A construction of a complete bounded null curve in $\mathbb{C}^3$

0.19 MB·English

by  Leonor Ferrer

COMPLETE BOUNDED NULL CURVES
9 IMMERSED IN C3 AND SL(2,C)
0
0
2
FRANCISCOMARTIN,MASAAKIUMEHARA,ANDKOTAROYAMADA

y
a
M ABSTRACT. Weconstructasimplyconnectedcompleteboundedmeancurvatureonesur-
face in the hyperbolic 3-space H3. Such a surface in H3 can be lifted as acomplete
boundednullcurveinSL(2,C). UsingatransformationbetweennullcurvesinC3 and
7 nullcurvesinSL(2,C),weareabletoproducethefirstexamplesofcompletebounded
nullcurvesinC3. Asanapplication, wecanshowtheexistenceofacompletebounded
] minimalsurfaceinR3whoseconjugateminimalsurfaceisalsobounded. Moreover,we
G
canshowtheexistenceofacompleteboundedimmersedcomplexsubmanifoldinC2.
D
.
h
t
a 1. INTRODUCTION
m
Theexistenceofcompletenonflatminimalsurfaceswithboundedcoordinatefunctions,
[
has been the instigator of many interesting articles on the theory of minimal surfaces in

6 R3 and C3 over the last few decades. The question of whether there exists a complete
v bounded complex submanifold in Cn was proposed by P. Yang in [Y] and answered by
0
P.Jonesin[J]wherethisauthorpresentashortandelegantmethodtoconstructbounded
3
5 (embedded) complex curves X : D1 C3, where D1 means the open unit disc of the
3 complex plane. Although these curve→s are minimal in C3 (they are holomorphic), their
0 respectiveprojectionsReX andImX arenotminimalinR3. Ifwepursuethis,weneed
6 toimposethatthecomplexcurveX :D C3alsosatisfies
0 1 →
/ (1.1) (X′)2+(X′)2+(X′)2 =0 X =(X ,X ,X ) ,
h 1 2 3 1 2 3
t where ′ denotesthe derivativewith respectto the complexcoordinateon D . From now
a (cid:0) (cid:1) 1
m on,curvesofthiskindwillbecalledholomorphicnullcurves.
ThepreviousquestioniscloselyrelatedtoanearlierquestionbyE.Calabi,whoaskedin
:
v 1965[C]whetherornotitispossibleforacompleteminimalsurfaceinR3tobecontained
Xi inaballinR3. Twoarticles,inparticular,havemadeveryimportant,ifnotfundamental,
contributionsto this problem. The first one was by L. P. Jorge and F. Xavier [JX], who
r
a constructedexamplesofcompleteminimalsurfacesinaslab. ThesecondonewasbyN.
Nadirashvili[N],whomorerecentlyproducedexamplescontainedinaball.Inbothcases,
thekeystepwastheingenioususeofRunge’sclassicaltheorem.
InrespecttocompleteboundedminimalnullcurvesinCn,theexistenceofsuchcurves
has been an open problemfor n = 3. For the case n = 2, J. Bourgain[Bo] provesthat
these curvescannotexist. Moreover,Jonesin[J] provedthatfor n 4 itis possibleto
constructcompleteboundednullcurvesinCn. ≥
In this paper we give a positive solution to the existence of complete bounded null
curvesinC3andobtainsomeinterestingconsequences. Tobemoreprecise,weprovethe
followingtheorem:
TheoremA. ThereisacompleteholomorphicnullimmersionX: D C3whoseimage
1
isbounded.Inparticular,thereisacompletebounded(immersed) min→imalsurfaceinR3
suchthatitsconjugateminimalsurfaceisalsobounded.
Date:May06,2008.
TheFirstauthorispartiallysupportedbyMEC-FEDERGrantno.MTM2007-61775. Thesecondandthe
thirdauthorsarepartially supportedbyGrant-in-AidforScientificResearch(A)No.19204005andScientific
Research(B)No.14340024,respectively,fromtheJapanSocietyforthePromotionofScience.
1
2 FRANCISCOMARTIN,MASAAKIUMEHARA,ANDKOTAROYAMADA
Here,wedenotebyD (resp.D )theopen(resp.closed)ballinCofradiusrcentered
r r
at0.
Since the projection of X into C2 gives a holomorphic immersion, we also get the
followingresult,seeSection3.2:
CorollaryB. ThereisacompleteholomorphicimmersionY : D C2 whoseimageis
1
→
bounded.
We remark that the existence of complete bounded complex submanifolds in C3 has
beenshownin[J].
TheoremAisequivalenttotheexistenceofcompleteboundednullcurvesinSL(2,C),
andalsoequivalenttocompleteboundedmeancurvature1surfaces(i.e.CMC-1surface)in
thehyperbolic3-space 3. Hereaholomorphicmap : M SL(2,C)fromaRiemann
surface M to the compHlexLie groupSL(2,C) is callBed null→if the determinantdet ′ of
′ = d /dz vanishes, that is det ′ = 0, where z is a complex coordinate of MB. A
B B B
projectionβ = π : M 3 ofanullholomorphiccurveisaCMC-1surfacein 3,
whereπ: SL(2,C◦)B 3→=SHL(2,C)/SU(2)istheprojection,see(2.12)inSectionH2.2.
→H
Then Theorem A is a corollary to the existence of complete bounded null curve in
SL(2,C) as in Theorem C, see Section 3.1. To state the theorem, we define the matrix
norm as
|·|
A :=√traceAA∗ (A∗ :=tA),
| |
for 2 2-matrix A (see Appendix A). Note that if A SL(2,C), A √2, and the
× ∈ | | ≥
equalityholdsifandonlyifAistheidentitymatrix.
TheoremC. Foreachrealnumberτ >√2,thereisacompleteholomorphicnullimmer-
sion Y : D SL(2,C) such that Y < τ. In particular, there is a complete CMC-1
1
surface in 3→= SL(2,C)/SU(2)of|ge|nuszero with one endcontainedin a givengeo-
desicball(Hofradiuscosh−1(τ2/2),seeLemmaA.2inAppendixA).
AprojectionofimmersednullholomorphiccurvesinC3(resp.SL(2,C))ontoLorentz-
Minkowski3-spaceL3(resp. deSitter3-spaceS3)givesmaximalsurfaces(resp. CMC-1
1
surfaces),whichmayadmitsingularpoints.Recently,Alarcon[A]constructedaspace-like
maximalsurfaceboundedbyahyperboloidinL3,whichisweaklycompleteinthesense
of[UY3]butmaynotbebounded. Itshouldberemarkedthatourboundednullcurvein
C3 inTheoremAinducesaboundedmaximalsurfaceinL3 asarefinementofAlarcon’s
result:
CorollaryD. Thereareaweaklycompletespace-likemaximalsurfaceinL3andaweakly
completespace-likeCMC-1surfaceinS3whoseimagearebounded.
1
The definition of weak completeness for maximal surfaces and for CMC-1 surfaces
(withsingularities)arementionedintheproofinSection3.3.
Our procedure to prove Theorem C is similar in style to that used by Nadirashvili in
[N] (see also [MN] for a more general construction). However, we have to improvethe
techniquesbecauseNadirashvili’smethoddoesnotallowustocontroltheimaginarypart
oftheresultingminimalimmersion. Inordertodothis,weworkonaCMC-1surfacein
hyperbolic3-space 3 instead of a minimalsurface in Euclidean3-space. On each step
H
ofconstruction,wewillapplyRungeapproximationforverysmallregionofthesurface,
and so we can treat such a small part of the CMC-1 surface like as minimal surface in
the Euclidean3-space, which is the first crucialpoint. We shall give an error estimation
betweenminimalsurfaceandtheCMC-1surfacebyusingthewell-knownODE-technique
(see A.2in AppendixA). Next, we willliftthe resultingboundedCMC-1 surfaceintoa
null curve SL(2,C). Since 3 is a quotient of SL(2,C) by SU(2), the compactness of
H
SU(2)yieldstheboundednessoftheliftednullcurveassociatedwiththeboundedCMC-
1 surface. Finally, using a transformation between null curves in C3 and null curves in
COMPLETEBOUNDEDNULLCURVES 3
SL(2,C), we can get a complete bounded null immersed curve in C3 from the one in
SL(2,C). Section3.2isdevotedtoexplainthisequivalence.
ToproveTheoremC,thefollowinglemmaplaysacrucialrole(seeSection3.4):
MainLemma. Letρandτ bepositiverealnumbersandX: D SL(2,C)aholomor-
1
→
phicnullimmersionsuchthat
1 0
(1) X(0)=id= ,
0 1
(cid:18) (cid:19)
(2) (D ,ds2 )containsthegeodesicdiscofradiusρwithcenter0,whereds2 isthe
1 X X
inducedmetricbyX definedin(2.11).
(3) X(z) τ forz D .
1
| |≤ ∈
Then for arbitrary positive numbers ε and s, there exists a holomorphic null immersion
Y =Y : D SL(2,C)suchthat
X,ε,s 1
→
(a) Y(0)=id,
(b) (D ,ds2)containsthegeodesicdiscofradiusρ+swithcenter0,
1 Y
(c) Y(z) τ√1+2s2+εforz D ,
1
(d) |Y X|≤< εand φY φX <∈εontheopendiscD1−ε,whereφX = X−1X′,
φ| −=Y|−1Y′and|′ =d−/dzf|orthecomplexcoordinatezinD .
Y 1
Remark. Acrucialdifferencebetweenourmainlemmaandthemainlemmain[N]isthe
estimation of extrinsic radius. In [N], the extrinsic radius is estimated as √r2+s2 +ε
andtheboundednessoftheextrinsicradiusoftheresultingsurfacereducestothefactthat
∞ n−2converges.However,ourestimationoftheextrinsicradiusismultiplicativeand
rednu=c1estothefactthat ∞ (1+n−2)converges.
P n=1
AsanapplicationofQTheoremA,theexistenceofhighergenusexamplescorresponding
toCorollaryBandCorollaryDisrecentlyshownin[MUY].
Wewouldliketofinishthisintroductionbymentioningarelatingresult. P.F.X.Mu¨ller
in[M]introducedaremarkablerelationshipbetweencompleteboundedminimalsurfaces
inR3andmartingales.
2. PRELIMINARIES
2.1. NullcurvesinC3. LetM beaRiemannsurfaceand : M C3 anullholomor-
F →
phicimmersion,andset
d
(2.1) Φ=φdz, φ=φF = ′ ′ = ,
F dz
(cid:18) (cid:19)
wherez isalocalcomplexcoordinateonM. Thenφ = (φ ,φ ,φ )isa locallydefined
1 2 3
C3-valuedholomorphicfunctionsuchthat
(2.2) φ φ=0 and φ >0,
· | |
where istheinnerproductofC3definedby
·
(2.3) x y =x y +x y +x y x=(x ,x ,x ),y =(y ,y ,y ) ,
1 1 2 2 3 3 1 2 3 1 2 3
·
and x = √x x isthe HermitiannormofC(cid:0)3. Conversely,ifa C3-valued1-fo(cid:1)rmΦ =
| | ·
φdzonasimplyconnectedRiemannsurfaceM satisfies(2.2),
z
(2.4) (z):= Φ: M C3
Φ
F −→
Zz0
isaholomorphicnullimmersion,wherez M isabasepoint.
0
∈
Wedefinetheinducedmetricds2 of as
F
F
1 1 1 1
(2.5) ds2F = Φ2 = φ2 dz 2 = λ2F dz 2 = ∗ , (λF := φ),
2| | 2| | | | 2 | | 2F h i | |
4 FRANCISCOMARTIN,MASAAKIUMEHARA,ANDKOTAROYAMADA
where , isthecanonicalHermitianmetricofC3.Notethatds2 isahalf ofthepull-back
F
∗ ,h,aindcoincideswiththeinducedmetricoftheminimalimmersion
F h i
Re : M R3.
F −→
Since isanullcurve,wecanwrite
F
1
(2.6) Φ= (1 g2),i(1+g2),2g ηdz, i=√ 1
2 − −
wheregandηdzareamero(cid:0)morphicfunctionanda(cid:1)holomorphic1-form,respectively.We
call(g,ηdz)theWeierstrassdataof . UsingtheseWeierstrassdata,wecanwrite
F
1
(2.7) λF = (1+ g 2)η .
√2 | | | |
Here, g: M C can be identified with the Gauss map by the stereographic
→ ∪ {∞}
projection.
2.2. NullcurvesinSL(2,C). Aholomorphicmap fromaRiemannsurfaceM intothe
complex Lie group SL(2,C) is called null if det B′ = 0 holds on M, where ′ denotes
B
thederivativewithrespecttoacomplexcoordinatez. Takeanullholomorphicimmersion
: M SL(2,C)andlet
B →
(2.8) Ψ=ψdz, ψ =ψB = −1 ′.
B B
Since isanullimmersion,Ψisaholomorphicsl(2,C)-valued1-formwith
B
(2.9) detψ =0 and ψ >0,
| |
where denotes the matrix norm defined in (A.1) in the Appendix. Conversely, if an
sl(2,C|)-·v|alued holomorphic1-form Ψ = ψdz on a simply connected Riemann surface
M satisfies(2.9),thesolution ofanordinarydifferentialequation
B
(2.10) −1d =Ψ, (z )=id
0
B B B
isanullholomorphicimmersionintoSL(2,C).
Wedefinetheinducedmetricds2 of as
B
B
1 1 1 1
(2.11) ds2B = Ψ2 = ψ 2 dz 2 = λ2B dz 2 = ∗ , (λB := ψ ),
2| | 2| | | | 2 | | 2B h i | |
where , is the canonicalHermitianmetric of SL(2,C)inducedfromthe matrixnorm
h i
(A.1)intheAppendix.Identifyingthehyperbolic3-space 3withtheset
H
3 = aa∗; a SL(2,C) =SL(2,C)/SU(2) (a∗ =ta¯)
H { ∈ }
asin(A.3)inAppendixA,
(2.12) β :=π = ∗: M 3
◦B BB −→H
givesa conformalmeancurvatureoneimmersion(aCMC-1 surface[Br1, UY1]), where
π: SL(2,C) 3 = SL(2,C)/SU(2)istheprojection. Theinducedmetricofβ coin-
→ H
cideswithds2, whichisthereasonwhyweaddthecoefficient1/2in(2.11). Since is
B
B
null,wecanwrite
1 g g2
(2.13) Ψ= − ηdz,
√2 1 g
(cid:18) − (cid:19)
wheregandηdzareameromorphicfunctionandaholomorphic1-form,respectively.We
call(g,ηdz)theWeierstrassdataof . Thenwecanwrite
B
1
(2.14) λB = (1+ g 2)η .
√2 | | | |
ThemeromorphicfunctiongiscalledthesecondaryGaussmapof ([UY2]).
Ifanullcurve inC3 andanullcurve inSL(2,C)areobtainBedbythesameWeier-
F B
strassdata(g,ηdz),theirinducedmetricscoincide,andthentheyhavethesameintrinsic
COMPLETEBOUNDEDNULLCURVES 5
behavior.Inthiscase,wecall thecousinof . TheformsΦandΨin(2.1)and(2.8)are
B F
relatedas
1 φ φ +iφ
(2.15) ψˆ:=φ=(φ ,φ ,φ ) ψ = 3 1 2 ,
1 2 3 ←→ √2 φ1 iφ2 φ3
(cid:18) − − (cid:19)
inwhich φ = ψ holds.RemarkthattheWeierstrassdata(g,ω)in[UY1]coincideswith
| | | |
(g,√2ηdz)here.
WecallaC3-valuedholomorphic1-formΦ=φdzonM aW-dataif(2.2)holds.IfM
issimplyconnected,itprovidesanullcurveinC3by(2.4),whileanullcurveinSL(2,C)
by(2.15)and(2.10).
3. PROOFS OF THETHEOREMS
3.1. Correspondence of null curves in C3 and SL(2,C). Firstly, we give a proof of
TheoremAintheintroductionusingTheoremC. Itshouldberemarkedthatevenwhena
nullcurveinC3 iscompleteandbounded,itscousininSL(2,C)maynotbeboundedin
general.SoweconsideranothertransformationtoproveTheoremA.
Let
(3.1) : (x ,x ,x ) C3; x =0 (y ) SL(2,C); y =0
1 2 3 3 ij 11
T { ∈ 6 }→{ ∈ 6 }
1 1 x +ix
(x ,x ,x )= 1 2 ,
T 1 2 3 x3 (cid:18)x1−ix2 (x1)2+(x2)2+(x3)2(cid:19)
whichisbirational,anditcanbeeasilycheckedthat mapsnullcurvesinC3 x =0
3
tonullcurvesinSL(2,C). Themap isessentiallyTthesameasBryant’stran\sf{ormatio}n
ofnullcurvesbetweenC3andthecomTplexquadricQ3[Br2].
ProofofTheoremAviaTheoremC. Let : M SL(2,C) be a complete boundednull
immersiondefinedona RiemannsurfaceBM. B→y replacing bya (a SL(2,C)), we
may assume = 0 without loss of generality, where B= ( B). T∈hen −1 is
11 ij
a bounded hoBlomo6rphic null immersion of M into C3. OBn anyBbounded setTin C◦3,Bthe
pull-backof canonicalHermitian metric of SL(2,C) by is equivalentto the canonical
Hermitian metric of C3 by the following well-known LTemma 3.1. Hence the induced
metricds2 of −1 iscompletebecausesoistheinducedmetricds2 of . (cid:3)
T−1◦B T ◦B B B
Lemma3.1. Letg andg betwoRiemannianmetricsonamanifoldN.Foreachcompact
1 2
subsetK ofN,thereexistconstantsa,b>0suchthatag g bg onK.
1 2 1
≤ ≤
Remark3.2. BoundednessoftherealpartofanullimmersioninC3 (aminimalsurface)
does notimply the boundednessof its imaginarypart (the conjugatesurface) in general.
Incontrasttothisfact,aCMC-1surfaceβ isboundedifandonlyifsoisitsholomorphic
lift ,because β = ∗ 2 andtraceβ = trace ∗ = 2,where denotes
B | | |BB | ≤ |B| BB |B| |·|
thematrixnormasinAppendixA. OrthecompactLiegroupSU(2)isconsideredasthe
“imaginarypart”in 3 =SL(2,C)/SU(2).
H
3.2. ProofofCorollaryB. Next,wegiveaproofofCorollaryBintheintroduction. Let
= (F ,F ,F ) : D C3 beanullholomorphicimmersionobtainedbyTheoremA,
1 2 3 1
F →
and(g,ηdz)theWeierstrassdataasin(2.6). ThentheprojectionFˆ := (F ,F ): D
1 2 1
C2of isaboundedholomorphicmap.Moreover,itisacompleteimmersion.Infact,→
F
2
dFˆ
4 = 1 g2 2+ 1+g2 2 η 2 =2(1+ g 4)η 2 (1+ g 2)2 η 2 =ds2.
F
(cid:12)dz (cid:12) | − | | | | | | | | | ≥ | | | |
(cid:12) (cid:12)
(cid:12) (cid:12) (cid:0) (cid:1)
Henc(cid:12)edFˆ(cid:12)/dznevervanishesandFˆ iscomplete. (cid:3)
(cid:12) (cid:12)
6 FRANCISCOMARTIN,MASAAKIUMEHARA,ANDKOTAROYAMADA
3.3. ProofofCorollaryD. LetX =(X ,X ,X ): D C3beaboundednullimmer-
1 2 3 1
→
sionasinTheoremA. Then
f :=Re(iX ,X ,X ): D L3
X 1 2 3 1
−→
providesamaximalsurface(i.e.withzeromeancurvature)withsingularities. SinceX is
an immersion, f is considered as a maxface in the sense of [UY3]. Moreover, the lift
X
metric asin[UY3, Definition2.7]coincideswith ds2 givenin(2.5), whichis complete.
X
Theninthesenseof[UY3,Definition4.4],f isweaklycomplete.
X
Similarly,takeanullimmersionY : D SL(2,C)asinTheoremC,andlet
1
→
f :=Y 1 0 Y∗: D S3 =SL(2,C)/SU(1,1),
Y 0 1 1 −→ 1
(cid:18) − (cid:19)
where S3 is the de Sitter 3-space. Then f gives a CMC-1 (i.e., mean curvature one)
1 Y
surfacewith singularities. SinceY isan immersion, f is a CMC-1 facein the sense of
Y
[F]and[FRUYY]. Moreover,sincetheinducedmetricofds2 iscomplete,f isweakly
Y Y
completeinthesenseof[FRUYY]. (cid:3)
3.4. ProofofTheoremCviaMainLemma. TheproofofTheoremCviaMainLemma
follows an standard inductive argument. We construct a suitable sequence X ∞ of
nullimmersionsofD intoSL(2,C)asfollows:Takeaninitialnullimmersion{Xn}:nD=0
1 0 1
SL(2,C) such that (D ,ds2 ) is the geodesic disc of radius ρ with center 0, and fix→a
1 X0 0
positiveintegerk 1. Foreachintegern > 0, weapplyourMainLemmainductively
0
≥
supposingthatXn−1hasbeenalreadyconstructed.Let
1 1 1
ε , s= .
≤ (n+k )2 n+k ≤ 8
0 0 (cid:18) (cid:19)
Since2s2+ε<3/(n+k )2,wecanconstructX suchthat
0 n
(1) (D ,ds2 )containsageodesicdisccenteredattheoriginandofradius
1 Xn
n
1
ρ :=ρ + ,
n 0
k+k
0
k=1
X
(2) theinequality
n
3
X 2 τ2 1+
| n| ≤ 0 (k+k )2
k=1(cid:18) 0 (cid:19)
Y
holdsonD , whereτ isapositiveconstantdependingonlyontheinitialchoice
1 0
ofthenullcurveX ,
0
(3) and X convergestoacompletenullimmersionX: D SL(2,C)uniformly
n 1
ona{nyco}mpactsetofD . →
1
Asaconsequence,X satisfies
∞ 3 ∞ 2 2 sinh(2π)
X 2 τ2 1+ <τ2 1+ =τ2
| | ≤ 0 n=1(cid:18) (k+k0)2(cid:19) 0 n=1 (cid:18)k(cid:19) ! 0 2π
Y Y
onD .WecanchoosetheinitialdataoftheinitialcurveX suchthatτ isarbitrarilyclose
1 0 0
to√2. Sincek isalsoarbitrary,wecanlet X <τ foranarbitraryτ >√2. (cid:3)
0
| |
4. PROOF OF MAIN LEMMA
4.1. Labyrinth. Toprovethemainlemma,wewillworkonNadirashvili’slabyrinth([N,
MN,CR]). WefixthedefinitionsandnotationsontheLabyrinth:LetN bea(sufficiently
large)positivenumber.Fork=0,1,2,...,2N2,weset
k 1 2
(4.1) r =1 r =1,r =1 ,...,r =1 ,
k − N3 0 1 − N3 2N2 − N
(cid:18) (cid:19)
COMPLETEBOUNDEDNULLCURVES 7
andlet
(4.2) D = z C; z <r and S =∂D = z C; z =r .
rk { ∈ | | k} rk rk { ∈ | | k}
Wedefineanannulardomain as
A
(4.3) :=D D =D D ,
A 1\ r2N2 1\ 1−N2
and
N2−1 N2−1
A:= D D , A:= D D ,
r2k \ r2k+1 r2k+1 \ r2k+2
k=0 k=0
[ [
N−1 e N−1
L= l , L= l ,
2kπ (2k+1)π
N N
k=0 k=0
[ [
wherel istherayl = reiθ; r 0 . LetΣebeacompactsetdefinedas
θ θ
{ ≥ }
2N2 2N2
Σ:=L L S, S = ∂D = S ,
∪ ∪ rj rj
j=0 j=0
[ [
anddefineacompactsetΩby e
Ω= U (Σ),
1/(4N3)
A\
where U (Σ) denotesthe ε-neighborhood(of the Euclidean plane R2 = C) of Σ. Each
ε
connectedcomponentofΩhaswidth1/(2N3).
Foreachnumberj =1,...,2N,weset
ω := l connectedcomponentsofΩwhichintersectwithl ,
j jπ jπ
N ∩A ∪ N
̟j :=U(cid:0)1/(4N3)((cid:1)ωj)(cid:0). (cid:1)
Thenω ’sarecompactsets.
j
4.2. TransformationofHolomorphicdata. Theconstructionofcompleteboundedmin-
imalsurfacesinR3providesthefollowingassertion(see[N,MN]):
Lemma4.1. LetN( 4)beaninteger,andΦ=φdzaW-dataonD (inthesenseofthe
1
≥
endofSection2). Thenforeachε > 0andeachintegerj satisfying1 j 2N,there
existsanewW-dataΦ˜ =φ˜dzonD satisfyingthefollowingproperties:≤ ≤
1
(a) OnD ̟ ,itholdsthat
1 j
\
ε
φ φ˜ < ,
| − | 2N2
(b) thereexistsaconstantC >0dependingonlyonΦsuchthat
φ˜ CN3.5 onω ,
j
| |≥
(φ˜ CN−0.5 on̟j,
| |≥
(c) there existsa realunitvectoru = (u ,u ,u ) inR3 suchthat u > 1 2/N
1 2 3 3
| | −
and
u (φ φ˜)=0
· −
holds,where istheinnerproductasin(2.3).
·
Proof. Let f be the realpartofnullimmersion : D C3 asin (2.4)forz = 0.
Φ Φ 1 0
F →
We think in fΦ as the minimal immersion j−1 in [N, p. 463]. Then we can construct
F
anewminimalimmersion imitatingthecorrespondingprocedureasin[N]. However,
j
F
ourconstructionof ismucheasier. Actually,wedonotneedtoadjustthat (̟ )to
j j j
becontainedinacerFtainconewithsmallcone-anglecenteredatx -axisinR3.F
3
8 FRANCISCOMARTIN,MASAAKIUMEHARA,ANDKOTAROYAMADA
Theconditions(a)and(b)followfrom[MN,p.292-3,items(A.1 ),(A.2 ),and(A.3 )].
i i i
Thecondition(c)followsfrom[N,(11)]. However,togettheestimate φ˜ CN−0.5 on
| | ≥
̟ ,weneedthepropertythat
j
2 √N
g
√N ≤| |≤ 2
asin[MN,(B.3)orp.293],wheregisthemeromorphicfunctionin(2.6). Forthispurpose,
the axis for the Lopez-Rosdeformationcorrespondingto the deformationof Weierstrass
data(g,ηdz) (g/h,hηdz)withrespecttoacertainholomorphicfunctionasin[N,(4)]
7→
andin[MN,p.294]mightbeslightlymovedfromthex -axiswiththeangle θ 2/√N,
3
| |≤
thatis
2
cosθ 1 .
≥ − N
Finally,weletΦ˜ tobetheWeierstrassdataofour . Thenitsatisfiesthedesiredproper-
j
ties. F (cid:3)
4.3. AreductionofMainLemma. Inthissubsection,weshallreduceourmainlemma
tothefollowing
KeyLemma. Let = : D SL(2,C)beanullholomorphicimmersionsatisfying
0 1
(0) = id such thBat (DB,ds2) c→ontains a geodesic disc of radius ρ with center 0. For
1 B
B
eachpositivenumbersεandswiths < 1/8,thereexistasufficientlylargeintegerN and
asequenceofnullimmersions
:D SL(2,C) (j =1,...,2N)
j 1
B →
satisfyingthefollowingproperties:
(1) (0)=id,
j
(2) Bψj ψj−1 <ε/(2N2)holdsonD1 ̟j,where
| − | \
Ψ =ψ dz :=( )−1d (j =0,1,...,2N),
j j j j
B B
(3) thereexistsaconstantc>0dependingonlyon suchthat
0
B
ψ cN3.5 onω ,
j j
| |≥
(ψj cN−0.5 on̟j,
| |≥
(4) D containsaclosedgeodesicdiscD centeredat0withradiusρ+swithrespect
1 g
tods2 . Moreover,itholdsthat
B
2N
b
(p) max (z) 1+2s2+ for p ∂D ,
2N 0 g
|B |≤(cid:18)z∈D1|B |(cid:19)s √N ∈
where∂D istheboundaryofD ,andb>0isaconstantdependingonlyon .
g g 0
B
Proof. Weconstructthesequenceofnullimmersions ,..., inSL(2,C)inductively.
1 2N
B B
Assumethat j−1 (j 1)constructedalready. Then j isconstructedasfollows: We
B ≥ B
set
2 1
ζ := 1 eiπj/N (j =1,2,...,2N).
j − N − 4N3
(cid:18) (cid:19)
Thenζ ∂̟ ,and ζ attainstheEuclideandistancebetweentheorigin0 D and̟ ,
j j j 1 j
∈ | | ∈
seeFigure1.
Letβj−1 = Bj−1Bj∗−1 : D1 → H3 beaCMC-1surfaceassociatedwithBj−1,andwe
set
(4.4) H(z):=a j−1(ζj) −1 j−1(z)a∗, and h=HH∗: D1 3.
B B →H
(cid:8) (cid:9)
COMPLETEBOUNDEDNULLCURVES 9
∂D
1
γ
D
1
γ
0
p
0 1−N2 −4N13
̟
j
ζ
j ω
j
p¯
FIGURE 1. Thepointζj andthecurveγ
Here a SU(2) is chosen so that the geodesic line passing throughh(0) and h(ζ ) lies
j
∈
inx x -planein 3 L4. (Here,weconsider 3ahyperboloidintheMinkowskispace
0 3
H ⊂ H
L4,see(A.2)inAppendixA). Weset
1 ψ3 ψ1+iψ2 =ψ :=H−1dH.
√2 ψ1 iψ2 ψ3 dz
(cid:18) − − (cid:19)
Thenonecaneasilycheckthat
(4.5) ψj−1 =aψa∗.
ApplyingLemma4.1totheW-data(ψ ,ψ ,ψ ),wegetnewaW-dataφ˜=(φ˜ ,φ˜ ,φ˜ )
1 2 3 1 2 3
satisfying(a)–(c).ThenwedefineanewnullimmersionH :D SL(2,C)suchthat
1
→
(4.6) H−1dH =ψ˜= 1 φ˜3 φ˜1+iφ˜2 ,e and H(ζ )=id.
dz √2(cid:18)φ˜1−iφ˜2 −φ˜3 (cid:19) j
e
Afterthat,weeset e
∗ −1
(z):=a H(0) H(z)a.
j
B
ThenBj(0)=id,whichproves(1),andif(cid:8)weeset(cid:9)ψjdez :=Bj−1dBj,itholdsthat
(4.7) ψ =aψ˜a∗.
j
By(a),(b)inLemma4.1and(4.5)and(4.7),wegettheassertions(2)and(3).
Inthisway,wecanget inductively. Notethat, by(2)andCorollaryA.6,itholds
2N
B
that
c ε
(4.8) distH3 βj(z),βj−1(z) ≤ 2N1 2, on D1\̟j,
whereβj = BjBj∗,βj−1(cid:0)= Bj−1Bj∗−1 a(cid:1)nddistH3 standsforthecanonicaldistancefunc-
tionof 3,seeA.1inAppendixA. Here,theconstantc dependsonlyon . Throughout
1 0
H B
thisproof,weshalldenoteby
c ,c ,...
1 2
constants which depend only on . The properties(2) and (3) yield that D contains a
0 1
B
closedgeodesicdisc D ofradiusρ+scenteredat0with respecttothe inducedmetric
g
ds2 by (see[N,p.463]).Letp ∂D andwewillshow(4)inthestatementofthe
B 2N g
2N B ∈
Lemma. Ifp ̟ forallj = 1,...,2N,theinequalityiseasytoshowusing(4.8)and
j
6∈
LemmaA.2inAppendixA,see[N,MN].
10 FRANCISCOMARTIN,MASAAKIUMEHARA,ANDKOTAROYAMADA
β (p)
j
Πj βj(ζj) q
1 0
o =
0 1
(cid:18) (cid:19)
FIGURE 2. theplaneΠj
Otherwise,weassumep ̟ forsomej. Letγ betheds2 -geodesicjoining0and
j 0 B
∈ 2N
p,andtakep¯ γ ∂̟ suchthatthegeodesicjoiningp¯andpliesin̟ ,seeFigure1.
0 j j
∈ ∩
Thentheds2 -distanceofpandp¯satisfies
B
2N
c
2
dist (p¯,p) s+ ,
ds2B2N ≤ √N
wherec isaconstantdependingonlyon (see[MN,p.296]).Thus,bytakingasuitable
2 0
B
pathγ(asinFigure1)joiningζ andp¯inthecomplementof̟ ,wehave
j j
c
(4.9) dist (ζ ,p) s+ 3 ,
ds2B2N j ≤ √N
sincedist (ζ ,p¯)isoforder1/N2.
ds2B2N j
LetΠ bethetotallygeodesicplanein 3 passingthroughβ (ζ )whichisperpendic-
j j j
H
ulartothegeodesicjoiningoandβ (ζ ),whereo 3 isthepointcorrespondingtothe
j j
∈ H
identitymatrixid(asin(A.2)inAppendixA). Letq ( Π )bethefootofperpendicular
j
∈
from β (p) to the plane Π , see Figure 2. Then dist q,β (p) gives the distance of
j j H3 j
β (p)andtheplaneΠ ,and(4)isobtainedasaconclusionofthefollowingLemma:
j j
(cid:0) (cid:1)
Lemma 4.2. Under the situations above, namely, for p ∂D ̟ and for q Π
g j j
∈ ∩ ∈
satisfying
dist (β (p),q)=dist (β (p),Π ),
H3 j H3 j j
itholdsthat
c
dist β (p),q 14s2+ 4 ,
H3 j ≤ √N
wherec isaconstantdependingo(cid:0)nlyon (cid:1).
4 0
B
Weshallprovethislemmalater,andnowfinishtheproofofKeyLemma:
ProofofKeyLemma,continued. Assume Lemma 4.2 is true. By (4.8) and Lemma A.2,
wehave
c ε
(ζ )2 =2cosh dist o,β (ζ ) 2cosh dist o,β (ζ ) + 1
|Bj j | H3 j j ≤ H3 0 j N
max (cid:0)β (z) 2(cid:0) 1+ c5(cid:1)(cid:1), o:=(cid:16) 1 0(cid:0) 3 (cid:1), (cid:17)
≤(cid:0)z∈D1| 0 |(cid:1) (cid:16) N(cid:17) (cid:18) (cid:18)0 1(cid:19)∈H (cid:19)