Minimal surfaces in $\mathbb{R}^3$ properly projecting into $\mathbb{R}^2$

0.74 MB·English

by  Antonio Alarcon

MINIMAL SURFACES IN R3 PROPERLY PROJECTING INTO R2
ANTONIOALARCO´NANDFRANCISCOJ.LO´PEZ
ABSTRACT. ForallopenRiemannsurfaceN andrealnumberθ∈(0,π/4),weconstructaconformal
minimalimmersionX =(X1,X2,X3):N →R3suchthatX3+tan(θ)|X1|:N →Rispositiveand
proper.Furthermore,Xcanbechosenwitharbitrarilyprescribedfluxmap.
0 Moreover,weproduceproperlyimmersedhyperbolicminimalsurfaceswithnonemptybound-
1 aryinR3lyingaboveanegativesublineargraph.
0
2

b
e
F
1. INTRODUCTION

9
1 Theconformalstructureofacomplete minimalsurfaceplaysafundamentalroleinitsglobal

properties. Itisthenimportanttodeterminetheconformaltypeofagivenminimalsurface. An
]
G openRiemannsurfaceissaidtobehyperbolicifanonlyifitcarriesanegativenon-constantsub-
harmonic function. Otherwise, itissaidto beparabolic. CompactRiemann surfaceswith empty
D
boundaryaresaidtobeelliptic.
.
h
Completeminimalsurfaceswithfinitetotalcurvatureorcompleteembeddedminimalsurfaces
t
a withfinitetopologyinR3areproperlyimmersedandhaveparabolicconformaltype(forfurther
m
information, see[Os,JM,CM,MPR,MP2]). Ontheotherhand,thereexistsproperlyimmersed
[ hyperbolicminimalsurfacesinR3 witharbitrarynoncompacttopology[Mo,AFM,FMM].

2
It is then interesting to elucidate how properness and completeness influence the conformal
v
geometryofminimalsurfaces. In[Lo2]itisshownthatanyopenRiemannsurfaceadmitsacon-
4
2 formal complete minimal immersion in R3, even with arbitrarily prescribed flux map. In this
1 paper we extend this result to the family of proper minimal immersions, proving considerably
4 more(seeTheorem4.4):
.
0
1 TheoremI.ForallopenRiemannsurfaceN,groupmorphismp: H1(N,Z)→R3and
9 realnumberθ ∈ (0,π),thereexistsaconformalminimalimmersionX = (X ,X ,X ) :
4 1 2 3
0 N →R3 satisfyingthat:
v: • X3+tan(θ)|X1| : N →Rispositiveandproper,and
i • ∂X =ip(γ)forallγ ∈ H (N,Z),where∂isthecomplexdifferentialoperator.
X γ 1
R
r
a This result is sharp in the sense that the angle θ cannot be zero. Indeed, by the Strong Half
SpaceTheorem[HM] properlyimmersed minimalsurfacesinahalfspaceareplanes. Contrari-
wise, TheoremIshows thatanywedgeofangle greaterthan π inR3 containsminimal surfaces
properlyimmersedinR3,evenofhyperbolictype. Inparticular,neitheropenwedgesnorclosed
wedgesofanglegreaterthanπ areuniversalregionsforsurfaces(see[MP1]foragoodsetting).
OtherPicardconditionsforproperlyimmersedminimalsurfacesinR3guaranteeingparabolicity
canbefoundin[Lo1].
2000MathematicsSubjectClassification. 53A10;49Q05,49Q10,53C42.
Keywordsandphrases. Properminimalsurfaces,Riemannsurfacesofarbitraryconformalstructure.
ResearchofbothauthorsispartiallysupportedbyMCYT-FEDERresearchprojectMTM2007-61775andJuntadeAn-
daluc´ıaGrantP09-FQM-5088.
2 A.ALARCO´NANDF.J.LO´PEZ
FromTheoremIfollowsomeremarkableresultsconcerningnotonlyminimalsurfaces.Weare
goingtomentionthreeofthemrelatedtoproperharmonicmapsintoC,properholomorphicnull
curvesinC3andmaximalsurfacesintheLorentz-MinkowskispaceR3.
1
SchoenandYauconjecturedthattherearenoproperharmonicmapsfromDtoCwithflatmet-
rics,andconnectedthisquestionwiththeexistenceofhyperbolicminimalsurfacesinR3properly
projectingintoR2[SY,p.18].Acounterexampletothisconjecturefollowsfromtheresultsin[DF],
which imply the existence of proper harmonic maps from any finite bordered Riemann surface
(that is to say, a compact Riemann surface minus a finite collection of pairwise disjoint closed
discs)intoR2.ItremainsopenwhetherornotahyperbolicminimalsurfaceinR3canbeproperly
projectedintoR2.TheoremIanswerspositivelythisquestionforminimalsurfaceswitharbitrary
openconformalstructure(justnoticethattheharmonicmap(X ,X ) : N →R2 isproper).
1 3
It is well known that any open Riemann surface properly holomorphically embeds in C3
and immerses in C2 [Bi, Nar, Re]. Moreover, there are proper null immersions in C3 of the
unit disc [Mo], and of any open parabolic Riemann surface of finite topology [Pi, Lo2]. The-
orem I also shows that any open Riemann surface admits a proper null immersion in C3, and
a holomorphic immersion in C2 properly projecting into R2. Indeed, choosing p = 0 in The-
orem I and labeling X∗ = (X∗,X∗,X∗) as the conjugate minimal immersion of X, the map
1 2 3
X+iX∗ = (F ,F ,F ) : N → C3 isaproperholomorphicnullimmersion,and(F ,F ) : N → C2
1 2 3 1 3
isaholomorphicimmersionwhichproperlyprojectsintoR2.
Finally, fromTheorem I follows the existence of proper Lorentziannull holomorphic immer-
sions in C3 (see [UY]) and proper conformal maximal immersions in the Lorentz-Minkowski
space, with singularities and arbitrary conformal structure. See [Al] for the hyperbolic simply
connectedcase.
The lastpartof the paper is devotedto properlyimmersed minimal surfacesin R3 with non
emptyboundary.ARiemannsurfaceMwithnonemptyboundaryissaidtobeparabolicifbounded
harmonic functions on M are determined by their boundary values, or equivalently, if the har-
monic measureof Mrespecttoapoint P ∈ M−∂(M)isfullon∂(M).Otherwise, thesurfaceis
saidtobehyperbolic(see[AS,Pe]foragoodsetting). Forinstance,D−{1}isparabolicwhereas
D+ := D∩{z ∈ C|Im(z) > 0} is hyperbolic. Properly immersed minimal surfaces with non
emptyboundarylyinginahalfspaceofR3 areparabolic[CKMR],andthesameresultholdsfor
properminimalgraphsinR3[Ne]. Itisalsoknownthatanyproperlyimmersedminimalsurface
inR3withnonemptyboundarylyingoveranegativesublineargraphinR3andwhoseGaussian
imageiscontainedinahyperbolicdomainoftheRiemannsphereisparabolic[LP]. Weprovethe
followingcomplementaryresult(seeTheorem5.1),whichalsoshowsthattheconditionaboutthe
sizeoftheGaussmapin[LP]playsanimportantrole:
Theorem II. Thereexists a conformalminimalimmersion X = (X1,X2,X3) : D+ →
R3 such that (X1,X3) : D+ → R2 isproperand limn→∞min{|XX1(3p(np)n|)+1,0} = 0 for
alldivergentsequence{pn}n∈N inD+.
TheoremIIcontributestotheunderstandingofMeeks’conjectureaboutparabolicityofminimal
surfaces with boundary. This conjecture asserts that any properly immersed minimal surface
lyingaboveanegativehalfcatenoidisparabolic.
Thetechniquesdevelopedinthispapermaybeappliedtoawiderangeofproblemsonmini-
malsurfacestheory. Inthepaper[AFL]completeminimalsurfacesinR3witharbitraryconformal
structure and whose Gauss map misses two points are constructed. Our tools come from deep
resultsonapproximationtheorybymeromorphicfunctions[Sc1,Sc2,Ro]. Themostusefulone
istheApproximationLemmainSection2,whereaccurateuseofRunge’sapproximationtheorem
andclassicaltheoryofRiemannsurfaces[AS,FK]ismade.Inthisway,wecanrefinetheclassical
MINIMALSURFACESINR3PROPERLYPROJECTINGINTOR2 3
construction methodsof completeminimal surfaces(see, amongothers, [JX,Nad,LMM,MUY]
foragoodsetting).
Thepaperislaidoutasfollows. InSection2weintroducethenecessarybackgroundonRie-
mannsurfacesandapproximationtheory,andtherequirednotations. Furthermore,weprovethe
Approximation Lemma. Section 3 is devoted to some preliminaries on minimal surfacesin R3.
Finally,TheoremsIandIIareprovedinSections4and5,respectively.
2. RIEMANNSURFACES ANDAPPROXIMATION RESULTS
GivenatopologicalsurfaceW,∂(W)willdenotetheonedimensionaltopologicalmanifoldde-
terminedbytheboundarypointsofW.GivenS⊂W,callbyS◦andStheinteriorandtheclosure
ofSinW,respectively.
A Riemann surface M is said to be open if it is non-compact and ∂(M) = ∅. As usual, C =
C∪{∞}willdenotetheRiemannsphere. Wedenote ∂ astheglobalcomplexoperatorgivenby
∂| = ∂ dzforanyconformalchart(U,z)on M.
U ∂z
Remark2.1. ThroughoutthispaperN willdenoteafixedbutarbitraryopenRiemannsurface.
Let S denote a subset of N, S 6= N. We denote by F (S), respectively F(S), as the space of
0
continuous functions f : S → C, respectively f : S → C, which are holomorphic, respectively
meromorphic, onanopenneighborhood of S inN,and f−1(∞) ⊂ S◦.Likewise, F∗(S),respec-
0
tively F∗(S), willdenote the spaceof continuous functions f : S → C, respectively f : S → C,
beingholomorphic,respectivelymeromorphic,onS◦ andsatisfyingthat f−1(∞) ⊂S◦.
As usual, a 1-form θ on S is said to be of type (1,0) if for any conformal chart (U,z) in N,
θ| = h(z)dzforsomefunctionh : U∩S → C.WedenotebyΩ (S),respectivelyΩ(S),asthe
U∩S 0
space of holomorphic, respectivelymeromorphic, 1-forms on an open neighborhood of S in N,
andwithoutpolesonS−S◦.WecallΩ∗(S)asthespaceof1-formsθ oftype(1,0)onSsuchthat
(θ| )/dz ∈ F∗(S∩U)foranyconformalchart(U,z)onN.LikewisewedefineΩ∗(S).
U 0
LetDiv(S)denotethefreecommutativegroupofdivisorsofSwithmultiplicativenotation. If
D = ∏n Qni ∈ Div(S),wheren ∈ Z−{0}foralli,theset{Q ,...,Q }issaidtobethesup-
i=1 i i 1 n
portofD,writtensupp(D).AdivisorD ∈Div(S)issaidtobeintegralifD = ∏n Qni andn ≥0
i=1 i i
for all i. Given D , D ∈ Div(S), D ≥ D if and only if D D−1 is integral. For any f ∈ F(S)
1 2 1 2 1 2
wedenoteby(f)0and(f)∞ itsassociatedintegraldivisorsofzeroesandpolesinS,respectively,
and label (f) = ((ff))∞0 as the divisor associated to f on S. Likewise we define (θ)0, (θ)∞ for any
θ ∈ Ω(S),andcall(θ)= (θ)0 asthedivisorofθonS.
(θ)∞
InthesequelwewillassumethatSisacompactsubset of N andW ⊂ N anopensubsetcon-
tainingS.
Bydefinition,aconnectedcomponentVofW−SissaidtobeboundedinWifV∩Wiscompact,
whereV istheclosureofV inN.
Definition2.2. AcompactsubsetS⊂W issaidtobeadmissibleinW ifandonlyif:
• W−ShasnoboundedcomponentsinW,
• M := S◦ consistsofafinitecollectionofpairwisedisjointcompactregionsinW withC0
S
boundary,
• C :=S−M consistsofafinitecollectionofpairwisedisjointanalyticalJordanarcs,and
S S
• anycomponent α of C withanendpoint P ∈ M admitsananalyticalextension β inW
S S
suchthattheuniquecomponentofβ−αwithendpointPliesin M .
S
4 A.ALARCO´NANDF.J.LO´PEZ
Observe that if S is admissible in N then it is admissible inW as well, but the contrary is in
generalfalse.
We shall say that a function f ∈ F∗(S), respectively f ∈ F∗(S), can be uniformly approxi-
0
mated on S by functions in F(W), respectively in F0(W), if there exists {fn}n∈N ⊂ F(W), re-
spectively {fn}n∈N ⊂ F0(W), such that {|fn − f|}n∈N → 0 uniformly on S. We also say that
{fn|S}n∈N → f in the ω-topology. In particular all fn have the same set Pf of poles on S◦. A
1-form θ ∈ Ω∗(S), respectively θ ∈ Ω∗(S), can be uniformly approximated on S by 1-forms in
0
Ω(W),respectivelyinΩ0(W),ifthereexists{θn}n∈N ⊂ Ω(W),respectively{θn}n∈N ⊂ Ω0(W),
such that {θnd−zθ}n∈N → 0 uniformly on S∩U, for any conformal closed disc (U,dz) on W. In
particularallθn havethesamesetofpolesPθ onS◦.Asabove,wesaythat{θn|S}n∈N → θinthe
ω-topology.
RecallthatacompactJordanarcinN issaidtobeanalyticalifitiscontainedinanopenana-
lyticalJordanarcinN.
GivenanadmissiblecompactsetS ⊂ W,afunction f : S → Cn,n ∈ N,issaidtobesmooth
if f| admits a smooth extension f to an open domain V in W containing M , and for any
MS 0 S
component αof C andanyopenanalyticalJordanarc βinW containingα, f admitsansmooth
S
extension f to β satisfying that f | = f | . A function f ∈ F∗(S) is said to be smooth if
β β V∩β 0 V∩β
f| issmooth,whereV isanyopenneighbourhoodof f−1(∞)suchthatS−V isadmissiblein
S−V
W− f−1(∞).Analogously,a1-formθ ∈ Ω∗(S)issaidtobesmoothif(θ| )/dz ∈F∗(S∩U)is
S∩U
smooth for any closed conformal disk (U,z) on W such that S∩U is an admissible set. Given
a smooth f ∈ F∗(S), we set df ∈ Ω∗(S) as the smooth 1-form given by df| = d(f| )
MS MS
and df| = (f ◦α)′(x)dz| , where (U,z = x+iy) is a conformal chart on W such that
α∩U α∩U
α∩U = z−1(R∩z(U)).Obviously,df| (t) = (f ◦α)′(t)dtforanycomponentαofC ,wheretis
α S
anysmoothparameteralongα.
Asmooth1-formθ ∈ Ω∗(S)issaidtobeexactifθ = df forsomesmooth f ∈ F∗(S),orequiv-
alentlyif θ =0forallγ ∈ H (S,Z).
γ 1
R
Severalextensions of classicalRunge’s Theoremcanbe found in [Ro, Sc1, Sc2]. For our pur-
poses,weneedonlythefollowingcompilationresult:
Theorem2.3. LetS⊂W beanotnecessarilyconnectedadmissiblecompactsubsetinW.
Thenanyfunction f ∈ F∗(S)canbeuniformlyapproximatedonSbyfunctionsinF(W)∩F (W−
0
P ),whereP = f−1(∞).Furthermore,ifD ∈ Div(S)isanintegraldivisorsatisfyingthatsupp(D)⊂
f f
S◦,thentheapproximationsequence{fn}n∈N inF(W)canbechosensatisfyingthat(f − fn|S)0 ≥ D.
2.1. The Approximation Lemmas. Throughout this section, W ⊂ N will denote an open con-
nectedsubsetoffinitetopology,andSanadmissiblecompactsubsetinW.
Lemma 2.4. Consider f ∈ F∗(S) such that f never vanishes on S−S◦. Then f can be uniformly ap-
proximatedonSbyfunctions{fn}n∈NinF(W)satisfyingthat(fn) =(f)onW foralln.Inparticular,
f isholomorphicandnevervanishingonW−Sforalln.
n
Proof. LetµandbdenotethegenusofW andthenumberoftopologicalendsofW−supp((f)).
It is well known (see [FK]) that there exist 2µ+b−1 cohomologically independent 1-forms
in Ω(W)∩Ω (W −supp((f))) generating the first holomorphic De Rham cohomology group
0
H1 (W −supp((f))). Furthermore, the 1-forms can be chosen having at most single poles at
hol
points of supp((f)).Thus, the map H1 (W−supp((f))) −→ C2µ+b−1, τ 7→ τ , where
hol c c∈B0
B isanyhomologybasisofW−supp((f)),isalinearisomorphism,andtheree(cid:0)xRists(cid:1)τ ∈ Ω(W)∩
0
MINIMALSURFACESINR3PROPERLYPROJECTINGINTOR2 5
Ω W−supp((f)) with atmost single polesatpoints of supp((f)) such that 1 τ ∈ Z for
0 2πi γ
all(cid:0)γ ∈ H W−sup(cid:1)p((f)),Z anddf/f −τ ∈Ω∗(S)isexact. Set f = fe− τ ∈ F∗R(S).
1 0 0
R
Itisclea(cid:0)rthatlog(f ) ∈ F∗(cid:1)(S),andso f isholomorphic andnevervanishingonS.ByTheo-
0 0 0
rem2.3,thereexistsasequence{hn}n∈N ⊂ F0∗(W)convergingtolog(f0)intheω-topologyonS.
Thesequence fn =ehn+ τ,n∈N,solvesthelemma. (cid:3)
R
Lemma2.5. Considerθ ∈ Ω∗(S)suchthatθ nevervanishesonS−S◦.
Thenθ canbeuniformlyapproximatedonSby1-forms{θn}n∈N inΩ(W)satisfyingthat(θn) = (θ)
onW.Inparticular,θ isholomorphicandnevervanishingonW−Sforalln.
n
Proof. Let τ be a non zero 1-form in Ω (W) with finitely many zeroes, none of them lying in
0
S−S◦.Label f = θ/τ ∈ F∗(S).ByLemma2.4, f canbeapproximatedintheω-topologyonSby
asequence{fn}n∈NinF(W)satisfyingthat(fn) =(f)onWforalln.Itsufficestotakeθn := fnτ,
n ∈N. (cid:3)
Lemma 2.6 (The Approximation Lemma). Let Φ = (φ ) be a smooth triple in Ω∗(S)3 such
j j=1,2,3 0
that ∑3 φ2 = 0, ∑3 |φ |2 never vanishes on S, and Φ| ∈ Ω (M )3. Then Φ can be uniformly
j=1 j j=1 j MS 0 S
approximatedonSbyasequence{Φn = (φj,n)j=1,2,3}n∈N ⊂ Ω0(W)3satisfyingthat:
(i) ∑3 φ2 =0and∑3 |φ |2 nevervanishesonW,
j=1 j,n j=1 j,n
(ii) Φ −ΦisexactonSforalln.
n
Proof. Labelg = φ3 ,η = 1φ = φ −iφ andη = gφ = −φ −iφ ∈ Ω∗(S).Westartwith
φ1−iφ2 1 g 3 1 2 2 3 1 2 0
thefollowingclaim:
Claim2.7. Withoutlossofgenerality,wecanassumethatg| isnotconstant.
MS
Proof. Supposeforamomentthat g| isconstant,anduptoreplacingΦbyΦ·Aforasuitable
MS
orthogonalmatrix A ∈ O(3,R),assumethatg 6= ∞.Foreachh ∈ F (W),setη (h) = (g+h)2η
0 2 1
and φ (h) = η (g+h).LetB beahomology basisof H (M ,Z),labelµ asitscardinalnumber
3 1 1 S
andconsider theholomorphic map T : F (W) → C2µ, T(h) = ( (η (h)−η ,φ (h)−φ )) .
0 c 2 2 3 3 c∈B
Note that the analytical subset T−1(0) is conical, that is to say, Rif T(h) = 0 then T(λh) = 0
for all λ ∈ C. Furthermore, since F (W) has infinite dimension we can choose a non constant
0
h ∈ T−1(0).Take{λn}n∈N ⊂ Cconvergingtozero,sethn := λnh ∈ T−1(0)foralln,andnotice
that{hn}n∈N →0intheω-topologyonS.
SetΨ ≡(ψ ,ψ ,ψ ) :=(1(η −η (h )), i(η +η (h )),φ (h ))∈ Ω∗(S)∩Ω (M )3,and
n 1,n 2,n 3,n 2 1 2 n 2 1 2 n 3 n 0 0 S
observethat∑3 ψ2 =0,∑3 |ψ |2nevervanishesonSandg = ψ3,n isholomorphicand
j=1 j,n j=1 j,n n ψ1,n−iψ2,n
nonconstanton M .ItisclearthatΨ −Φisexacton M ,n ∈ N.Furthermore,wecanslightly
S n S
deformΨ | sothatΨ −ΦisexactonSaswell, n ∈ N(weleavethedetailstothereader). If
n CS n
thelemmaholdsforΨn foralln,wecanconstructasequence{Ψˆn,m}m∈N ⊂ Ω0∗(S)3 converging
to Ψ in the ω-topology on S and satisfying that Ψˆ −Ψ is exact on S for all n. A standard
n n,m n
(cid:3)
diagonalargumentprovestheclaim.
Claim2.8. Without lossofgenerality, wecanassumethat g,1/g and dg never vanishon ∂(M )∪C
S S
(hencethesameholdsforη,i =1,2,andφ ,j=1,2,3).Inparticular,g∈ F∗(S)anddg∈ Ω∗(S).
i j
Proof. Take a sequence M ⊃ M ⊃ ... of tubular neighborhoods of M in W such that M ⊂
1 2 S n
Mn◦−1 foranyn,∩n∈NMn = MS,Φ(andsog)meromorphicallyextends(withthesamename)to
M ,∑3 |φ |2 6= 0on M ,and g,1/g,anddgnevervanishon∂(M )foralln(takeintoaccount
1 j=1 j 1 n
6 A.ALARCO´NANDF.J.LO´PEZ
Claim2.7).ChooseM insuchawaythatS := M ∪C isanadmissiblesubsetofWandγ−M◦
n n n S n
isa(non-empty)JordanarcforanycomponentγofC .Inparticular,C = C −M◦,n ∈N.
S Sn S n
Let(h ,ψ )∈ F∗(S )×Ω∗(S )besmoothdatasuchthat
n 3,n n 0 n
• (h ,ψ )| =(g,φ )| and∑3 |ψ |2nevervanishesonS ,whereΨ =(ψ ) =
n 3,n MSn 3 MSn j=1 j,n n n j,n j=1,2,3
1(1/h −h ), i(1/h +h ),1 ψ ∈Ω∗(S )3,n∈N,
2 n n 2 n n 3,n 0 n
• h(cid:0)n,1/hn anddhnnevervanish(cid:1)on∂(MSn)∪CSn,
• Ψ | −ΦisexactonS,and
n S
• thesequence{Ψn|S}n∈N ⊂Ω0∗(S)3convergestoΦintheω-topologyonS.
Theconstructionofthesedataisstandard,weomitthedetails.
Label T ⊂ Ω (W)3 as the subspace of data Ψ formally satisfying (i) and (ii). If the Lemma
0
heldforanyofthedatain{Ψ | n ∈N},Ψ wouldlieintheclosureofT inΩ∗(S )3withrespect
n n 0 n
totheω-topologyonS foralln ∈N,hencethesamewouldoccurforΦandwearedone. (cid:3)
n
Let B be a homology basis of H (S,Z) and label ν its cardinalnumber. Endow F∗(S) with
S 1 0
themaximumnorm,andconsidertheFre´chetdifferentiablemap:
P : F∗(S)×F∗(S)→C3ν, P((h ,h ))= ((eh2−h1 −1)η ,(eh2+h1−1)η ,(eh2−1)φ ) .
0 0 1 2 Zc 1 2 3 c∈BS
(cid:0) (cid:1)
The meromorphic data inside the integrals arise frommultiplying g by eh1 and φ3 by eh2. Label
A : F∗(S)×F∗(S) →C3ν astheFre´chetderivativeofP at(0,0).
0 0 0
Claim2.9. A | issurjective.
0 F0(W)×F0(W)
Proof. ReasonbycontradictionandassumethatA (F (W)×F (W))liesinacomplexsubspace
0 0 0
U = { (x ,y ,z ) ∈C3ν| ∑ A x +B y +D z =0},where A ,B andD ∈Cforall
c c c c∈BS c∈BS c c c c c c c c c
c ∈ B (cid:0)and∑ (cid:1) |A |+|B |+|D | (cid:0)6=0.Thissimplyme(cid:1)ansthat:
S c∈BS c c c
(cid:0) (cid:1)
(2.1) − hη + hη = hη + hη + hφ =0
1 2 1 2 3
ZΓ ZΓ ZΓ ZΓ ZΓ
1 2 1 2 3
forallh ∈F (W),whereΓ =∑ A c,Γ = ∑ B candΓ =∑ D c.
0 1 c∈BS c 2 c∈BS c 3 c∈BS c
LabelΣ = {f ∈ F (W)| (f) ≥ (φ )2}.ByTheorem2.3,thefunctionh = df/φ ∈ F∗(S)lies
0 0 3 3 0
intheclosureofF (W)intheω-topologyonF∗(S)forany f ∈ Σ .Therefore,equation(2.1)can
0 0 0
beappliedformallytoh = df/φ ,gettingthat 1df = gdf = 0forall f ∈ Σ .Integrating
3 Γ1 g Γ2 0
byparts, R R
dg
(2.2) f = f dg=0
ZΓ g2 ZΓ
1 2
forall f ∈ Σ .
0
LetusshowthatΓ =0.
1
Letµ and b denote the genusofW andthe number of endsofW. Itiswellknown (see[FK])
that there exist 2µ+b−1 cohomologically independent 1-forms in Ω (W) generating the first
0
holomorphic DeRhamcohomologygroupH1 (W)ofW.Thus,themap H1 (W) −→ C2µ+b−1,
hol hol
τ 7→ τ , where B is any homology basis of W, is a linear isomorphism. Assume that
c c∈B0 0
Γ 6=0(cid:0)Rand(cid:1)take[τ] ∈ H1 (W)suchthat τ 6=0.SinceWisanopensurface,F (W)hasinfinite
1 hol Γ1 0
dimensionandwecanfind F ∈ F (W)sRuchthat(τ+dF) ≥ (dg) (g)2 (φ )2.Seth := (τ+dF)g2
0 0 0 ∞ 3 dg
andnotethat(h) ≥ (φ )2.ByTheorem2.3,hliesintheclosureofΣ inF∗(S)withrespecttothe
3 0 0
MINIMALSURFACESINR3PROPERLYPROJECTINGINTOR2 7
ω-topology,henceequation(2.2)canbeformallyappliedtoh,givingthat τ+dF = τ =0,
Γ Γ
1 1
acontradiction. R R
ByasimilarargumentΓ =0andequation(2.1)becomes:
2
(2.3) hφ =0
3
ZΓ
3
forallh ∈F (W).
0
Since∑ |A |+|B |+|D | 6= 0,thenΓ 6= 0.Reasonasaboveandchoose[τ] ∈ H1 (W)
c∈BS c c c 3 hol
andF ∈ F (W)(cid:0)suchthat τ 6=(cid:1)0and(τ+dF) ≥(φ ).Seth := τ+dF andnotethath ∈ F∗(S).
0 Γ3 0 3 φ3 0
ByTheorem2.3,hliesinthReclosureofF (W)inF∗(S)withrespecttotheω-topology,andequa-
0 0
tion(2.3)givesthat τ+dF = τ =0,acontradiction. Thisprovestheclaim. (cid:3)
Γ Γ
3 3
R R
Let{e ,...,e } beabasisof C3ν,fix H = (h ,h ) ∈ A−1(e)∩(F (W)×F (W)) foralli,
1 3ν i 1,i 2,i 0 i 0 0
andsetQ :C3ν →C3ν astheanalyticalmapgivenby
0
Q ((z ) ) =P( ∑ z H).
0 i i=1,...,3ν i i
i=1,...,3ν
ByClaim2.9d(Q ) isanisomorphism,sothereexistsaclosedEuclideanballU ⊂ C3ν centered
0 0
at the origin such that Q : U → Q (U) is an analytical diffeomorphism. Furthermore, notice
0 0
that0=Q (0)∈ Q (U)isaninteriorpointofQ (U).
0 0 0
On the other hand, byLemmas2.4and 2.5there existsa sequence {(fn,ψn)}n∈N ⊂ F(W)×
Ω0(W) such that (fn) = (g) and (ψn) = (φ3) for all n, and {(fn,ψn)}n∈N → (g,φ3) in the ω-
topologyonF∗(S)×Ω∗(S).
0
LabelP : F∗(S)×F∗(S)→C3νastheFre´chetdifferentiablemap
n 0 0
Pn((h1,h2))= Zc(eh2−h1η1,n−η1,eh2+h1η2,n−η2,eh2ψn−φ3) c∈BS,
(cid:0) (cid:1)
where η = 1ψ (1/f − f ) and η = iψ (1/f + f ), and call Q : C3ν → C3ν as the ana-
1,n 2 n n n 2,n 2 n n n n
lyticalmapQn((zi)i=1,...,3ν) = Pn(∑i=1,...,3νziHi)foralln ∈ N.Since{Qn}n∈N → Q0 uniformly
oncompactssubsetsofC3ν,withoutlossofgeneralitywecansupposethatQ : U → Q (U)is
n n
ananalyticaldiffeomorphismand0 ∈ Q (U)foralln.Labely = (y ,...,y )astheunique
n n 1,n 3ν,n
pointinUsuchthatQn(yn) =0andnotethat{yn}n∈N →0.Setting
gn =e∑3j=ν1yj,nh1,jfn, φ3,n = e∑3j=ν1yj,nh2,jψn
foralln∈N,thesequence{(gn,φ3,n)}n∈N solvesthelemma. (cid:3)
Corollary2.10. Inthepreviouslemmawecanchooseφ = φ foralln ∈ N,providedthatφ extends
3,n 3 3
holomorphicallytoW andφ nevervanishesonC .
3 S
Proof. Withoutlossofgenerality,wecansupposethatg,1/ganddgnevervanishon∂(M )∪C .
S S
Indeed, like in the proof of Claim 2.8 consider a sequence {Mn}n∈N of tubular neighborhoods
of M in W such that ∩∞ M = M , S := M ∪C is admissible in W, g, 1/g and dg never
S n=1 n S n n S
vanishes on ∂(M ) and γ−M◦ is a (non-empty) Jordan arc for any component γ of C , for all
n n S
n ∈N.Leth ∈ F∗(S )beasmoothdatumsuchthat:
n n
• h | = g| and∑3 |ψ |2nevervanishesonS ,whereΨ =(ψ ) = 1(1/h −
n MSn MSn j=1 j,n n n j,n j=1,2,3 2 n
h ), i(1/h +h ),1 φ ∈Ω∗(S )3,n∈N, (cid:0)
n 2 n n 3 0 n
• h ,1/h anddh ne(cid:1)vervanishon∂(M )∪C ,
n n n Sn Sn
• Ψ | −ΦisexactonS,and
n S
• thesequence{Ψn|S}n∈N ⊂Ω0∗(S)3convergestoΦintheω-topologyonS.
8 A.ALARCO´NANDF.J.LO´PEZ
ReasoningasintheproofofClaim2.8,iftheLemmaheldforanyofthedatain{Ψ | n ∈N}the
n
samewouldoccurforΦandwearedone.
Reasoning like in the proof of Lemma 2.6, we can prove that Aˆ : F (W) → Cν is sur-
0 0
jective, where Aˆ is the Fre´chet derivative of Pˆ : F∗(S) → C2ν, Pˆ(h) := P(h,0). Then take
0 0
Hˆ ∈ Aˆ−1(e)∩F (W) foralli,where {e ,...,e }isabasisofC2ν,anddefine Qˆ : C2ν → C2ν
i 0 i 0 1 2ν 0
byQˆ ((z) ) = Pˆ(∑ z Hˆ ).
0 i i=1,...,2ν i=1,...,2ν i i
Use Riemann-Roch Theorem to find a holomorphic function H ∈ F (W) such that (H) =
0
(φ3|W−S), and then Lemma 2.4 to get {fn}n∈N ⊂ F(W) such that (fn) = (g|S) for all n and
{fn}n∈N → g/Hintheω-topologyonF∗(S).
SetPˆ : F∗(S)→C2νbyPˆ (h) = (e−hη −η ,ehη −η ) ,whereη = 1φ ( 1 −
n 0 n c 1,n 1 2,n 2 c∈BS 1,n 2 3 fnH
f H)andη = iφ ( 1 +f H),and(cid:0)cRallQˆ :C2ν →C2νasthean(cid:1)alyticalmapQˆ ((z) )=
n 2,n 2 3 fnH n n n i i=1,...,2ν
Pˆ (∑ z Hˆ )foralln∈N.Tofinish,reasonasintheproofofLemma2.6. (cid:3)
n i=1,...,2ν i i
3. WEIERSTRASS REPRESENTATION AND FLUXMAPOF MINIMAL SURFACES
LetW be anopenRiemann surfaceand let X = (X ,X ,X ) : W → R3 be a conformalmin-
1 2 3
imal immersion. Denote by φ = ∂X , j = 1,2,3,and Φ = ∂X ≡ (φ ) . The 1-forms φ are
j j j j=1,2,3 k
holomorphic,havenorealperiodsandsatisfythat∑3 φ2 =0.Furthermore,theintrinsicmetric
k=1 k
inW isgivenbyds2 =∑3 |φ |2,henceφ ,k =1,2,3,havenocommonzeroes.
k=1 k k
Conversely, a vectorial holomorphic 1-form Φ = (φ ,φ ,φ ) on W without real periods and
1 2 3
satisfyingthat∑3 φ2 =0and∑3 |φ (P)|2 6=0forallP ∈W,determinesaconformalminimal
k=1 k k=1 k
immersionX :W →R3 bytheexpression:
X =Re Φ.
Z
By definition, the triple Φ is said to be the Weierstrass representation of X. The meromorphic
function g = φ3 corresponds to the Gauss map of X up to the stereographic projection (see
φ1−iφ2
[Os]).
Weneedthefollowingdefinition:
Definition 3.1. Given a proper region M ⊂ N with possibly non empty boundary, we denote
by M(M) the space of maps X : M → R3 extending as a conformal minimal immersion to an
openneighborhoodof MinN.ThisspacewillbeequippedwiththeC0 topologyoftheuniform
convergenceoncompactsubsetsof M.
Given X ∈ M(M) and an arclength parameterizedcurve γ(s) in M, the conormal vector of
X at γ(s) is the unique unitarytangent vector µ(s) of X at γ(s) such that {dX(γ′(s)),µ(s)} is a
positivebasis. Ifinadditionγisclosed,theflux p (γ)ofX alongγisgivenby µ(s)ds,andit
X γ
iseasytocheckthatp (γ)=Im ∂X.Thefluxmap p : H (M,Z)→R3isagrRoupmorphism.
X γ X 1
R
LetS ⊂ N beacompactadmissiblesubset. WedenotebyJ(S)thespaceofsmoothmapsX :
S →R3 suchthatX| ∈ M(M )andX| isregular,thatistosay,X| isaregularcurveforall
MS S CS α
α ⊂ C .ItisclearthatY| ∈ J(S)forallY ∈ M(N).
S S
Consider X ∈ J(S)andlet̟ : C → R3 beansmoothnormalfieldalongC respecttoX.This
S S
simplymeansthatforanyanalyticalarcα ⊂ C andanysmoothparameterton X| , ̟(α(t))is
S α
smooth, unitaryandorthogonalto(X| )′(t),̟ extendssmoothlytoanyopenanalyticalarcβin
α
Wcontainingαand̟istangenttoXonβ∩S.Thenormalfield̟issaidtobeorientablerespectto
MINIMALSURFACESINR3PROPERLYPROJECTINGINTOR2 9
Xifforanycomponentα ⊂ C havingendpointsP ,P lyingin∂(M ),andforanyarclengthpa-
S 1 2 S
rametersalongX| ,thebasisB = {(X| )′(s ),̟(s)}ofthetangentplaneofX| atP,i = 1,2,
α i α i i MS i
arebothpositiveornegative,wheres isthevalueofsforwhichα(s) = P,i =1,2.
i i i
Definition3.2. WecallM∗(S)asthespaceofmarkedimmersionsX := (X,̟),whereX ∈ J(S)
̟
and̟isanorientablesmoothnormalfieldalongC respecttoX,endowedwiththeC0topology
S
oftheuniformconvergenceofmapsandnormalfields.
Given X ∈ M∗(S), let ∂X = (φˆ ) be the complex vectorial “1-form” on S given by
̟ ̟ j j=1,2,3
∂X | = ∂(X| ),∂X (α′(s)) = dX(α′(s))+i̟(s),where α isacomponent of C and sisthe
̟ MS MS ̟ S
arclengthparameterof X| for which {dX(α′(s)),̟(s)} arepositive, whereas above s and s
α i i 1 2
are the values of s for which α(s) ∈ ∂(M ). If (U,z = x+iy) is a conformal chart on N such
S
that α∩U = z−1(R∩z(U)),itisclearthat(∂X )| = dX(α′(s))+i̟(s) s′(x)dz| ,hence
̟ α∩U α∩U
∂X ∈ Ω∗(S)3.Furthermore,gˆ = φˆ /(φˆ −iφˆ ) ∈ F∗(S)p(cid:2)rovidedthatgˆ−1(∞(cid:3))⊂ S◦.
̟ 0 3 1 2
It is clear that φˆ is smooth on S, j = 1,2,3, and the same occurs for gˆ on S−gˆ−1(∞). No-
j
tice that ∑3 φˆ2 = 0, ∑3 |φˆ |2 never vanishes on S and Real(φˆ ) is an “exact” real 1-form on
j=1 j j=1 j j
S, j = 1,2,3, hence we also have X(P) = X(Q)+Real P(φˆ ) , P, Q ∈ S. For these rea-
Q j j=1,2,3
sons,(gˆ,φˆ )willbecalledasthegeneralized“WeierstrassRdata”ofX .AsX| ∈ M(M ),then
3 ̟ MS S
(φ ) := (φˆ | ) ,and g := gˆ| arethe Weierstrassdataandthe meromorphicGauss
j j=1,2,3 j MS j=1,2,3 MS
mapofX| ,respectively.
MS
Thegrouphomomorphism
p : H (S,Z)→R3, p (γ)=Im ∂X ,
X̟ 1 X̟ Zγ ̟
is said to be the generalized flux map of X . Obviously, p = p | provided that X = Y|
̟ X̟Y Y H1(S) S
and̟ istheconormalfieldofYalonganycurveinC .
Y S
4. PROPERNESS AND CONFORMAL STRUCTURE OF MINIMAL SURFACES
¿From now on, we label x : R3 → R as the k-th coordinate function, k = 1,2,3. Given a
k
compact M ⊂ N andamapX : M → R3,wedenotekXk := max ∑3 (x ◦X)2 1/2 asthe
M j=1 j
maximumnorm. Foreach̺,a ∈R,wecall (cid:8)(cid:0) (cid:1) (cid:9)
Π (̺)= {(x ,x ,x ) ∈R3| x +tan(̺)x ≤ a}, Π∗(̺)=R3−Π (̺).
a 1 2 3 3 1 a a
Thefollowinglemmaconcentratesmostofthetechnicalcomputationsrequiredintheproofof
themaintheoremofthissection.
Lemma4.1. Let M, V ⊂ N be twocompactregions withanalyticalboundary such that M ⊂ V◦ and
N −MhasnoboundedcomponentsinN.
ConsiderX ∈ M(M)andlet p :H (V,Z)→Rbeanymorphismextensionof p .Supposethereare
1 X
θ ∈ (0,π/4)andδ ∈(0,+∞)suchthatX(∂(M))⊂ Π∗(θ)∪Π∗(−θ).
δ δ
Then, for any ǫ > 0 there exists Y ∈ M(V) such that p = p, kY−Xk < ǫ on M, Y(∂(V)) ⊂
Y
Π∗ (θ)∪Π∗ (−θ)andY(V−M) ⊂Π∗(θ)∪Π∗(−θ).
δ+1 δ+1 δ δ
Proof. Westartwiththefollowingclaim.
Claim4.2. ThelemmaholdswhentheEulercharacteristicχ(V−M◦)vanishes.
10 A.ALARCO´NANDF.J.LO´PEZ
Proof. Since M ⊂ V◦ and V◦−M has no bounded components in V◦, then V−M◦ = ∪k A ,
j=1 j
where A ,...,A arepairwisedisjointcompactannuli. Write∂(A ) = α ∪β ,whereα ⊂ ∂(M)
1 k j j j j
andβ ⊂ ∂(V)forallj.
j
Since X(∂(M)) ⊂ Π∗(θ)∪Π∗(−θ), each α can be divided into finitely many Jordanarcs αi,
δ δ j j
i = 1,...n ≥ 2,laidendtoendandsatisfyingthateitherX(αi) ⊂ Π∗(θ)orX(αi) ⊂ Π∗(−θ)for
j j δ j δ
alli.Uptorefiningthepartitions,wecanassumethatn =m ∈Nforallj.
j
Anarc αi is saidto be positiveif X(αi) ⊂ Π∗(θ),and negativeotherwise. Notice that X(αi) ⊂
j j δ j
Π∗(−θ) for any negative (and possibly for some positive) αi. We also label Qi and Qi+1 as the
δ j j j
endpointsofαi,insuchawaythatQi+1 =αi∩αi+1,i =1,...,m,(obviously,Qm+1 = Q1).
j j j j j j
Let{ri| i = 1,...,m}beacollectionofpairwisedisjointanalyticalJordanarcsin A suchthat
j j
ri has initial point Qi ∈ α , finalpoint Pi ∈ β , ri is otherwise disjoint from ∂(A ), and ri meets
j j j j j j j j
transversallyαi atQi,foralliandj.
j j
LetWbeatubularneighborhoodofVinN,thatistosay,anopenconnectedsubsetofN such
thatV ⊂W,W−V consistsofafinitecollectionofopenannuli,andtheclosureofanyannuliin
W−V intersects∂(V).
Set M = M∪ ∪ ri ,andnoticethat M isanadmissiblesubsetinN,andsoinW.CallΩi
0 j,i j 0 j
asthecloseddisc(cid:0)in A b(cid:1)oundedbyαi∪ri∪ri+1 andapiece,named βi,of β connecting Pi and
j j j j j j j
Pi+1.ObviouslyΩi∩Ωi+1 =ri+1,i < m,Ωm∩Ω1 =r1,andA =∪m Ωi.ThedomainΩi issaid
j j j j j j j j i=1 j j
tobepositive(respectively,negative)ifαi ispositive(respectively,negative).
j
FIGURE 1. Theannulus Aj.
TakeXˆ ∈ M∗(M )suchthatXˆ| = X,and
̟ 0 M
(4.1) dist Xˆ(Pi), Π (θ)∪Π (−θ) >1,
j δ δ
(cid:0) (cid:1)
foralliandj.Inaddition,wechooseXˆ insuchawaythat:
̟
(1+) IfΩij ispositivethenXˆ(rij∪rij+1)⊂ Π∗δ(θ).
(1−) IfΩij isnegativethenXˆ(rij∪rij+1) ⊂Π∗δ(−θ).