Real hypersurfaces and complex analysis

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1、jacobowitz.qxp8/27/988:20AMPage1480RealHypersurfacesandComplexAnalysisHowardJacobowitzhetheoryoffunctions(whatwenowacomplexn−1dimensionalsubmanifoldofcallthetheoryoffunctionsofacom-Cn.Thissaid,thedimensionsinstatementslikeplexvariable)wasoneofthegreat2n−1nMC

2、achievementsofnineteenthcenturyTmathematics.Itsbeautyandrangeofshouldnotcauseanyconcern.Thebestexam-applicationswereimmenseandimmediate.ThepletokeepinmindistheboundaryofanopendesiretogeneralizetohigherdimensionsmustsubsetofCn(wheneverthisboundaryishavebeencor

3、respondinglyirresistible.Inthisde-smooth).Indeed,muchoftheexcitementinthesiretogeneralize,thereweretwowaystopro-studyofrealhypersurfacescomesfromthein-ceed.Onewastofocusonfunctionsofseveralterplaybetweenthedomainandtheboundarycomplexvariablesasthegeneralizati

4、onoffunc-andbetweenthegeometryandtheanalysis.tionsofonecomplexvariable.TheotherwastoconsiderafunctionofonecomplexvariableasFunctionsamapofadomaininCtoanotherdomaininItisnaturaltobeginbyconsideringafunctionCandtostudy,asageneralization,mapsofdo-onCnasholomorph

5、icifitisholomorphicineachmainsinCn.Bothapproachesimmediatelyledvariableseparately(thatis,itisholomorphictosurprisesandbotharestillactiveandim-whenrestrictedtoeachofthespecialcomplexportant.Thestudyofrealhypersurfacesaroselinesfz=(z1;:::;zn)2Cnjzkfixedforallke

6、x-withinthesegeneralizations.Thisarticlesur-ceptfork=jandzjarbitraryg).Forcontinuousveyssomecontemporaryresultsaboutthesefunctionsthiscoincideswithanyotherreason-hypersurfacesandalsobrieflyplacesthesubjectablegeneralization(saybyconvergentpowerinitshistorical

7、context.WeorganizeoursurveyseriesorbythesolutionoftheCauchy-Riemannbyconsideringseparatelythesetworoadstoequations).Almostatonce,weencounterastrik-generalization.ingdifferencebetweenfunctionsofoneandWestartwithahypersurfaceM2n−1ofR2nmorecomplexvariables.(Cont

8、rastthistothethe-andconsideritasahypersurfaceofCn,usingoryoffunctionsofrealvariables,whereoneanidentificationofR2nwithCn.WecallMarealmustdelvedeeplybeforethedimensionisrel-hypersurfaceoft