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Variational Problems in Geometry.pdf

Variational Problems in Geometry

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简介:本文档为《Variational Problems in Geometrypdf》,可适用于高等教育领域,主题内容包含TranslationsofMATHEMATICALMONOGRAPHSVolumeVariationalProblemsinGeometrySe'符等。

TranslationsofMATHEMATICALMONOGRAPHSVolumeVariationalProblemsinGeometrySe''NishikawakAmericanMathematicalsocietyVariationalProblemsinGeometryTranslationsofMATHEMATICALMONOGRAPHSVolumeVariationalProblemsinGeometrySeikiNishikawaTranslatedbyKinetsuAbeAmerl=nMathematicalocistyProvidenceRhodeIslandEditorialBoardShoshichiKobayashi(Chair)MasamichiTakesakifol:FLM**KIKAGAKUTEKIHENBUNMONDAIbySeikiNishikawaOriginallypublishedinJapanesebyIwanamiShoten,Publishers,Tokyo,TanslatedfromtheJapanesebyKinetsuAbeMathematicsSubjectCiasaicationPrimary,C,C,E,JLibraryofCongressCatalogingInPublicationDataNishikawa,SeikiKikigakutekihenbunmondaiEnglishVariationalproblemsingeometrySeildNishikawatranslatedbyKinetsuAbepcm(Translationsofmathematicalmonographs,ISSNv)(Iwanamiseriesinmodemmathematics)IncludesbibliographicalreferencesandindexISBN(acidfreepaper)HarmonicmapsVariationalinequalities(Mathematics)RiemannianmanifoldsITitleItSeriesIIISeries:IwanamiseriesinmodernmathematicsQAN'dcbytheAmericanMathematicalSocietyAllrightsreservedTheAmericanMathematicalSocietyretainsallrightsexceptthosegrantedtotheUnitedStatesGovernmentPrintedintheUnitedStatesofAmericaThepaperusedinthisbookisacidfreeandfallswithintheguidelinesestablishedtoensurepermanenceanddurabilityInformationoncopyingandreprintingcanbefoundinthebackofthisvolumeVisittheAMShomepageatURL:http:vvvamsorgContentsPrefacetotheEnglishEditionPrefaceixOutlinesandObjectivesoftheTheoryChapterAreLengthofCurvesandGeodesicsArelengthandenergyofcurvesEuler'sequationConnectionsandcovariantdifferentiationGeodesicsMinimallengthpropertyofgeodesicsSummaryExercisesChapterFirstandSecondVariationFormulasThefirstvariationformulaCurvaturetensorThesecondvariationformulaExistenceofminimalgeodesicsApplicationstoRiemanniangeometrySummaryExercisesChapterEnergyofMapsandHarmonicMapsEnergyofmapsTensionfieldsThefirstvariationformulaHarmonicmapsThesecondvariationformulaSummaryExercisevviCONTENTSChapterExistenceofHarmonicMapsTheheatflowmethodExistenceoflocaltimedependentsolutionsExistenceofglobaltimedependentsolutionsExistenceanduniquenessofharmonicmapsApplicationstoRiemanniangeometrySummaryExercisesAppendixAFundamentalsoftheTheoryofManifoldsandFunctionalAnalysisAIFundamentalsofmanifoldsAFundamentalsoffunctionalanalysisProspectsforContemporaryMathematicsSolutionstoExerciseProblemsBibliographyIndexPrefacetotheEnglishEditionThisbook,publishedoriginallyinJapanese,isanoutgrowthoflecturesgivenatTohokuUniversityandattheSummerGraduateProgramoftheInstituteforMathematicsandItsApplications,UniversityofMinnesotaIntheselectures,throughadiscussiononvariationalproblemsofthelengthandenergyofcurvesandtheenergyofmaps,Iintendedtoguidetheaudiencetothethresholdofthefieldofgeometricvariationalproblems,thatis,thestudyofnonlinearproblemsarisingingeometryandtopologyfromthepointofviewofglobalanalysisItismypleasureandprivilegetoexpressmydeepestgratitudetoProfessorKinetsuAbewhogenerouslydevotedconsiderabletimeandefforttothetranslationIwouldalsoliketotakethisopportunitytoexpressmydeepappreciationtoProfessorPhillipeTondeurwhoinvitedmetojointheSummerGraduateProgram,andtomyfriendAndrejTreibergsformakinghisnotesavailabletotheorganizationofthelastchapterSeikiNishikawaAprilvHPrefaceItissaidthattechniquesforsurveyingweredevelopedfromtheneedtorestorelandsafterfrequentfloodsoftheNileRiverinancientEgyptGeometryistheareaofmathematicswhosenameoriginatesfromthismethodofsurveyingnamely,"tomeasurelands"(geo=lands,metry=measure)Assuch,itisanancientpracticetostudyfiguresfromtheviewofpracticalapplicationsItisalsosaidthattheancientGreeksalreadyknewofthemethodofindirectsurveyingusingthecongruenceconditionsoftrianglesAminimallengthcurvejoiningtwopointsinasurfaceiscalledageodesicOnemaytracetheoriginoftheproblemoffindinggeodesicsbacktothebirthofcalculusManycontemporarymathematicalproblems,asinthecaseofgeodesics,maybeformulatedasvariationlproblemsinsurfacesorinthemoregeneralizedformofmanifoldsOnemaycharacterizethegeometricvariationalproblemsasafieldofmathematicsthatstudiestheglobalaspectsofvariationalproblemsrelevantinthegeometryandtoplogyofmanifoldsForexample,theproblemoffindingasurfaceofminimalareaspanningagivenframeofwireoriginallyappearedasamathematicalmodelforsoapfilmsIthasalsobeenactivelyinvestigatedasageometricvariationalproblemWithrecentdevelopmentsincomputergraphics,totallynewaspectsofthestudyonthesubjecthavebeguntoemergeThisbookisintendedtobeanintroductiontosomeofthefundamentalquestionsandresultsingeometricvariationalproblems,studyingthevariationalproblemsonthelengthofcurvesandtheenergyofmapsThefirsttwochaptersapproachvariationalproblemsoflengthandenergyofcurvesinRiemannianmanifoldswithanindepthdiscussionoftheexistenceandpropertiesofgeodesicsviewedasthesolutiontovariationalproblemsInaddition,aspecialemphasisisixxPREFACEplacedonthefactthattheconceptsofconnectionandcovariantdifferentiationarenaturallyinducedfromthefirstvariationformulaofthisvariationalproblem,andthatthenotionofcurvatureisobtainedfromthesecondvariationalformulaThelasttwochapterstreatthevariationalproblemontheenergyofmapsbetweentwoRiemannianmanifoldsanditssolutions,namelyharmonicmapsTheconceptofharmonicmapsincludesgeodesicsandminimalsubmanifoldsasexamplesItsexistenceandpropertieshavesuccessfullybeenappliedtovariousproblemsingeometryandtopologyThisbooktakesuptheexistencetheoremofEellsSampson,whichisconsideredtobethemostfundamentalamongexistencetheoremsforharmonicmapsTheproofusestheinversefunctiontheoremforBanachspacesItispresentedtobeasselfcontainedaspossibleforeasyreadingEachchapterofthisbookmaybereadindependentlywithminimalpreparationforcovariantdifferentiationandcurvatureonmanifoldsThefirsttwochapters,throughthediscussionofconnectionsandcovariantdifferentiation,aredesignedtoprovidethereaderwithabasicknowledgeofRiemannianmanifoldsAsprerequisitesforreadingthisbook,theauthorassumesafewelementaryfactsinthetheoryofmanifoldsandfunctionalanalysisTheyareincludedintheformofappendicesattheendofthebookDetailsinfunctionalanalysismaybeskippedThereader,however,isencouragedtodotheexerciseproblemsattheendofeachchapterbyhimselforherselffirstThesolutionsmaybeconsultedifnecessary,sincemanyoftheexerciseproblemscomplementthecontentsofthebookThisbookisanoutgrowthoflecturesdeliveredatTohokuUniversityandtheSummerGraduateProgramsheldatTheInstituteforMathematicsandItsApplications,UniversityofMinnesotaThefirsthalfofthebookaimsatajuniorandseniorlevel,andthelasthalfatafirstandsecondyeargraduatelevelEachhalfroughlyconsistsoftheamountoftopicsthatmaybecoveredinonesemesterIntheactuallectures,theauthoralsodiscussestheharmonicmapsbetweenRiemannsurfacesThisportionisnotincludedinthisbookduetothelimitedspaceThereaderwhoisinterestedinthestudyofharmonicmapsisadvisedtofirststudytheharmonicmapsbetweenRiemannsurfacesItwouldbethisauthor'swishaswellaspleasureifthisbookcouldinterestmanyreadersinvariationalproblemsingeometryPREFACExiLastbutnotleast,theauthorexpresseshissinceregratitudetotheeditorialstaffofIwanarniShotenfortheirvaluablehelpinthepublicationofthisbookSeikiNishikawaDecemberOutlinesandObjectivesoftheTheoryAmonggeometricvariationalproblems,theextremevalueproblemregardingthelengthofcurvesisasoldasthoseincalculusChapterofthisbookisdevotedtodiscussionsofvariationalproblemsofcurvesinmanifoldsAsiswellknown,thelengthofacurvejoiningtwopointsinaplaneisgivenbyintegratingthemagnitudeoftangentvectorsSimilarly,onecandefinethelengthandenergyforcurvesinmoregeneralRiemannianmanifoldsbymeasuringthemagnitudeofthetangentvectorsusingRiemannianmetricsInChapter,Euler'sequationiscalculatedItcharacterizesthecriticalpointsofthelengthandenergyofcurveswhentheyareconsideredasfunctionalsdefinedinthespaceofcurvesConsequently,theequationofgeodesicsisobtainedTheconceptsofconnectionsandcovariantdifferentiationarenaturallyinducedfromtheequationofgeodesicsinamanifoldCovariantdifferentiation,anessentialtoolforstudyingvariationalproblemsinmanifolds,isanoperationthatdefinesthederivativeofavectorfieldbyavectorfieldinamanifoldThemostfundamentalconnection,calledtheLeviCivitaconnection,isuniquelydeterminedinamanifoldequippedwithaRiemannianmetric,ie,aRiemannianmanifoldThenotionofparalleltransportisinducedfromthisconnectionThediscoveryofthenotionofparalleltransportinRiemannianmanifolds()andEinstein'suseofgeometrybasedonafourdimensionalindefinitemetricforhisgeneralrelativity()greatlyadvancedthestudyofRiemanniangeometryGeodesicsinRiemannianmanifoldscorrespondtostraightlinesintheplaneandtheyarelocallycharacterizedasthecurvesofminimallengthbetweenpointsOnecanconstructaspeciallocalcoordinatesystem,calledanormalcoordinatesystem,usingtheseminimalgeodesicsabouteachpointinaRiemannianmanifoldParalleltransportandnormalcoordinatesystemsarethemostbasictoolsinXiixivOUTLINESANDOBJECTIVESOFTHETHEORYcomparingthegeometryofaRiemannianmanifoldwiththegeometryofamodelspace(forexample,Euclideanspace)InChapter,usingcovariantdifferentiation,thefirstvariationformula(Euler'sequation)forthevariationalproblemregardingtheenergyofcurvesinRiemannianmanifoldsiscomputedinthegeneralcasewheretheimageofacurveisnotalwayscontainedinalocalcoordinateneighborhoodThesecondvariationformulaissubsequentlycomputedJustasthenotionofconnectionsisderivedfromthefirstvariationformula,itisseenthatthesecondvariationformulapossessesanintimaterelationshiptothenotionofcurvatureinRiemannianmanifoldsInotherwords,thenotionsofcurvaturetensorandthecurvatureofaRiemannianmanifoldarenaturallyinducedfromthesecondvariationformulafortheenergyofcurvesGiventwopointsinaRiemannianmanifold,thedistancebetweenthesetwopointsisgivenbytheleastupperboundofthelengthsofpiecewisesmoothcurvesconnectingthemWhetheraRiemannianmanifoldbecomesacompletemetricspacewithrespecttothisdistanceisanimportantquestionItwasrelativelyrecently()thatHopfRinowgavenecessaryandsufficientconditionsforthequestionTheresultsbyHopfandRinowaresignificantnotonlyinmakingthenotionofcompletenesssuccinct,butalsoinshowingthatthiscompletenessistheconditionthatguaranteestheexistenceofaminimalgeodesicjoiningtwogivenpointsAsstatedabove,thesecondvariationformulafortheenergyofcurvesiscloselyrelatedtothecurvatureofRiemannianmanifoldsUsingthis,onecanstudytheeffectsofthecurvatureofaRiemannianmanifoldonitstopologicalstructureMyers'theoremandSynge'stheoremarediscussedastypicalexamplesofsuchapplicationsTheformerstatesthatthefundamentalgroupofacompactandconnected,Riemannianmanifoldofpositivecurvatureisafinitegroup,andthelatterstatesthatanevendimensionalcompact,connectedandorientableRiemannianmanifoldofpositivecurvatureissimplyconnectedResearchonRiemannianmanifoldsusingexistenceandpropertiesofgeodesicsisbeingactivelypursuedInChapter,harmonicmapsandtheenergyofmapsarediscussedTheygeneralizethevariationalproblemoftheenergyofcurvesinRiemannianmanifoldsNamely,afunctionalcalledtheenergyofmapsisdefinedinthemappingspaceconsistingofsmoothmapsbetweenRiemannianmanifolds,andharmonicmapsgivenasitsOUTLINESANDOBJECTIVESOFTHETHEORYxvcriticalpointsareinvestigatedTheenergyofmapsisanaturalgeneralizationoftheenergyofcurvesExamplesofharmonicmapsappearinvariousaspectsofdifferentialgeometryHarmonicfunctions,geodesics,minimalsubmanifolds,isometricmaps,andholomorphicmapsareafewtypicalexamplesThefirstvariationformula,whichcharacterizesthecriticalpointsoftheenergyfunctional,canbeobtainedbyessentiallythesameapproachasinthecaseofgeodesicsHowever,thecomputationsbecomeunnecessarilycomplicatedandonlyyieldresultsofalocalnaturewithoutuseofthecovariantdifferentiationthatisnaturallyinducedfromtheLeviCivitaconnectionofRiemannianmanifoldsToalleviatethesedifficulties,itisdesignedinthischaptertoderive,throughdiscoveriesintheprocess,thecomputationalrulesforthecovariantdifferentiationthatisinducedfromtheLeviCivitaconnectionintangentbundlesandtheirtensorproductsoverRiemannianmanifoldsThisroutemaynotbethemostdirectone,buttheauthorbelievesthatitismoreeffectiveinfamiliarizingthereaderwiththedefinitionandtherulesofcomputationsforcovariantdifferentiationthantheaxiomaticapproachAtfirst,thereadermayfeeluneasy,especiallyabouttheportionoftheinducedconnectionsNonetheless,actualcomputationshelppromoteunderstandingofthenotionThefastestwaytograsptherulesofcomputationinvolvingcovariantdifferentiationisactuallytoengageinthecomputationsThecomputationsofthefirstvariationformulafortheenergyfunctionalofmapsyieldavectorfieldcalledthetensionfieldItisgivenasthetraceofthesecondfundamentalformofthemapsAharmonicmapisthencharacterizedasamapwhosetensionfieldisidenticallyChapterisdevotedtotheexistenceproblemofharmonicmapsbetweencompactRiemannianmanifoldsWhetherornotagivenmapishomotopicallydeformabletoaharmonicmapisoneofthemostfundamentalquestionsamonggeometricvariationalproblemsItmayberegardedasageneralizationoftheexistenceproblemofclosedgeodesicsTothisend,the"heatflowmethod"isfirstintroducedThisisaneffectivetechniquefordeformingagivenmaptoaharmonicmapThen,usingthistechnique,itisprovedthatanymapfromacompactRiemannianmanifoldMintoacompactRiemannianmanifoldNofnonpositivecurvatureisfreehomotopicallydeformabletoaharmonicmapThistheoremwasfirstprovedbyEellsSampsoninxviOUTLINESANDOBJECTIVESOFTHETHEORYTheproofofthistheoremusingtheheatflowmethodfirstrequirestheexistenceofatimedependentsolutiontoaninitialvalueproblemwithanyinitialmapoftheparabolicequationforharmonicmapsTheoriginalproofusessuccessiveapproximationstoconstructasolutionafterconvertingtheproblemtoaproblemofintegralequationsviathefundamentalsolutionoftheheatequationInthisbook,thesolutionisconstructedthroughuseoftheinversefunctiontheoreminBanachspacesinanefforttominimizetheamountofpreparationTheexistenceoftimedependentlocalsolutionsisalwaysguaranteed,buttheexistenceofglobaltimedependentsolutionsisnotselfevident,sincetheparabolicequationforharmonicmapsisnonlinearInfact,provingtheexistenceofglobaltimedependentsolutionsentailssomeestimatesofthegrowthrateofsolutionsintimeThecurvatureoftheRieinannianmanifoldNplaysacrucialroleinestimatingtheinfluenceofnonlineartermsAnestimationformulathatguaranteestheexistenceandconvergenceoftimedependentglobalsolutionsisobtainedusingtheWeitzenbockformulafortheheatoperatorundertheconditionthatNisofnonpositivecurvatureTheWeizenbockformula,ingeneral,givestherelationshipbetweensecondorderpartialdifferentialoperatorsnaturallyactingontensorfieldsonRiemannianmanifoldsandtheLaplaceorheatoperatoractingonfunctionsItisrevealedthattheRiemanncurvatureanditsRicciidentityplayessentialrolesforexistenceofsolutionstothosedifferentialoperatorsInthischapter,anaprioriestimateregardingthegrowthrateofsolutionsisobtainedusingtheWeizenbockformulafortheenergydensityofsolutionstotheparabolicequationforharmonicmapsandtheheatoperatorThisideaisoriginallyduetoBochnerIthasbecomeaneffectiveandfundamentaltechniquefortheproofsoftheoremssuchastheKodairavanishingtheoremandmorerecentlyingaugetheoryAsinthecaseofgeodesics,onecanalsoinvestigatethestructuresofRiemannianmanifoldsusingtheexistenceandpropertiesofharmonicmapsThetheoremofPreissman,oneofthetypicalapplicationsofharmonicmaps,isdiscussedThetheoremstatesthatanontrivial,AbeliansubgroupofthefundamentalgroupofacompactmanifoldofnegativecurvatureisinfinitelycyclicTheresearchofRiemannianmanifoldsusingtheexistenceandpropertiesofbarmonicmapsseemstopossessapromisingfutureForexample,newproofsfromamoreanalyticalpointofviewforthetopologicalspheretheoremandtheFrankelconjecturewererecentlygivenbyexploitingOUTLINESANDOBJECTIVESOFTHETHEORYxviitheexistencetheoremofharmonicspheresduetoSacksandUhlenbeckAstrongrigiditytheoremregardingcomplexstructuresinKahlermanifoldsofnegativecurvaturewasalsoobtainedusingtheexistencetheoremofEellsandSampsonCHAPTERArcLengthofCurvesandGeodesics"Giventwopointsinasurface,findacurvejoiningthemoftheminimumarclength"AsolutiontothisquestioniscalledageodesicFindinggeodesicsisatypicalprobleminthecalculusofvariationItsorigincouldbetracedbacktothebirthofcalculusInthischapter,thevariationalproblemofarclengthandtheenergyofcurvesinRiemannianmanifoldsisdiscussedasanintroductiontogeometricvariationalproblemsThecriticalpointsinthisvariationalproblemsatisfyadifferentialequationcalledtheEulerequationTheconceptofcovariantdifferentiationisnaturallyinducedfromthisequationThefirstvariationalformulaoftheenergyofcurvesisobtainedGeodesicsarecharacterizedasthecriticalpointsofthisvariationalproblemArelengthandenergyofcurvesThereader,whohasalreadylearnedthetheoryofsurfaces,k

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