THE COMPLEX MONGE-AMP ERE EQUATION FOR COMPLEX HOM

THECOMPLEXMONGE-AMPEREEQUATIONFORCOMPLEXHOMOGENEOUSFUNCTIONSINCnRafalCzy_zIMUJpreprint2000/19Abstract.WeprovesomeexistenceresultsforthecomplexMonge-Ampereequa-tioninCn:(ddcu)n=gdinthecertainclassofhomogeneousfunctionsinCn,i.e.weshowthatforsomenonegativecomplexhomogeneousfunctionsgthereex-istsplurisubharmoniccomplexhomogeneousfunctionu-asolutionofthecomplexMonge-Ampereequation.0.Introduction.Inthispaperweconsiderthefollowingproblem:forwhichnonnegativecomplexhomogeneousfunctionsginCnthereexistsacomplexho-mogeneousplurisubharmonicfunctionuinCnsolvingthecomplexMonge-Ampereequation:(0.1)(ddcu)n=gd?whereddenotestheLebesguemeasureinCn.TheproblemoftheexistenceofglobalsolutionsofthecomplexMonge-AmpereequationsinCnhasbeentreatedonlyinseveralcases.In[K1]KolodziejshowedsomesucientconditionswhichguaranteethatanitemeasureinCnadmitsasolutionoftheequation(ddcu)n=dintheclassL+.Uniqueness,uptoanadditiveconstant,inthiscasewasprovedbyBedfordandTaylorin[BT2].In[J]JeuneprovedthataperturbationoftheLebesguemeasureinCnbyasmoothfunctionwhich,togetherwithallitsderevatives,tendsto0fastenoughatinnity,admitsasmoothsolutionofthecomplexMonge-Ampereequation.Monn[M]2000MathematicsSubjectClassication.32U15;32W20.Keywordsandphrases.plurisubharmonicfunctions,complexMonge-Ampereoperator.TypesetbyAMS-TEX1provedtheexistenceofthesolutionofthecomplexMonge-AmpereequationintheclassofradialfunctionsinCn,i.e.thereexistsaradial,entireplurisubharmonicfunctionasolutionof(0:1)fornonnegativeradialfunctionginCn.Kolodziejin[K3]showedthatforgiventwoentirelocallyboundedplurisubharmonicfunctionsvandwsatisfyingwv,(ddcv)n(ddcw)nandlimjzj!1(v(z)w(z))=0,onecansolvetheMonge-Ampereequationforanymeasuresuchthat:(ddcv)nd(ddcw)n:Furthermore,thesolutionuisuniqueamongfunctionssatisfyingwuv.InthispaperweprovetheexistenceofasolutionofthecomplexMonge-AmpereequationsforacertainclassofhomogeneousfunctionsinCn.Inthecomplexplaneeverycomplexhomogeneousfunctionisoftheformcjzjandasimpleobservationshowsthatthefunctionu(z)=2(+2)2jzj+2isthesolutionofthefollowingequationddcu=jzjd,where0.Forthisreasoninthispaperwealwaysassumethatn2.Intherstsectionweprovethatforanynonnegative,smooth(outsidetheori-gin),complexhomogeneousfunctiongoforderofhomogenityn(t2),where0t1n1thereexistsasmooth(outsidetheorigin)solutionuoftheequa-tion(0:1).Wealsoestablishaconnection,whichplaysthemostimportantroleinprovingthetheoremmentionedbefore,betweentheexistenceofasolutionoftheequationofcomplexMonge-AmperetypeinthecomplexprojectivespacePn1andtheexistenceofasolutionoftheMonge-Ampereequationintheclassofho-mogeneousfunctionsinCn.Namely,weshowthatasolutioninPn1allowsustoconstructacorrespondingsolutioninCn.TheexistenceofasolutionforsomeequationsoftheMonge-AmperetypeinaspecialcompactKahlermanifoldswasprovedbyBenAbdesselemin[BA].Attheendofthissectionweprovethat,undertheadditionalassumptiononthefunctiong,itispossibletosolvetheequation(0:1)withaweakerrestrictionontheorderofhomogenity.Inthesecondsectionweshallprovetheexistenceofasolutionof(0:1)forlocallyboundeddatag.ToprovethisweneedageneralizationofTian’stheoremfrom[T].TiansolvedthefollowingequationoncompactKahlermanifolds(M;!)withapositiverstChernclass:(0.2)(ddc’+!)n=et’+f!n;whereddc’+!0,fisC1smoothand0t1.Fort=1thisequationprovidestheexistenceofaKahlermetriconM.Weprovethattheequation(0:2)hasasolutionforaboundedfunctionfand0t(M),where(M)isaglobalholomorphicinvariantonMintroducedbyTian.1.Existenceofasolutionforsmoothdata.Denition1.1.Wesaythatafunctionf:Cn!Riscomplexhomogeneousoforderifitsatisesthefollowingcondition:f(z)=jjf(z);forall2Candz2Cn:2WeshalldenotebyHC(Cn)thespaceofallcomplexhomogeneousfunctionsoforderinCn.Sometimesinthispaperweshallcallforacomplexhomogeneousfunctionsimplyhomogeneousfunction.WeshalldenotebyL+thesetofallentireplurisubharmonicfunctionsuinCnforwhichthereexistconstantsC1andC2(dependingonu)suchthatC1+log(1+jzj)u(z)C2+log(1+jzj):WeshalldenotebyH+thesetofallentireplurisubharmonicfunctionsuinCnwhichsatisfyu(z)=logjj+u(z)forall2Candz2Cn:TherearewellknownfactsthatH+L+andZCn(ddcu)n=(2)nforallu2L+:NowwerecallthatforafunctionfromH+itismuchmoreknownaboutit’sMonge-Amperemeasure.Proposition1.2.Ifu2H+then(ddcu)n=(2)n0,where0isDiracmeasureatzero.Proof.Firstweshallproveourpropositionforsmoothfunctions.Supposethatu2H+\C1(Cnnf0g).Thendierentiatingby@2@zj@zktheequationu(z)=logjj+u(z)forz6=0weobtainujk(z)=jj2ujk(z)for6=0andz6=0;whereujk(z):=@2u@zj@zk(z).Takingz=zjzjand=jzjwehaveujk(z)=jzj2ujk(zjzj):RecallthatifaplurisubharmonicfunctionuisC2smooththen(ddcu)n=4nn!det@2u@zj@zkd:UsingthisequationwecanobtainthatforanyR0ZB(0;R)nf0g(ddcu)n=n!4nZB(0;R)nf0gdet(ujk(z))d=n!4nZB(0;R)nf0gjzj2ndet(ujk(zjzj))d=lim!0n!4nZRr1drZ@B(0;1)det(ujk(zjzj))d=(0;ifR@B(0;1)det(ujk(zjzj))d=0+1;ifR@B(0;1)det(ujk(zjzj))d6=0:3HoweverweknowthatRCn(ddcu)n=(2)n+1,sodet(ujk(z))mustvanishon@B(0;1).Fromthatweconcludethat(ddcu)n=0inCnnf0g.Thelaststatementimpliesthatmeasure(ddcu)nissupportedattheo