Holomorphic quadratic differentials and the Bernstein proble

We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosen

Holomorphicquadraticdi erentialsand

theBernsteinprobleminHeisenbergspace

arXiv:0705.1436v2 [math.DG] 9 Oct 2007IsabelFern´andezaandPabloMirabaDepartamentodeMatem´aticas,UniversidaddeExtremadura,E-06071Badajoz,Spain.e-mail:isafer@ugr.esbDepartamentodeMatem´aticaAplicadayEstad´ stica,UniversidadPolit´ecnicadeCar-tagena,E-30203Cartagena,Murcia,Spain.e-mail:pablo.mira@upct.esAMSSubjectClassi cation:53A10Keywords:minimalgraphs,Bernsteinproblem,holomorphicquadraticdi erential,Hei-senberggroup.ResumenWeclassifytheentireminimalverticalgraphsintheHeisenberggroupNil3endowedwithaRiemannianleft-invariantmetric.Thisclassi cation,whichprovi-desasolutiontotheBernsteinprobleminNil3,isgivenintermsoftheAbresch-Rosenbergholomorphicdi erentialforminimalsurfacesinNil3.1.Introduction

Arecentmajorachievementinthetheoryofconstantmeancurvature(CMC)sur-facesinRiemannian3-manifoldshasbeenthediscoverybyAbreschandRosenberg[AbRo1,AbRo2]ofaholomorphicquadraticdi erentialforeveryCMCsurfaceinanyhomogeneous3-manifoldwitha4-dimensionalisometrygroup.Themostremarkableconsequenceofthisconstructionistheclassi cationgivenin[AbRo1,AbRo2]oftheimmersedCMCspheresinthesehomogeneousspacesasembeddedrotationalones,thussolvinganopenproblemproposedin[HsHs].

OurpurposeinthispaperistosolvebymeansoftheAbresch-Rosenbergdi erentialanotherpopularopenprobleminthetheoryofCMCsurfacesinhomogeneousspaces.

We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosen

Namely,theBernsteinproblemintheRiemannianHeisenberg3-spaceNil3,i.e.theclassi cationoftheentireminimalverticalgraphsinNil3.

Letus xsometerminology.Byusingexponentialcoordinatesanduptore-scaling,weshallregardtheHeisenbergspaceNil3asR3endowedwiththeRiemannianmetric

2 1dσ2:=dx2+dy2+dz+(ydx xdy).2

ThisRiemannianmetricisleft-invariantfortheLiegroupstructureofHeisenbergspace.InthiswayNil3becomesoneofthe vesimply-connectedcanonicalhomogeneous3-manifoldswitha4-dimensionalisometrygroup.

A(vertical)graphz=f(x,y)inNil3issaidtobeminimalifitsmeancurvaturevanishesidentically,andentireifitisde nedforevery(x,y)∈R2.TheBernsteinprobleminNil3asksfortheclassi cationofallentireverticalminimalgraphsofNil3.Equivalently,wewishto ndalltheC2solutionsgloballyde nedonR2ofthequasilinearellipticPDE

(1+β2)fxx 2αβfxy+(1+α2)fyy=0,(1.1)

whereα:=fx+y/2andβ:=fy x/2(see[ADR,IKOS]forinstance).Thesimplestentiresolutionsto(1.1)arethelinearones,whichgiverisetoentireminimalgraphsinNil3thatarecalledhorizontalumbrellas[AbRo2].Inthismodeltheyarejustnon-verticalplanes,buttheyarenottotallygeodesic.Othersimplesolutionisf(x,y)=xy/2,andmoregenerally,the1-parameterfamily[FMP]

xy cy1+y2+logy+1+y2,c∈R.(1.2)f(x,y)=2

Theresultingentireminimalgraphsarecalledsaddle-typeminimalexamples[AbRo2].Thereareotherexplicitentiresolutionsto(1.1)foliatedbyEuclideanstraightlines(see

[Dan2]),whatsuggeststhattheclassofentireminimalgraphsinNil3isquitelarge.TheBernsteinprobleminNil3isausualdiscussiontopicamongpeopleworkingonCMCsurfacesinhomogeneousspaces,andappearsexplicitlyasanopenprobleminseveralworkslike[FMP,AbRo2,Dan2,IKOS,MMP].LetusalsopointoutthattheBernsteinprobleminthesub-RiemannianHeisenbergspaceH1hasbeenintenselystudiedinthelastfewyears(see[BSV,CHMY,GaPa,RiRo]andreferencestherein).InthisworkweshowthatthemodulispaceofentireminimalgraphsinNil3canbeparametrizedintermsoftheirAbresch-Rosenbergdi erentials.Speci cally,weprove:Theorem1(MainTheorem)LetQdz2beaholomorphicquadraticdi erentialinΣ≡CorDwhichisnotidenticallyzeroifΣ≡C.Thenthereexistsa2-parameterfamilyof(genericallynon-congruent)entireminimalgraphsinNil3whoseAbresch-Rosenbergdi erentialisQdz2.

Conversely,anyentireminimalgraphinNil3isoneoftheaboveexamples.

Itisworthmentioningthatiftwoholomorphicquadraticdi erentialsonCorDonlydi erbyaglobalconformalchangeofparameter,thenthefamiliesofminimalgraphs

We classify the entire minimal vertical graphs in the 3 dimensional Heisenberg group Nil endowed with a Riemannian left-invariant metric. This classification, which provides a solution to the Bernstein problem in Nil, is given in terms of the Abresch-Rosen

thattheyinducearethesame(thecorrespondingimmersionsdi erbythisconformalglobalchangeofparameter).

WealsoremarkthattheconformallyimmersedminimalsurfacesX:Σ→Nil3withΣ≡CandAbresch-Rosenbergdi erentialQdz2=cdz2,c=0,areexactlythesaddle-typeminimalexamples(1.2),seeSection2.Inthatcasesomeoftheelementsofthe2-parameterfamilyaremutuallycongruent.Ontheotherhand,theminimalsurfacesinNil3withvanishingAbresch-Rosenbergdi erentialonΣ=Caretheverticalplanes,whicharenotentiregraphs.

ThesolutiontotheBernsteinproblemexposedinTheorem1issharp,inthefollowingsense.First,ittellsthatthespaceofentiresolutionsto(1.1)isextremelylarge.So,asthesesolutionswillnotbeexplicitingeneral,themostreasonableanswertotheBernsteinprobleminNil3thatonecanlookforisadescriptionofthemodulispaceofentireminimalgraphsintermsofsomealreadyknownclassofobjects.Withthis,Theorem1tellsthatthespaceofentireminimalgraphsinNil3canbedescribedbythespaceofholomorphicquadraticdi erentialsonCorD,whichisawellknownclass.TheproofofTheorem1isbasedonfourpillars.First,theAbresch-Rosenbergho-lomorphicdi erential,whoseroleissubstantialsinceitistheclassifyingobjectinthesolutiontotheBernsteinproblem.Second,theconstructionbytheauthorsin[FeMi1]ofaharmonicGaussmapintothehyperbolicplaneH2forCMCsurfaceswithH=1/2inH2×R,andtheglobalimplicationsofthisconstruction.Actually,thepresen …… 此处隐藏：33380字，全部文档内容请下载后查看。喜欢就下载吧 ……