Riemann Surfaces By Way of Analytic Geometry英文学习材料

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1、RiemannSurfacesByWayofAnalyticGeometryDrorVarolinPrefaceThepresentbookarosefromtheneedtobridgewhatIperceivedasarathersubstantialgapbetweenwhatgraduatestudentsatStonyBrookknowaftertheyhavepassedtheirqualifyingexams,andhigherdimen-sionalcomplexanalyticgeometryinitspres

2、entstate.Atpresent,thegenericpost-qualstudentatStonyBrookisrelativelywell-preparedinalgebraictopologyanddifferentialgeometry,butfarfromsoinrealandcomplexanalysis,orpartialdifferentialequations.Fortunately,theamountofrealanalysisneededintheapproachtoRiemannsurfacespre

3、sentedinthisbookisratherminimal.ThedeepestresultsneededaretheHahn-BanachTheoremandtheSpectralTheoremforcompact,self-adjointoperators(andthelatterisnotusedinafundamentalway).Coursesinpartialdifferentialequationsmayoftenpointindirectionsthattypicallydonotleadtocom-plex

4、geometry,eveninsofarastheL2-methodsoriginatedbyBochnerandKodairainthecompactsettingandbyAndreottiandVesentinni,Hormander,Kohn,andMorreyingeneral,andlaterdevelopedbyBombieri,¨Catlin,Demailly,Siu,Skoda,andmanyothers.Amazingly,evencoursesincomplexanalysisdonottypicallye

5、mphasizethepointsmostimportantinthestudyofRiemannsurfaces,focusinginsteadonissuesofminimalregularityforsolvingtheCauchy-Riemannequations.OneexceptionistheRiemannMappingTheorem,oneproofofwhichisrathersimilartotheproofoftheUniformizationTheoremwegiveinChapter10.Inthisb

6、ook,theRiemannMappingTheorem,andanyotherformofclassification,takeabackseatwhilethedriversareresultsbasedontechnique,andespeciallytheapplicationsofsolvingthe@(andsometimes@@)equation.Wepresentasmanymethodsaspossibleforsolvingtheseequations,introducinganddis-cussingGr

7、een’sFunctionsandRunge-typeapproximationtheoremsforthispurpose,andgivingaproofoftheHodgeTheoremusingbasicHilbertandSobolevspacetheory.PerhapsthecenterpieceisHormander’s¨Theoremonsolutionof@withL2estimates.TheProofofHormander’sTheoreminonecomplexdimension¨simplifiesgr

8、eatly,becauseacertainboundaryconditionthatarisesinthefunctionalanalyticformulationofthe@-problemonHilbertspacesisaDirichletboundar