Willmore two-spheres in the four-sphere

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1、TRANSACTIONSOFTHEAMERICANMATHEMATICALSOCIETYVolume352,Number10,Pages4469{4486S0002-9947(00)02571-XArticleelectronicallypublishedonJune13,2000WILLMORETWO-SPHERESINTHEFOUR-SPHERESEBASTIANMONTIELAbstract.GenuszeroWillmoresurfacesimmersedinthethree-sphereS3correspondviathe

2、stereographicprojectiontominimalsurfacesinEuclideanthree-spacewithnitetotalcurvatureandembeddedplanarends.ThecriticalvaluesoftheWillmorefunctionalare4k,wherek2N,withk6=2;3;5;7.Whentheambientspaceisthefour-sphereS4,theregularhomotopyclassofimmersionsofthetwo-sphereS2i

3、sdeterminedbytheself-intersectionnumberq2Z;hereweshallprovethatthepossiblecriticalvaluesare4(jqj+k+1),wherek2N.Moreover,ifk=0,thecorrespondingimmersion,oritsantipo-dal,isobtained,viathetwistorPenrosebrationP3!S4,fromarationalcurveinP3and,ifk6=0,viastereographicproject

4、ion,fromaminimalsurfaceinR4withnitetotalcurvatureandembeddedplanarends.AnimmersionliesinbothfamilieswhentherationalcurveiscontainedinsomeP2P3or(equivalently)whentheminimalsurfaceofR4iscomplexwithrespecttoasuitablecomplexstructureofR4.1.IntroductionAlthoughthestudyofco

5、nformalinvariantsofsubmanifoldsreceivedsomeatten-tionintheclassicalgeometryofsurfaces,andlaterfromBlaschke[Bl]andothers,itremainedalmostunknownuntil1965.Inthatyear,Willmore[Wi1]proposedthestudyofaglobalconformalinvariantforcompactsurfacesinEuclideanspaces,oftencalledthe

6、Willmorefunctional,thatassociatestoanimmersion:!Rnfromacompactsurface,theconformallyinvariantintegralZ2W()=jHjdA;whereHdenotesthemeancurvaturevectoroftheimmersionanddAistheareameasureoftheinducedmetric.OneeasilyobtainsthattheabsoluteminimumforWis4,whichisattainedo

7、nlybytheumbilicalspheres.Whenthegenusofisone(thatis,whenisatorus)theminimumvalueofWwasconjecturedbyWillmoretobe22andtobereachedonlyby(stereographicprojectionof)theCliordtorus.Thisconjectureremainsopen,althoughtherearesomegoodapproacheswhichhavebeensourcesofnewideas,

8、suchas[Wi2],[LY],[K1],[K2],[MRo].Veryrecently,A.Roshassolvedtheconjectureincasethetorusisinvariantunderanantip