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简介:InthisarticlewestudyholomorphicisometriesofthePoincar'ediskintoboundedsymmetricdomains.EarlierwesolvedtheproblemofanalyticcontinuationofgermsofholomorphicmapsbetweenboundeddomainswhichareisometriesuptonormalizingconstantswithrespecttotheBergmanmetric,showinginparticularthatthegraphV0ofanygermofholomorphicisometryofthePoincar'ediskΔintoanirreducibleboundedsymmetricdomainΩ€CNinitsHarish-Chandrarealizationmustextendtoanaffine-algebraicsubvarietyVC×CN=CN+1,andthattheirreduciblecomponentofV∩(Δ×Ω)containingV0isthegraphofaproperholomorphicisometricembeddingF:Δ→Ω.Inthisarticlewestudyholomorphicisometricembeddingswhichareasymptoticallygeodesicatageneralboundarypointb∈Δ.Startingwiththestructuralequationforholomor-phicisometriesarisingfromtheGaussequation,weobtainbycovariantdifferentiationanidentityrelatingcertainholomorphicbisectionalcurvaturestotheboundarybehaviorofthesecondfundamentalformσoftheholomorphicisometricembedding.Usingthenonpositivityofholomorphicbisectionalcurvaturesonaboundedsymmetricdomain,weprovethatσmustvanishatageneralboundarypointeithertotheorder1ortotheorder21,calledaholomorphicisometryofthefirstresp.secondkind.Wedealwithspecialcasesofnon-standardholomorphicisometricembeddingsofsuchmaps,showingthattheymustbeasymptoticallytotallygeodesicatageneralboundarypointandinfactofthefirstkindwheneverthetargetdomainisaCartesianproductofcomplexunitballs.WealsostudytheboundarybehaviorofanexampleofholomorphicisometricembeddingfromthePoincar'ediskintoaSiegelupperhalf-planebyanexplicitdeterminationoftheboundarybehaviorofholomorphicsectionalcurvaturesinthedirectionstangenttotheembeddedPoincar'edisk,showingthatthemapisindeedasymptoticallytotallygeodesicatageneralboundarypointandofthefirstkind.Fo