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1、BULLETINOFTHEAMERICANMATHEMATICALSOCIETYVolume83,Number5,September1977THEEXISTENCEOFMINIMALIMMERSIONSOFTWO-SPHERESBYJ.SACKSANDK.UHLENBECK1CommunicatedbyS.S.Chern,February18,1977Inthisarticleweannounceaseriesofresultsontheexistenceofharmonicmapsfromsurfac
2、estoRiemannianmanifoldsand,ascorollariesoftheseresults,obtaintheoremsontheexistenceofminimalimmersionsof2-spheres.LetNbeacompactconnectedRiemannianmanifoldand,forconvenience,assumethatNisisometricallyimbeddedinRkforsomesufficientlylargek.LetMbeaclosedRie
3、mannsurfacewithanymetriccompatiblewithitsconformaistructure.AmapsGL(M,Rk)OC°(M,N)iscalledharmonicifitisanextre•malmapoftheenergyintegralE(s)=JM'*
4、2*%=JMtraceKx)dtxMwhereki(x)=2**®**(*)Grçcw)®^(w)-/=iHarmonicmapssatisfyanEuler-LagrangeequationAs+A(s)(dsf
5、ds)=0inaweaksense,whereAisthesecondfundamentalformoftheimbeddingNCRfc.ItthenfollowsfromregularitytheoremsthatharmonicmapsareC°°.Ifsisharmonicandaconformaiimmersion,itisalsoanextremalfortheareainte•gral.ProvingtheexistenceofharmonicmapsofMintoNbydirectmet
6、hodsfromglobalanalysissuchasMorsetheoryorLjusternik-Schnirelmantheoryap•pliedtoEdefinedonsomefunctionspacemanifoldisdifficult,becauseEisin•variantundertheconformaigroupofMyandtheextremalmapsofEformanon-compactsetwhenM=S2.Inparticular,Edoesnotsatisfycondi
7、tionCofPalais-Smale.However,fora>1,aslightlydifferentintegral,AMS(MOS)subjectclassifications(1970).Primary53A10,58E05.1ResearchsupportedbyNSFgrantsMCS76-06319andMCS76-07541,Copyright©1977,AmericanMathematicalSociety10331034J.SACKSANDK.UHLENBECKforsGL2a(M
8、,Rk)OC°(M,TV)=L2ot(M,TV),isC2andsatisfiesthePalais-SmaleconditionCinacompleteFinslermetriconL^M,TV).IfwenormalizetheareaofMtobe1then,asa—»1,Ea(s)—•E(s)+1.ByexaminingtheconvergenceofasequencesaofcriticalmapsofEaasa—>1,variousresultsontheexistenceofharmoni
9、cmapsareobtained.MAINCONVERGENCETHEOREM.Letsa(/)beasequenceofcriticalmapsofE^ya(z)>1,lim^^a(z)=1.Thenthereexistasubsequencei,aharmon•icmaps:M—>Nandafinitenumberofpoints{xt,...,xt}suchthatsa^—•sinCX(M-{xj,...,xt},TV).Moreov
