Minimal Surfaces in Euclidean Space Lecture Notes,

MinimalSurfacesinEuclideanSpaceLectureNotes,UAB,Spring1992GilbertWeinsteinContents1Introduction11.1Plateau’sProblem.................................11.2Dirichlet’sPrinciple................................41.3RelatedQuestions.................................61.4Exercises......................................82AnalyticPreliminaries102.1FunctionalAnalysis.................................102.2SobolevSpaces...................................112.3SobolevInequalities................................162.4EllipticRegularity.................................182.5Exercises......................................223VariationalPrinciples233.1Poincar´eInequalities................................233.2Dirichlet’sPrinciple................................263.3AGeneralVariationalPrinciple..........................263.4Poisson’sEquation.................................273.5Exercises......................................304Plateau’sProblem314.1ExistenceofaMinimalSurfaceSpanningΓ...................314.2TheIsoperimetricInequality............................404.3Uniformization...................................434.4Exercises......................................47Bibliography49iChapter1IntroductionInthisintroductorylecture,Iwishtopresentanoverviewofthequestionsandresultswhichwillbeaddressedduringthisterm.Thepresentationisnotrigorousandmanydetailsareomitted,buthopefullythischaptershouldserveasaroadmap.1.1Plateau’sProblemOurfirstgoalistoprovetheexistenceofasolutiontoPlateau’sProblem[12,4,13]:Problem1.1GivenasimpleclosedJordancurveΓ⊂Rn,whichistheboundaryofafiniteareasurfaceofdisktype,findasurfaceofdisktypeSwithboundaryΓ,suchthatShasleastareaamongallsuchsurfaces.Wewouldliketosaythatasurfaceisofdisktypeifitishomeomorphictoadisk,butascanbeseenfromsimpleexamples,thiswouldbetoorestrictive.Unlessratherstringentconditionsareputontheboundarycurve,thesolutionmayhaveselfintersections[2].Thus,wedefineStobeofdisktypeifitistheimageofasmoothmap:X:D→Rn,whereD={(u,v):u2+v21}⊂R2istheunitdiskintheplane,andX=(X1,...,Xn)areCartesiancoordinatesinn-dimensionalEuclideanspace.Bysmooth,wemeanX∈C2(D)∩C0(D).Furthermore,wewillrequirethat∇Xisnon-degenerateinD.Thisexcludesthepresenceofso-calledbranchpoints.FinallyfortheboundaryofX(D)tobeΓ,wewillaskthatX|∂Dmapsthecircle∂DcontinuouslyandmonotonicallyontoΓ.TheareaofthesurfaceX(D)iseasilycalculatedtobe:A(X)=ZDpef−g2,(1.1)wheree=|Xu|2,f=|Xv|2,andg=Xu·Xv.IfthesurfaceXisofleastarea,thenastandardcalculationshowsthatitsatisfiesthefollowingsystemofpartialdifferential1CHAPTER1.INTRODUCTION2equations,seeExercise1.1:∂∂ufXu−gXvpef−g2!+∂∂veXv−gXupef−g2!=0.(1.2)AsurfaceXsatisfyingEquations(1.2)iscalledaminimalsurface,eventhoughitmaynotbeasurfaceofleastarea.TheareaA(X)isinvariantunderre-parameterizations,thatis,ifψ:D→DisanysmoothinjectivemapofDontoitself,thenA(X)=A(X◦ψ).Thisleadstosomedifficulty,forifXisasolutionofProblem1.1,andψsuchamap,thenclearlyX◦ψisanothersolution.Therefore,thesolutionoftheproblem,intermsofthemapsXisnotunique,andtheproblemisnotwell-posed.Anotherwaytoseethis,istonotethatEquations(1.2)aredegenerate.Inordertoovercomethisdifficulty,werecallhowthisisdealtwithinthe1-dimensionalproblem:thegeodesicproblem.ConsideradomainΩ⊂RnwithaRiemannianmetricgij,i.e.asmoothsymmetricn×nmatrixvaluedfunctiondefinedonΩ.Givenacurve,γ=(γ1,...,γn):I→ΩdefinedonaclosedintervalI⊂R,thelengthofγis:L(γ)=ZI|˙γ|g,(1.3)where˙γ=dγ/dt=(˙γ1,...,˙γn)isthetangentofγ,anditsnormistakenwithrespecttotheRiemannianmetricg:|˙γ|2g=g(˙γ,˙γ)=Xi,jgij˙γi˙γj.(1.4)ThelengthL(γ)iseasilyseentobeinvariantunderre-parameterizations,i.e.ifξisanysmoothmonotonicmapξ:I→J,then:L(γ)=L(γ◦ξ).(1.5)Thereforethesamenon-uniquenessproblemariseswhenwetrytominimizethelengthofacurve,saybetweentwopointspandq∈Ω.Definetheenergyofγtobe:E(γ)=12ZI|˙γ|2g.Forsimplicity,assumethatI=[0,1].ThentheSchwartzinequalitygivesL(γ)≤p2E(γ),CHAPTER1.INTRODUCTION3withequalityifandonlyif|˙γ|gisconstant,i.e.γisparameterizedproportionallytoar-clength.Sinceeverycurve,inparticularthesolution,canbere-parameterizedproportion-allytoarclength,wehaveinfL=inf√2E.Furthermore,minimizingEyieldsasolutionparameterizedproportionallytoarclength.Thesolutionisnowunique,andthevariationalproblemiswell-posed.NotethatwithoutfixingI,thesolutionisinvariantunderanon-compactgroupoftransformations,thelinearmaps,andanyminimizationschemeisdoomedtofailure.The2-dimensionalanalogofparameterizationsproportionaltoarclengtharethecon-formalmaps.ThemapX:D→Rnissaidtobeconformalif|Xu|2=|Xv|2,Xu·Xv=0.(1.6)IfthemapXisinvertibleinaneighborhoodUofapointp∈X(D),X−1:U→Ddefinestwofunctions(u,v)onUcalledlocalisothermalcoordinates.ThefirstresultweneedistheexistenceoflocalisothermalcoordinatesonanysmoothsurfaceSofdisktype.Thisisadeepresult,generallyknownasauniformizationtheorem.DefinetheenergyofXby:E(X)=12ZD|∇X|2,(1.7)where|∇X|2=e+f.ThefunctionalE(X)isinvariantunderconformalmapsofthedomainD,seeExercise1.3.IfXminimizesE,thenitiseasilyseentosatisfythefollowingsystemoflinearellipticpartialdifferentialequations:ΔX=0.(1.8)AmapXsatisfyingEquations(1.8)issaidtobeharmonic.NotethatifXis