Teichmuller theory and handle addition for minimal

arXiv:math/9806089v2[math.DG]10Jun2005AnnalsofMathematics,156(2002),713–795Teichm¨ullertheoryandhandleadditionforminimalsurfacesByMatthiasWeberandMichaelWolf*1.Introduction1.1.Thesurfaces1.2.Theproof2.Background,notationandasketchoftheargument2.1.Minimalsurfaces2.2.Teichm¨ullertheory2.3.Abriefoutlineoftheproof3.Thegeometryoforthodisks3.1.Orthodisks3.2.FromorthodiskstoRiemannsurfaces3.3.FromorthodiskstoWeierstrassdata3.4.GeometricsignificanceoftheformalWeierstrassdata3.5.Examplesofsimpleorthodisks3.6.OrthodisksfortheCostatowers3.7.Moreorthodisksbydrillingholes4.Thespaceoforthodisks4.1.Introduction4.2.GeometriccoordinatesfortheDH1,1-surface4.3.HeightfunctionfortheDH1,1-surface4.4.GeometriccoordinatesfortheDHm,n-surfaces4.5.HeightfunctionsfortheDHm,n-surfaces4.6.PropernessoftheheightfunctionsfortheDHm,n-surfaces4.7.Amonodromyargument5.Thegradientflow5.1.Overallstrategy5.2.DeformationsofDH1,15.3.Infinitesimalpushes6.Regeneration7.NonexistenceoftheDHm,n-surfaceswithnm*ThesecondauthorwaspartiallysupportedbyNSFgrantnumberDMS-9626565andtheSFB.714MATTHIASWEBERANDMICHAELWOLF8.Extensionsandgeneralizations8.1.Highersymmetry8.2.Deformationswithmorecatenoidalends8.3.EmbeddednessaspectsofDHm,n9.References1.IntroductionInthispaper,wedevelopTeichm¨ullertheoreticalmethodstoconstructnewminimalsurfacesinE3byaddinghandlesandplanarendstoexistingminimalsurfacesinE3.Weexhibitthismethodonaninterestingclassofminimalsurfaceswhicharelikelytobeembedded,andhavealowdegreeGaußmapfortheirgenus.Inparticular,weexhibitatwo-parameterfamilyofcompleteminimalsurfacesintheEuclideanthree-spaceE3;thesesurfacesareembedded(atleast)outsideacompactsetandareindexed(roughly)bythenumberofendstheyhaveandtheirgenus.Theyhaveatmosteightself-symmetriesdespitebeingofarbitrarilylargegenus,andareinterestingforanumberofreasons.Moreover,ourmethodsalsoextendtoprovethatsomenaturalcandidateclassesofsurfacescannotberealizedasminimalsurfacesinE3.Asaresultofbothaspectsofthiswork,weobtainaclassificationofafamilyofsurfacesaseitherrealizableorunrealizableasminimalsurfaces.Thispaperisacontinuationofthestudyweinitiatedin[WW];inastrongsenseitisanextensionofthatpaper,astheessentialorganizationoftheproof,togetherwithmanydetails,havebeenretained.Indeed,partofourgoalinwritingthispaperwasademonstrationoftherobustnessofthemethodsof[WW],inthathereweproduceminimalsurfacesofaverydifferentcharacterthanthoseproducedin[WW],yettheproofchangesonlyinafewquitetechnicalways.(Inparticular,thepresentproofhandlesthepreviouscaseofChen-Gackstattersurfacesofhighgenusasanelementarycase.)Indeedintheinterveningyearsbetweenourinitialpreparationofthismanuscriptanditsfinalrevisionforpublication,thismethodhasbeenappliedtoproduceotherfamiliesofsurfacesofsubstantivelydifferentcharacteristicsortoprovetheirnonexistence([WW2],[MW]).1.1.Thesurfaces.HoffmanandMeeks(see[Ho-Me])haveconjecturedthatanycompleteembeddedminimalsurfaceinspacehasgenusatleastr−2,whererdenotesthenumberofendsofthesurface.Inthispaper,wepro-videsignificantevidenceforthisconjectureinthesituationwherethesurfaceshaveeightsymmetries.Thisisanimportantcasefortworeasons:first,itispresentlyunknownwhetherthereareanycompleteembeddedminimalsur-TEICHM¨ULLERTHEORY715faceswhichhavenosymmetries1,andsecond,thereareveryfewfamiliesofexamplesknownwheretherearemorethanfourends.(Indeed,theonlysuchconstructionsavailablearefromtherecentworkofKapouleas[Kap],wherethegenusisbothhighandinestimable.)Inparticular,weconsidertwofamiliesofsurfaces,withthefirstincludedinthesecond.ThefirstcaseconsistsofsurfacesCTgwhichgeneralizeCosta’sexample[Cos].WeproveTheoremA.Foralloddgenerag,thereisacompleteminimalsur-faceCTg⊂E3whichisembeddedoutsideacompactsurfacewithboundaryofgenusg,withgparallel(horizontal)planarendsandtwocatenoidends.ThesymmetrygroupofCTgisgeneratedbyreflectivesymmetriesaboutapairoforthogonalverticalplanesandarotationalsymmetryaboutahorizontalline.Thesesurfacesrepresenttheborderlinecasefortheconjecture.(Theevengenuscaseshavesubstantiallydifferentcombinatorics,andrequireadifferenttreatment.)ConsidertheRiemannsurfaceunderlyingsuchanexample:itisafundamentaltheoremofOsserman[Oss1]thatsuchasurfaceisconformallyacompactsurfaceofgenusg,puncturedatpointscorrespondingtotheends.LetZdenotetheverticalcoordinateofsuchaminimalsurface:clearly,Ziscriticalatthegpointscorrespondingtotheplanarends,thetwopointscorrespondingtothecatenoidends,andginteriorpointswherethetworeflectiveplanesmeetthesurface.Wegeneralizethesesurfacesasfollows,imaginingDrillingadditionalHolestoobtainsurfacesDHm,n(see§3.7).TheoremB.(i)Foreverypairofintegersn≥m≥1,thereexistsacompleteminimalsurfaceDHm,n⊂E3ofgenusm+n+1whichisembeddedoutsideacompactsetwiththefollowingproperties:ithas2n+1verticalnormals,2m+1planarends,andtwocatenoidends.ThesymmetrygroupisasinTheoremA.(ii)Fornm,thereisnocompleteminimalsurfacewiththosesymme-triesofthetypeDHm,n(and2n+1verticalnormals,2m+1planarends,andtwocatenoidends).Inthesecondstatement,thesurfacesforwhichweprovenonexistenceareinpreciseanalogywiththesurfacesforwhichweproveexistence.Therearemanyconfigurat