Generalized Calabi's correspondence and complete spacelike surfaces

0.18 MB·English

by  Hojoo Lee

GENERALIZED CALABI’S CORRESPONDENCEAND
COMPLETE SPACELIKE SURFACES
3
1 HOJOOLEEANDJOSE´ M.MANZANO
0
2
Abstract. ExtendingCalabi’scorrespondencebetweenminimalgraphs
n
intheEuclideanspaceR3andmaximalgraphsintheLorentz-Minkowski
a
J spacetimeL3 toawideclassof3-manifoldscarryingaunitKillingvec-
torfield,weconstructatwincorrespondencebetweengraphswithpre-
0
3 scribedmeancurvatureHintheRiemannianGeneralizedBianchi-Cartan-
Vranceanu(GBCV)spaceE3(M,τ)andspacelikegraphswithprescribed
] mean curvature τ in the GBCV spacetime L3(M,H). For instance, the
G prescribed mean curvature equation in L3 can be transformed into the
D minimalsurfaceequationinthegeneralizedHeisenbergspacewithpre-
. scribed bundle curvature. We present several applications of the twin
h correspondenceand studythe modulispace of completespacelikesur-
t
a facesintheGBCVspacetimes.
m
[

1
1. Motivationandmainresults
v
1
1.1. History of Calabi problems. The significant role played by maximal
4
2 and constant mean curvature submanifolds in spacetimes is widely rec-
7 ognized. Maximal submanifolds immersed in a Lorentzian manifold are
.
1 spacelikesubmanifoldswithvanishingmeancurvature,andnaturallyarise
0
as solutions of the variational problem of locally maximizing the induced
3
volumefunctionalwithintheclassofspacelikesubmanifolds.
1
: Among various interesting applications, the issue of the existence of
v
i maximal submanifolds appears in the Schoen-Yau proof of the Positive
X
Mass Theorem and also in the analysis of solutions of the Einstein-Yang-
r
a Millsequation. Veryrecently,BonsanteandSchlenker[8]usedthegeometry
ofmaximal surfacesin theantideSitterspacetimetoestablishavariant of
Schoen’sconjectureontheuniversalTeichmu¨llerspace.
Determining the existence, uniqueness, and regularity of maximal and
constantmean curvature submanifolds in various spacetimesis oneofthe
importantproblemsinbothmathematical relativityanddifferentialgeom-
etry. Indeed,suchsubmanifoldsareRiemannianmanifoldsreflectingsome
oftheRiemannianfeaturesoftheirambientspacetimes[6].
In1970,Calabi[9]presentedaremarkableuniquenessresultontheglobal
behavior of maximal surfaces. He proved that, in the Lorentz-Minkowski
space L3 = (R3,dx2 + dy2 dz2), the only entire maximal graphs defined
−
2010MSC:Primary49Q05,53A10;Secondary53C50,35B08.
1
2 HOJOOLEEANDJOSE´ M.MANZANO
overthewholexy-plane arespacelikeplanes,whichimpliesthatspacelike
planesaretheonlycompletemaximalsurfacesinL3. Ontheotherhand,the
beautiful theorem of Bernstein states that the only entire minimal graphs
intheEuclideanspaceE3 = (R3,dx2 +dy2 +dz2)areplanes. However,itis
well-knownthatE3admitsdifferentcompletenon-planarminimalsurfaces
(e.g.,catenoids,Costa’ssurface,etc).
ChengandYau[11]extendedCalabi’s Theoremtohigherdimensionby
provingthattheonlycompletemaximalhypersurfaceinLorentz-Minkowski
space Ln+1 3 are the spacelike hyperplanes. Nonetheless, Bombieri, De
≥
Giorgi and Giusti [7] disproved the Riemannian analog by finding entire
non-planarminimalhypersurfacesintheEuclideanspaceEn+1 9.
≥
Duringthelastfewdecades,specialattentionhasbeengiventotheexis-
tence, uniqueness, and non-existenceof constant mean curvature surfaces
inmoregeneralambientspaces. Here,wesketchsomeofknownresults.
Ferna´ndezandMira[19,Theorem3.1]constructedalargefamilyof
•
completemaximalsurfacesinthestaticRobertson-Walker(RW)space
H2 R endowed with the Lorentzian product metric , H2 dz2.
The×y are entire graphs over the hyperbolic base H2, handi so−they
inducefoliationsoftheambientspace. Ontheotherhand,Albujer[1,
Examples3.1and3.3]provedthatH2 Ralsoadmitsnon-complete
×
entiremaximalgraphs.
Albujer and Al´ıas [2, Theorem 3.3] proved that, given an arbitrary
•
Riemannian surface M with Gaussian curvature K 0, complete
M
maximal surfaces in the product space M R (endo≥wed with the
×
Lorentzianproductmetric , dz2)aretotallygeodesic. Further-
M
h i −
more,ifK doesnotidenticallyvanish,thentheyshowthatΣmust
M
beaspacelikehorizontalsliceM t forsomet R.
0 0
×{ } ∈
Calabi type problems in the Generalized Robertson-Walker (GRW)
•
spacetimes are topics of increasing interest in recent years [31,33].
Given a smooth warping function φ : R (0,+ ) and a base Rie-
mannianmanifoldM,theGRWspacetime→M R∞isdefinedasM R
φ
× ×
equipped with the Lorentzian warped metric φ(z)2 , dz2, and
M
h i −
carriesaglobaltimelikeconformalvectorfieldφ(z)∂ .
z
GRW spacetimes are particular cases of Conformally Stationary
(CS) spacetimes which are time-orientable spacetimes admitting a
timelikeconformalfield. Forinstance,seeMontiel’sobservation[31,
Section 3]. Several existence results of spacelike immersions in CS
spacetimesarealsoknown[22,24].
Calabi’s original proof [9] of the fact that spacelike planes are the only
entiremaximalgraphsinL3ismotivatedbyhisinterestingdualitybetween
minimalgraphsinE3andmaximalgraphsinL3. Thistwincorrespondence
is extended to a correspondence between graphs of constant mean curva-
ture H in the Riemannian Bianchi–Cartan–Vranceanu (BCV) space E3(κ,τ)
GENERALIZEDCALABI’SCORRESPONDENCEANDCOMPLETESPACELIKESURFACES 3
and spacelike graphs of constant mean curvature τ in the BCV spacetime
L3(κ,H) [29]. In the particular case τ = H = 0, the twin correspondence
reducestotheAlbujer-Al´ıasduality[2].
Since the twin correspondence in BCV spaces sends entire graphs to
entire graphs, it becomes a natural and useful tool to study Bernstein-
Calabi type problems. For instance, entire spacelike graphs of constant
meancurvature 1 in L3 = L3(E2,0)arethetwinsurfacesofentireminimal
2
graphs in the Riemannian Heisenberg space Nil3 = E3(E2, 1). Since the
2
moduli space of entire spacelike graphs of constant mean curvature 1 in
2
L3 is large (Treibergs [35, Theorem 2] showed that such spacelike graphs
canbeasymptotictoanarbitrary 2perturbationofthelightconeinL3),it
C
followsthatthereexistlotsofentireminimalgraphsinNil3.
Calabi’s dualitycanalsobeextendedtohighercodimension[27].
1.2. Sketch of main results. The key point of this article is to study pre-
scribed mean curvature surfaces in a manifold via their twin surfaces in
differentambientmanifolds.
The ambient manifolds we will deal with are introduced in Section 2,
and will be called Riemannian and Lorentzian Generalized Bianchi–Cartan–
Vranceanu(GBCV in thesequel)spaces,which will bedenotedbyE3(M,τ)
andL3(M,τ),respectively,whereMwillbeaRiemanniansurface,andτwill
beanysmoothfunctiononM. TheseGBCVspacesarelocallycharacterized
by admitting a Riemannian submersion over M with bundle curvature
τ, and whose fibers are the integral curves of a unit Killing vector field
(timelike in the Lorentzian case), so GBCV spaces are often called Killing
submersions [17,30]. The main technical tool in this section will be the
concept of Calabi potential, which yields an explicit way of recovering the
metric of the GBCV spaces E3(M,τ) and L3(M,τ) in terms of M and τ.
Moreover, Killing submersions are spaces where graphs arise in a natural
wayassmoothsectionsoftheaforementionedRiemanniansubmersion,and
theirmeancurvaturesadmitdivergence-formexpressions(seeLemma3.5)
GBCV spaces can be understood as twisted product spaces since if the
bundle curvature is identically zero, then the GBCV spaces recover the
Riemannian and Lorentzian product spaces M R. If M is the simply-
×
connectedconstant curvature κ surface and the bundle curvature τ is also
constant,theyreducetotheclassicalBCVspacesE3(κ,τ)andL3(κ,τ).
Section 3 is devoted to establish our Calabi type twin correspondence,
andtoshowthattwinsurfacesadmitsimultaneousconformalcoordinates.
Moreexplicitly,graphswithprescribedmeancurvatureHintheRiemann-
ian GBCV space E3(M,τ) correspond to spacelike graphs with prescribed
meancurvatureτintheLorentzianGBCVspaceL3(M,H). Inotherwords,
thetwincorrespondenceswapsthemeanandbundlecurvatures.
4 HOJOOLEEANDJOSE´ M.MANZANO
Asaparticularcase,ifthecommonbaseMistheEuclideanplaneE2and
H = τ = 0, the twin correspondence reduces to the classical Calabi corre-
spondence[9]. When the curvature of the base, the bundle curvature and
themeancurvatureareallconstants,itreducestothetwincorrespondence
intheclassicalBCVspaces[27,Theorem2].
In Section 4, we will make use of the twin correspondences to investi-
gate complete spacelike surfaces in the GBCV spacetimes. We recall that
completenessofentirespacelikegraphsinLorentzianmanifoldsisnotguar-
anteedingeneral[1].
We establish that complete spacelike surfaces in the GBCV spacetime
L3(M,τ) with simply-connected base M are entire graphs. In particular,
thestudyofcompletespacelikesurfacesintheGBCVspacetimereducesto
thatoftheentirespacelikesolutionsofthecorrespondingprescribedmean
curvature equation, where classical PDE techniques can be applied. It is
importanttoremarkthatcompletespacelikesurfacesinaGBCVspacetime
areincorrespondencewithnon-trivialspacelikefoliationsoftheirambient
spaces modulo the 1-parameter group of isometries associated to the unit
verticalKillingvectorfield.
We find that there exists a large family of GBCV spacetimes carrying
no complete spacelike surfaces, and this is achieved by obtaining a sharp
boundonthemeancurvaturefunctionofanentiregraphintheRiemannian
GBCV spaces. More explicitly, we prove that if the bundle curvature τ of
the GBCV spacetime L3(M,τ) satisfies inf τ > 1Ch(M), where Ch(M)
M| | 2
denotes the Cheeger constant of the non-compact simply-connected base
M,thenL3(M,τ)admitsnocompletespacelikesurfaces. Asaconsequence,
ifwedenoteby c 0theinfimum oftheGaussian curvature ofM andthe
≤
estimateinf τ > 1√ cholds,thenitispossibletoconcludethatL3(M,τ)
M| | 2 −
doesnotadmit completespacelikesurfaces, either. This is duetothenon-
existencetheoremaboveandacomparisonresultforCheegerconstants. It
isimportanttonoticethatthereisnoassumptiononthemeancurvatureof
thecompletespacelikesurface.
Asaparticularcase,wededucethattheredonotexistcompletespacelike
surfaces in the classical BCV spacetime L3(κ,τ) with constant base curva-
tureκ 0andconstantbundlecurvature τ > 1√ κ. Moreover,weshow
≤ | | 2 −
that the lower bound 1√ κ is sharp, for the moduli space of complete
2 −
maximal surfaces in the anti de Sitter spacetime L3(κ, 1 √ κ) is large. In
2 −
fact,weprovethatamaximalsurfaceinL3(κ, 1√ κ) = L3(H2(κ), 1√ κ)is
2 − 2 −
completeifandonlyifitisanentirespacelikegraphdefinedoverthewhole
hyperbolic base H2(κ). This extends the Cheng-Yau result [11] that entire
constantmeancurvaturespacelikegraphsinL3 = L3(E2,0)arecomplete.
As a final consequence of the non-existence result, we prove that the
GBCV spacetime L3(M,τ) is not distinguishing (in the sense of causality)
wheninf τ > 1Ch(M).
M| | 2
GENERALIZEDCALABI’SCORRESPONDENCEANDCOMPLETESPACELIKESURFACES 5
2. ConstructionofGBCVspaces
We will begin by defining models for the Riemannian and Lorentzian
Generalized Bianchi–Cartan–Vranceanu spaces (GBCV in the sequel) in
termsofthebundlecurvatureandaconformalparametrizationofthebase
surfaceasfollows.
Definition 2.1 (GBCV spaces E3(M,τ) and L3(M,τ)). Let M = (Ω,g =
δ(x,y) 2(dx2 + dy2)) denote a Riemannian surface, where Ω R2 is open
−
⊆
andstar-shapedwithrespecttotheorigin,andδ (Ω)ispositive. Given
∞
∈ C
τ 2(Ω),wewillcallCalabipotentialthefunctionC 2(Ω)definedby
δ,τ
∈ C ∈ C
1 τ(tx,ty)
(2.1) C (x,y) = 2 t dt,
δ,τ Z0 δ(tx,ty)2
ThustheCalabipotentialC satisfiesthedivergence-formequation
δ,τ
2τ ∂ ∂
(2.2) = xC + yC ,
δ2 ∂x δ,τ ∂y δ,τ
(cid:0) (cid:1) (cid:0) (cid:1)
andallowsustodefinetwo3-dimensionalmanifoldsasfollows:
(a) The Riemannian GBCV space E3(M,τ) is the product 3-manifold
Ω RendowedwiththeRiemannianmetric
×
1
dx2 +dy2 + dz+C (x,y) ydx xdy 2.
δ(x,y)2 δ,τ −
(cid:16) (cid:17) (cid:0) (cid:0) (cid:1)(cid:1)
(b) The GBCV spacetime L3(M,τ) is the 3-manifold Ω R endowed
×
withtheLorentzianmetric
1
dx2 +dy2 dz C (x,y) ydx xdy 2.
δ(x,y)2 − − δ,τ −
(cid:16) (cid:17) (cid:0) (cid:0) (cid:1)(cid:1)
We will call the 2-manifold M = (Ω,g) the base surface. The real-valued
functionτwillbecalledthebundlecurvaturefunctionoftheambientGBCV
spacesE3(M,τ)andL3(M,τ). ThisismotivatedbyRemark2.2below.
Remark 2.2 (GBCV spaces as Killing submersions). The projection map
π : Ω R Ω given by π(x,y,z) = (x,y) is a Riemannian submersion
× →
from the ambient GBCV spaces to the base surface, and its fibers are the
integral curves the unit Killing vector field ∂ . Spaces satisfying these
z
conditions are known as Killing submersions [17]. When M is simply-
connected,theRiemannianGBCVspacesasdefinedabovearecharacterized
astheonlysimply-connected3-manifoldswhichadmitastructureofKilling
submersionoverMwithbundlecurvaturefunctionτ[30]. HencetheCalabi
potentialcanbethoughtasawayofrecoveringthemetricoftheGBCVspace
intermsofitsbundlecurvature.
Remark 2.3 (GBCV spaces as twisted product spaces). If the bundle cur-
vaturefunctionτidenticallyvanishes,GBCVspacesreducetotheproduct
6 HOJOOLEEANDJOSE´ M.MANZANO
spaces E3(M,0) = M R with the Riemannian product metric , +dz2,
M
andL3(M,0) = M R×withtheLorentzianproductmetric , h idz2.
M
× h i −
On the other hand, if both the curvature of the base M and the bundle
curvature τ are constant, geometric meanings of the bundle curvature τ
arerevealedin[14,Section2.1]and[16,Proposition1.6.2],andtheirproofs
naturally extend to the GBCV spaces. Let 3 denote the GBCV space
K
E3(M,τ) or L3(M,τ) carrying the unitary Killing field ξ = ∂ and let be
z
∇
its associated Levi-Civita connection. Then the bundle curvature function
τsatisfiesthegeometricidentity
(2.3) ξ = τX ξ, forall X X 3 .
X
∇ ∧ ∈ K
(cid:16) (cid:17)
We note that the vector product in 3 is determined by an oriented
∧ K
orthonormalframe,whichcanbeexplicitlychosenbyusingtheCalabipo-
tential. Moreprecisely,theglobalorthonormalframe E ,E ,E inE3(M,τ)
1 2 3
{ }
willbeassumedpositivelyoriented,where
E = δ ∂ yC ∂ , E = δ ∂ +xC ∂ , E = ∂ .
1 x δ,τ z 2 x δ,τ z 3 z
−
(cid:0) (cid:1) (cid:0) (cid:1)
Likewise, the orientation in the Lorentzian case will be given by the or-
thonormalframe L ,L ,L inL3(M,τ),where
1 2 3
{ }
L = δ ∂ + yC ∂ , L = δ ∂ xC ∂ , L = ∂ .
1 x δ,τ z 2 x δ,τ z 3 z
−
(cid:0) (cid:1) (cid:0) (cid:1)
Another geometric interpretation of the bundle curvature function of the
Killing submersion π : 3 M can be found in [30, Proposition 3.3]. In
K →
theRiemanniancase,theproofof(2.3)isavailable in[17,Proposition2.6].
Remark2.4(GBCVspacesasextensionsofBCVspacesE3(κ,τ)andL3(κ,τ)).
ClassicalBCVspaceshaveconstantbasecurvatureandconstantbundlecur-
vature. Given aconstantκ R,weconsiderδ(x,y) = 1+ κ(x2 + y2),which
∈ 4
ispositiveinΩ = R2 forκ 0orΩ = (x,y) R2 : x2+ y2 < 4 forκ < 0.
≥ { ∈ −κ }
Then the metric g = δ(x,y) 2(dx2 +dy2) on Ω has constant curvature κ,
−
andM = (Ω,g)isisometrictoE2(forκ= 0),H2(κ)(forκ < 0),orS2(κ)minus
apoint(forκ > 0). Whenthebundlecurvaturefunctionτ(x,y)isconstant,
the associated Calabi potentialis nothing but C (x,y) = τδ(x,y) 1, so we
δ,τ −
get the classical BCV spaces E3(κ,τ) and L3(κ,τ), see [14]. Hence, in this
notation we obtain E3(κ,τ) = E3(M2(κ),τ) and L3(κ,τ) = L3(M2(κ),τ),
whereM2(κ)denotestheaforementionedsurfacesofconstantcurvatureκ.
3. TwincorrespondencesinGBCVspaces
We will now derive formulas for the mean curvature of vertical graphs
intheGBCVspaces. KeepingthenotationofDefinition2.1,thegraphofa
functionu 2(Ω ),whereΩ Ωisanopensubset,isthesurface
′ ′
∈ C ⊆
(x,y,z) Ω R : z= u(x,y) .
′
∈ ×
(cid:8) (cid:9)
GENERALIZEDCALABI’SCORRESPONDENCEANDCOMPLETESPACELIKESURFACES 7
In other words, we are considering graphs over the global zero-section
z = 0. IntheLorentziancase,thegraphissaidtobespacelike iftheinduced
metric from the ambient manifold is Riemannian. The following lemma
canbeeasilydeduced.
Lemma 3.5 (Mean curvature of graphs in GBCV spaces). Let u 2(Ω )
′
∈ C
denotetheheightfunctionofagraphdefinedonsomeopensubsetΩ Ω.
′
⊆
(1) ThemeancurvaturefunctionHofthegraphofuintheRiemannianGBCV
spaceE3(M,τ)satisfies
∂ α ∂ β G
(3.1) 2H = δ2 + = div  ,
M
∂x (cid:18)ω(cid:19) ∂y ω!! q1+kGk2M
where α = u + yC , β = u xC , ω = 1+δ2(α2+β2), and the
x δ,τ y δ,τ
−
vectorfieldGonΩ MisgivenbyG = δ2(α∂p+β∂ ).
′ x y
⊂
(2) If the graph of u is spacelike in the Lorentzian BCV space L3(M,τ), then
itsmeancurvaturefunctionH satisfies
e
e
∂ α ∂ β G
(3.2) 2H = δ2 + = div  ,
e ∂x ωee! ∂yωee M q1−ekGek2M
where α = u yC , β = u +xC , ω = 1 δ2(α2+β2), and the
x δ,τ y δ,τ
− q −
vector field G on Ω M is given by G = δ2(α∂ +β∂ ). Wenotice that
e ′ ⊂e e e e x ye e
thegraphofuisspacelikeinL3(M,τ)ifandonlyif
e e e e
2
1 G = 1 δ2(eα2+β2)> 0.
−k kM −
Example3.6(HelicoidsineGBCVspacesewitehrotationalsymmetry). Ver-
tical translations (i.e., the elements of the 1-parameter group of isometries
associatedtotheKilling vectorfield∂ )are generically theonlyisometries
z
in the GBCV spaces. If we additionally assume that both the conformal
factor and the bundle curvature are radial with respect to the origin (i.e.,
δ(x,y)andτ(x,y)arefunctionsofx2+y2),thentheinducedCalabipotential
C isalsoradial,androtationsabouttheaxisx = y= 0arealsoisometries.
δ,τ
Itiseasytocheckthat,givenµ ,µ R,thehelicoid
1 2
∈
= (ρcos(θ),ρsin(θ),µ θ+µ ) : ρ,θ R
Hµ1,µ2 { 1 2 ∈ }
has zero mean curvature in the GBCV spaces when radial symmetry is
assumed, though in the Lorentzian case, the spacelike condition is not
guaranteed. Notethat isahorizontalplaneforµ = 0andconverges
Hµ1,µ2 1
toaverticalplanewhenµ . Infact,eachlevelcurvel = z=
1 →∞ z0 Hµ1,µ2∩{
z isanambientgeodesic,so isalsoaruledsurface. Moreover,except
0} Hµ1,µ2
8 HOJOOLEEANDJOSE´ M.MANZANO
attheaxisx = y= 0,theycanbeexpressedlocallyasthegraphs
y
z= µ arctan +µ .
1 2
x
(cid:18) (cid:19)
ItisworthtomentionthattheonlyruledminimalsurfacesintheHeisen-
berg space Nil = E3(E2,1) are (part of) planes, helicoids, and hyperbolic
3 2
paraboloids up to ambient isometries [25, Theorem 2.3]. It would be in-
teresting to extend this characterization to GBCV spaces with rotational
symmetry.
We will construct a Calabi-type correspondence between graphs with
mean curvature function H in the Riemannian GBCV space E3(M,τ) and
spacelike graphs with mean curvature function τ in the Lorentzian GBCV
spaceE3(M,H). Moreexplicitly,weobtain:
Theorem3.7(Twincorrespondence). LetMbeanopendomainΩ R2,star-
⊆
shapedwithrespecttotheorigin,endowedwiththemetricδ 2(dx2+dy2)forsome
−
positive δ C (Ω). Given τ,H 2(Ω) and a simply-connected open domain
∞
∈ ∈ C
Ω Ω,then:
′
⊆
(a) If the graph of a function f 2(Ω ) in E3(M,τ) has mean curvature
′
∈ C
function H, then there exists g 2(Ω ) such that the graph of g is
′
spacelike inL3(M,H)andhasmea∈ncCurvaturefunctionτ.
(b) Ifthegraphofafunctiong 2(Ω )inL3(M,H)isspacelikeandhasmean
′
∈C
curvature function τ, then there exists f 2(Ω ) such that the graph of
′
f inE3(M,τ)hasmeancurvaturefunctio∈nCH.
Thegraphsin(a)and(b)canbechosentosatisfythetwinrelations:
β α β α
(3.3) α,β = , , orequivalently, α,β = , ,
−ω ω! ω −ω
where (cid:16)e e(cid:17) (cid:0) (cid:1) e e
e e
α,β = f + yC , f xC , ω = 1+δ2 α2+β2 ,
x δ,τ y δ,τ
− q
(cid:0) (cid:1) (cid:16) (cid:17) (cid:16) (cid:17)
α,β = g yC ,g +xC , ω = 1 δ2 α2+β2 .
x δ,H y δ,H
− q −
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
Moreoveer, leet us parametrize the graphs as F(x,y)e= (x,y, f(x,ye)) andeG(x,y) =
(x,y,g(x,y)), (x,y) Ω . Then, the mapping F(x,y) G(x,y) is a conformal
′
∈ 7→
diffeormorphism betweenthetwographs. IfIandI denotetheinducedmetricson
∗
thegraphs z= f(x,y)inE3(M,τ)andz = g(x,y)inL3(M,H),respectively, then
1
I = I .
∗G(x,y) ω(x,y)2 F(x,y)
Intheconditionsabove,thegraphof f inE3(M,τ)andthegraphof gin
L3(M,H)arecalledtwinsurfaces.
GENERALIZEDCALABI’SCORRESPONDENCEANDCOMPLETESPACELIKESURFACES 9
Proof. Wewillbeginby proving(a) andtheproofof(b)will beanalogous.
Lemma3.5showsthatthemeancurvaturefunctionHofthegraphof f over
Ω intheGBCVspaceE3(M,τ)isthedivergence-formequation
′
2H ∂ α ∂ β
(3.4) = + ,
δ2 ∂x(cid:18)ω(cid:19) ∂y ω!
where (α,β) = (f + yC , f xC ) and ω = 1+δ2 α2+β2 . Equa-
x δ,τ y δ,τ
tion(2.2),tellsusthattheCalab−ipotentialC (x,y)qsatisfie(cid:16)sthediv(cid:17)ergence-
δ,τ
formequation
2τ ∂ ∂
(3.5) = xC + yC .
δ2 ∂x δ,τ ∂y δ,τ
(cid:0) (cid:1) (cid:0) (cid:1)
Likewise,theCalabi potentialC (x,y)satisfies
δ,H
2H ∂ ∂
(3.6) = xC + yC .
δ2 ∂x δ,H ∂y δ,H
(cid:0) (cid:1) (cid:0) (cid:1)
Hence, by using (3.6), we are able to rewrite the mean curvature equation
(3.4)asthezero-divergenceequation
∂ α ∂ β
0= xC + yC .
δ,H δ,H
∂x (cid:18)ω − (cid:19) ∂y ω − !
Since the domain Ω is simply connected, Poincare´’s Lemma yields the
′
existenceofafunction g 2(Ω )suchthat
′
∈C
β α
g ,g = + yC , xC .
x y δ,H δ,H
−ω ω − !
(cid:16) (cid:17)
Setting(α,β)= (g yC ,g +xC ),weobtainthespacelikecondition
x δ,H y δ,H
−
e e 1 δ2 α2+β2 = 1 > 0,
− ω2
(cid:16) (cid:17)
e e
so it makes senseto introduce ω = 1 δ2 α2+β2 , from where the twin
relations (3.3) easily follow. Finallqy, w−e e(cid:16)mploy t(cid:17)he twin relations, the
e e e
integrabilitycondition ∂ (f )= ∂ (f )and(3.5)todeducethat
∂x y ∂y x
∂ α ∂ β ∂ ∂
+ = β + (α)
∂x ω! ∂yω ∂x − ∂y
e e ∂ (cid:0) (cid:1) ∂
e e = f +xC + f + yC
y δ,τ x δ,τ
∂x − ∂y
(cid:16) (cid:17) (cid:0) (cid:1)
∂ ∂ 2τ
= xC + yC = .
∂x δ,τ ∂y δ,τ δ2
(cid:0) (cid:1) (cid:0) (cid:1)
Thismeansthatthespacelikegraphz = g(x,y)overthesamedomainΩ in
′
L3(M,τ)hasmeancurvaturefunctionτ,sothecorrespondenceholds.
10 HOJOOLEEANDJOSE´ M.MANZANO
Wewillnowdealwiththelastparagraphinthestatement. Letζ = ξ+iη
bealocalcomplexconformalcoordinateonthegraphz = f(x,y)andletus
considerthecoordinatetransformationφ : (x,y) ξ,η toobtainthelocal
isothermalparametrizationofthegraphz = f(x,→y)(cid:0)inE3(cid:1)(M,τ):
F = F φ 1 : ξ,η (x,y) (x,y, f(x,y)).
−
◦ 7→ 7→
(cid:0) (cid:1)
Here, φ 1 denobtes the local inverse coordinate transformation ξ,η
−
7→
(x,y) = p ξ,η ,q ξ,η ,soourgoalistoprovethat
(cid:0) (cid:1)
(cid:0) (cid:0) (cid:1) (cid:0) (cid:1)(cid:1)
G = G ϕ 1 : ξ,η (x,y) (x,y,g(x,y))
−
◦ 7→ 7→
(cid:0) (cid:1)
is an isothermalbparametrization of the twin graph z = g(x,y) in L3(M,H)
andthatthearisingconformalfactoris 1 . Takingthecomplexifiedoperator
ω2
∂ = 1 ∂ i ∂ ,wewillbedonebyprovingthat
∂ζ 2 ∂ξ − ∂η
(cid:16) (cid:17)
∂G ∂G ∂G ∂G 1 ∂F ∂F
, = 0 and , = , .
*∂ζ ∂ζ+ * ∂ζ ∂ζ + ω2 *∂ζ ∂ζ +
b b L3(M,H) b b L3(M,H) b b E3(M,τ)
We consider the global orthonormal frames E ,E ,E and L ,L ,L in
1 2 3 1 2 3
E3(M,τ)andL3(M,H),respectively,givenby { } { }
E = δ ∂ yC ∂ , E = δ ∂ +xC ∂ , E = ∂ ,
1 x δ,τ z 2 x δ,τ z 3 z
−
L = δ(cid:0)∂ + yC ∂(cid:1) , L = δ(cid:0)∂ xC ∂(cid:1) , L = ∂ .
1 x δ,H z 2 x δ,H z 3 z
−
(cid:0) (cid:1) (cid:0) (cid:1)
Thechainrulegives
∂F ∂p∂F ∂p∂F ∂p ∂ ∂f ∂ ∂p ∂ ∂f ∂
= + = + + +
∂ζ ∂ζ∂x ∂ζ∂y ∂ζ ∂x ∂x∂z! ∂ζ ∂y ∂y∂z!
b
1∂p 1∂q ∂p ∂q
= E + E + α +β E ,
1 2 3
δ∂ζ δ∂ζ ∂ζ ∂ζ!
and,analogously,
∂G 1∂p 1∂q ∂p ∂q
= L + L + α +β L .
1 2 3
∂ζ δ∂ζ δ∂ζ ∂ζ ∂ζ!
b
e e
Usingthetwinrelations(α,β) = ( β, α),itiseasytodeducefollowingtwo
−ω ω
equalitiesthatwillcomeinhandyforthenextcomputations:
e e
1 α2δ2 1 1 1 β2δ2 1 1
− = +α2 and − = +β2 .
δ2 ω2 (cid:18)δ2 (cid:19) δe2 ω2 (cid:18)δ2 (cid:19)
e