
wouldiryply that the
welding determines uniquely the curve when-the curve has zero area. In the case of the well- known quasiciröles -<---+quasisymmetric correspondence, recall that_the-proc.ess._9f recovering the quasicircle from
its
welding is also viaa
" pr,-trick". See Vainio [V]for
related work.l,'.,: L.o(f),
306 Y. Katznelson, S. Nug, and
D.P.
Sullivan References[AB]
AulroRs, L., and L. Bpns: Riemann's mapping theorem for variable metrics. - Ann. of Math. (2) 72, 1960, 385-404.tD]
DAvto, G.: Courbes corde-arc et espaces de Ilardy generalises. - Ann. Inst. Fourier (Greno- ble) 32:3, t982, 227-239.[EN]
EARLE, C.J., and S. N.l,c: Conformally natural reflections in Jordan curves with ap- plications to Teichmiiller spaces. - In Holomorphic functions and moduli II, edited by D. Drasin et al., Berkeley Mathematical Sciences Research Institute Conference.Springer-Verlag, New York, 1988, L79-194.
tG]
GonRINc, F.W.: Spirals and the universal Teichmiiller space. - Acta Math. 141, 1978, 99-113.[Go]
Goln,rur,t, B.R., and J.M.J. or,Msrso: counterexamples in analysis. - Holden-Day, Inc., San Francisco, 1964.FK]
JERISON, D., and C. Knmc: Ilardy spaces/*,
and singular integrals on chord-arc domains. - Math. Scand.50, 1982,22L-247.tN]
NEHARI, Z.: Conformal mapping. - McGraw-Ilill, New York, 1gb2.tO]
Oscooo, W.: A Jordan curve of positive area. - Tlans. Amer. Math. Soc. 4, 1.903, lO7-1L2.tS]
Sorvruos, S.W.: The Cauchy integral, chord-arc curves and quasiconformal mappings. - In Bieberbach conjecture symposium volume, edited by A. Baernstein et al., American Mathematical Society, Providence, R.I., 1986, 107-188.tV]
Vr,tttIo, J.V.: Conditions for the possibility of conformal sewing. - Ann. Acad. Sci. Fenn.Ser. A I Math. Dissertationes 53, 1985.
Y. Katznelson Stanford University
Departmeni of Mathematics Mathematics Unit
Subhashis Nag Dennis P. Sullivan
Indian Statistical
Institute
City University of New York Stanford, CAU.S.A.
and I.H.B.S.
Bures-sur-Yvette France
Received 2 November 1989
Bangalore-560 059 India
and I.H.E.S.
Bures-sur-Yvette France
Graduate Center New York, NY U.S.A.
and I.H.E.S.
Bures-sur-Yvette France