Global Existence for the Minimal Surface Equation on $\mathbb{R}^{1,1}$

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by  Willie Wai Yeung Wong

GLOBAL EXISTENCE FOR THE MINIMALSURFACE EQUATIONON R1,1
WILLIEWAIYEUNGWONG
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0 Abstract. Ina2004paper,Lindbladdemonstratedthattheminimalsurfaceequa-
2 tiononR1,1describinggraphicaltime-likeminimalsurfacesembeddedinR1,2en-
joysmalldataglobalexistenceforcompactlysupportedinitialdata,usingChristodoulou’s
n
conformal method. Here we give a different, geometric proof of the same fact,
a
J whichexposesmoreclearlytheinherentnullstructureoftheequations,andwhich
allowsustoalsoclosetheargumentusingrelativelyfewderivativesandmildde-
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cayassumptionsatinfinity.

]
P
A
1. Introduction
.
h
t The equation describing graphical timelike minimal surfaces in R1,d+1 can be
a
m writtenasthequasilinearwaveequation
[  mµνφ,ν 
1 (1) p1+mστφ,σφ,τ,µ=0.
v
In dimensions d ≥4 the small data global existence follows largely from the lin-
6
9 ear decay of solutions to the wave equation which has the rate ≈t−3/2, and is by
0 nowstandard[Sog08]. Indimensions d =2,3, thelinear decay ofthe waveequa-
1 tionhasgenericrates≈t−1/2,t−1 respectively,which,notbeingintegrableintime,
0
canleadtofinite-timesingularityformationforevensmalldataforgenericquasi-
.
1 linear wave equations; see the survey article [HKSW14]. Equation (1) however
0
exhibits the null condition, which is a structural condition on the nonlinearities
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identified first by Klainerman [Kla86] and Christodoulou [Chr86] in d = 3 and
1
: later generalized by Alinhac in d = 2 [Ali01a, Ali01b]. Using this fact Brendle
v
[Bre02]andLindblad[Lin04]establishedthesmalldataglobalexistencefor(1)in
i
X dimensionsd =3andd=2respectively. Brendle’sprooffollowedthecommuting
r vectorfieldmethodofKlainerman[Kla86]. Lindblad,however,gavetwodifferent
a proofsinhispaperusingrespectivelythecommutingvectorfieldmethodaswell
asChristodoulou’scompactificationmethod.
Inthispaperwefocusonthecased =1. Thelinearwaveequationexhibits no
decayind=1,andhencetheclassicalnullconditioncannotbeusedtoassertthat
theperturbationiseffectivelyshortrange,asinthecaseford≥2. Asasideeffect
thismeansthatadirectproofofglobalexistencefor(1)ind=1modeledafterthe
commutingvectorfieldmethodisnotpossible. Astrikingaspectof[Lin04]isthat
via the conformal compactification method, Lindblad was also able to prove the
2000MathematicsSubjectClassification. MSC,lookuponAMS.
TheauthorthankstheTsinghuaSanyaInternationalMathematicsForum,Sanya,Hainan,People’s
RepublicofChina;aswellastheNationalCenterforTheoreticalSciences,MathematicsDivision,Na-
tionalTaiwaneseUniversity, Taipei,Taiwan,fortheirhospitalityduringtheperiodinwhichthisre-
searchwasperformed.
$Format:v:%h %d;lastedit:%aNon%ai.$.
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globalexistenceofsolutionsto(1)forsmallinitialdata. AsLindbladobserved,the
maintrade-offisthatfortheconformalcompactificationmethod,thedatamustbe
ofcompactsupport, whileinthe vectorfieldmethodthedata ismerely required
tohave“sufficientlyfast”decayatinfinity.Thepurposeofthispaperistoproduce
an alternative proofof the d =1 case allowing initial data that is notnecessarily
compactly supported. In the course of the discussion we will also extract some
moredetailedgeometricinformationconcerningthesolutionwhenthedata isof
compactsupport.
2. GeneralGeometricformulation
TheresultofBrendleandLindbladarebasedoncomparingthetime-likemin-
imalsurfacetoaflathyperplanebynormalprojection,andthescalarfunctionφ
in (1) describes the deviation, or height, of the minimal surface as a graph over
the hyperplane. The minimal surface equation can however by written intrinsi-
callybywayoftheGaussandCodazziequations. Toquicklyrecall: letM ⊆R1,d+1
be a time-like hypersurface, then the Gauss equation requires that the Riemann
curvaturetensor1fortheinducedLorentzianmetricg obey2
(Gauss) Riem =k k −k k
abcd ac bd ad bc
where k is the second fundamental form of the embedding of M. The Codazzi
equationontheotherhandrequires
(Codazzi) ∇ k −∇ k =0
a bc b ac
where∇istheLevi-Civita connectionassociatedtog. Itiswellknownthatequa-
tions(Gauss)and(Codazzi)aretheonlyobstructionstotheexistenceofanisomet-
ricembedding. Moreprecisely,wehavethefollowingtheorem.
Theorem1(Fundamentaltheoremofsubmanifolds). LetM beasimply-connected
manifold,withprescribedaLorentzianmetricg andasymmetricbilinearformk. Sup-
pose that connection and curvature of g satisfy (Gauss) and (Codazzi). Then there
exists an isometric immersion of M intoMinkowski space with one higher dimension,
suchthatk isthecorrespondingsecondfundamentalform.
Remark2. TheRiemannianversionofamoregeneraltheoremiswell-known. See
forexampleTheorem19inChapter7of[Spi79]. TheLorentzianversionfollows
mutatismutandisfromthe sameproof. Ontheother hand, sinceweareconsider-
ingtheCauchyproblem,onecanalsosimply“integrate”thesecondfundamental
form in time once to obtain the normal vector field to M, and integrate it once
moretogetaparametrizationofthehypersurface.
Theupshotofthisisthat,theinitialvalueproblemfor(1)canbereformulated
asfindingthe metric g andthe secondfundamental formk fromthe initial data,
requiringthattheyobey(Gauss)and(Codazzi),
Equations(Gauss)and(Codazzi)donotyetdescribeanevolution: theyareun-
derdetermined. Wegetawell-posedinitialvalueproblemifwecombine(Codazzi)
withtheminimalsurfaceequationtr k=0,whichgives
g
(Minimality) ∇ak =0.
ab
1WeusetheconventionRiemabcdXaYbZc=∇[X,Y]Zd−[∇X,∇Y]ZdwhichimpliesRicac=Riemabcb.
2Wewill,throughoutthispaper,freelyraiseandlowerindicesusingtheinducedmetric.
1+1GLOBALEXISTENCEOFTIMELIKEMINIMALSURFACE 3
Using that the Riemann curvature tensor is at the level of two derivatives of the
metric,theinitialdataforthesystem(Gauss),(Codazzi),and(Minimality)consists
ofthemetricg,theconnectioncoefficientsΓ,andthesecondfundamentalformk
restrictedtoaninitialslice. Theinitialvalueproblemfor(1)requiresthedatafor
φ and ∂ φ at time t =0; these determine fully the first jet of φ restricted to the
t
initialslice. Asthe(1)issecondorderhyperbolic,theformallyprovidesusthek-
jetofφforanyk∈Nrestrictedtotheinitialslice,andasimplecomputationshows
that the 2-jet of φ fully determines g, Γ, and k on the initial slice. And thus we
canindeedapproachthevanishing-mean-curvature evolution throughtheinitial
valueproblemwrittenintermsoftheGauss-Codazziequations.
3. Doublenullformulationwhend=1
For geometric partial differential equations, it is necessary to fix a gauge (i.e
make a coordinate choice). For hyperbolic equations it is convenient to use a
double-null formulation. Since M is a two-dimensional Lorentzian manifold, it
isfoliatedbytwotransversefamiliesofnullcurves. Equivalently,thereexiststwo
independentfunctionsu,v satisfying
(2) g(∇u,∇u)=g(∇v,∇v)=0
andg(∇u,∇v)>0. Notethatreplacingubyf(u)foranyf :R→Rwithf′,0gives
anequivalent reparametrization; wefixu,v byprescribingtheir initialvalues on
theinitialslice,satisfyinginparticularthat
(3) Initialslice={u−v=0}.
Sinceu and v aresolutions to the eikonal equation (2), as is well known Ld=ef∇u
andN d=ef∇v aregeodesicnullvectorfields,atatanypoint{L,N}formanullframe
ofthetangentspaceT M.
p
Themetrictakestheform
(4) g =(expψ)(du⊗dv+dv⊗du)
whereψ issomerealvaluedfunction. Thisimpliesthat
L(v)=g(∇u,∇v)=g(L,N)=N(u)=exp(−ψ)
sothatthecoordinatederivativesare
(5) ∂ =(expψ)L ∂ =(expψ)N
v u
andthattheinversemetricis
(6) g−1=exp(−ψ)(∂ ⊗∂ +∂ ⊗∂ )=(expψ)(L⊗N+N⊗L).
u v v u
Wecanalsocompute
∇ N =(expψ)g(L,∇ N)N =−L(ψ)N,
L L
(7)
∇ L=(expψ)g(N,∇ L)L=−N(ψ)L.
N N
The assumption that k is trace-free and symmetric requires that there exist
scalarfunctionsλ,ν suchthat
(8) k=λL⊗L+νN⊗N.
Then(Codazzi)implies
(LaNb−NaLb)∇ k =0
a bc
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and(Minimality)becomes
(LaNb+NaLb)∇ k =0.
a bc
Taking linear combinations we get finally the null propagation equations for the
secondfundamentalform
L(λ)=0,
(k-NP)
N(ν)=0.
Animmediateconsequenceof(k-NP)isthefollowingproposition.
Proposition3. Thescalarλisindependentofv;thescalarν isindependentofu.
Theevolutionofthemetricisnowascalarequationfortheconformalfactorψ.
Theequation(Gauss)implies
(9) Riem LaNbLcNd =exp(−4ψ)λν.
abcd
Combinedwiththedefinitionthat
Riem LaNbLcNd =g(∇ L−[∇ ,∇ ]L,N)
abcd [L,N] L N
wearriveat
L(N(ψ))+L(ψ)N(ψ)=exp(−3ψ)λν,
whichwecanrewriteas
(ψ-wave) ∂2 ψ=exp(−ψ)λν.
uv
Theequations(k-NP)and(ψ-wave)areevolution equationsontheMinkowski
spaceR1,1withu,vthecanonicalnullcoordinatesr±t,withinitialdataprescribed
att=0. Intheremainderofthispaperwediscusstheeasyconsequencesfromthe
formulationabove.
4. Spatiallycompactinitialdata
Suppose that the initial data for λ,ν is compactly supported, then for suffi-
ciently large u,v the functions λ,ν vanish respectively. In view of Proposition
3and(9),weimmediatelyseethat
Theorem 4. For compact initial data, the manifold M is intrinsically flat outside a
compactdomain.
Since (ψ-wave) is now a semilinear wave equation, with the nonlinearity only
supportedinafixedspace-timecompactregion,immediately byCauchystabilitywe
have
Theorem5. Forallsufficientlysmallcompactlysupportedinitialdata,wehaveglobal
existencefortheinitialvalueproblem.
Remark6. Thecompactsupportassumptionisonlynecessaryforλ,ν,andnotfor
ψ!
Remark7. Onecaneasilycheckthattheargumentclosesbystandardenergyargu-
mentsforinitialdatasatisfying
• λ,ν∈L∞({t=0})withcompactsupportandsufficientlysmallnorm;
• ψ∈H1/2+ǫ({t=0})and∂ ψ∈L2({t=0})withsufficientlysmallnorm.
t
1+1GLOBALEXISTENCEOFTIMELIKEMINIMALSURFACE 5
TheH1/2+ǫ isforSobolevembeddingintoL∞. Theregularityoutlinedaboveisfa-
vorablecomparedtotheclassicallocalwellposednesslevelatH5/2+ǫ forthequasi-
linearwaveequation (1). Recallingthatthesecondfundamentalformisroughly
twoderivatives ofthegraphicalfunctionφ andthemetricisatthelevelof|∂φ|2,
weseethatthecorrespondingregularityforλ,ν∈H1/2+ǫ andψ∈H3/2+ǫ ishigher
comparedtotheresultabove.
5. Non-compactinitialdata
We can also handle the case where λ,ν are not compactly supported initially,
butwithsufficiently strongdecay. Thisisthemaintheorem ofthis paper. Below
weletr=u+vbethecoordinatefunctionontheinitialslice,andt=u−vthetime-
functiononM;∂ and∂ refertothecoordinatederivativesinthe(r,t)coordinate
r t
system.
Theorem 8. Consider the initial value problem for the system (k-NP) and (ψ-wave),
with smooth initial data λ ,ν ,ψ =ψ| and ψ =∂ ψ| . Suppose there exists
0 0 0 {t=0} 1 t {t=0}
ǫ>0suchthatwhenevertheinitialdatasatisfies
kλ k +kν k ≤ǫ,
0 1 0 1
(cid:13)ψ (cid:13) +(cid:13)∂ ψ (cid:13) +(cid:13)ψ (cid:13) <ǫ,
(cid:13)(cid:13) 0(cid:13)(cid:13)∞ (cid:13)(cid:13) r 0(cid:13)(cid:13)1 (cid:13)(cid:13) 1(cid:13)(cid:13)1
wehaveauniqueglobalsolution.
Remark9. Asiswellknown,theregularitylevelW1,1 iscriticalforwaveequation
inonespatialdimension.
Proof. Using(k-NP) andthat λ andν propagatein differentdirections weimme-
diatelyobtainthatkλνk ≈kλνk .ǫ2. Usingthefundamentalsolutionforthe
L1t,r L1u,v
waveequationinonedimensionweget
(cid:13)ψ(t,·)(cid:13) .ǫ+(cid:13)λνexp(−ψ)(cid:13) .
(cid:13)(cid:13) (cid:13)(cid:13)∞ (cid:13)(cid:13) (cid:13)(cid:13)L1([0,t]:L1)
Thisweboundby
ǫ+exp((cid:13)(cid:13)(cid:13)ψ(cid:13)(cid:13)(cid:13)L∞([0,t]:L∞))·kλνkL1([0,t]:L1).ǫ+ǫ2exp((cid:13)(cid:13)(cid:13)ψ(cid:13)(cid:13)(cid:13)L∞([0,t]:L∞)).
Then aslongasǫ issufficiently smallwehaveby bootstrappingthat(cid:13)ψ(cid:13) <3ǫ.
(cid:13)(cid:13) (cid:13)(cid:13)L∞t,r
Thetheoremthenfollowsfromthisaprioriestimateandstandardarguments. (cid:3)
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MichiganStateUniversity,EastLansing,MI,48824
E-mailaddress:[email protected]