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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
