双曲空间形式中具有常平均曲率的超曲面-论文.pdf

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1、数学杂志Vo1．34(2014)J．ofMath．(PRC)NO．4HYPERSURFACESWITHCoNSTANTMEANCURVATUREINAHYPERBoLICSPACEHANYing～bo，FENGShu-xiang(Coll叼e。l，』ln￡．。ndInform。ti。竹Scie咒ce，Xiny。ngNor"。2UniversityXinyang464000，Chin。)Abstract：Inthepaper，westudyann—dimensionalcompleteconnectedandorientedhyper—surfaceM

2、Ⅳin日叶f一1)withconstantmeancurvatureandtwodistinctprincipalcurvatures，oneofwhichissimple．Byusingthemovingframes，weobtainthatifthesquarednormofsecondfun-damenta1formofMsatisfiesarigiditycondition(1．3)，theM”isisometrictohyperboliccylinder．Keywords：hypersurfaces；hyperbolicspaceform；

3、constantmeancurvature2010MRSubjectClassification：53C42；53A10Documentcode：AArticleID：0255-7797(2014)04—0633—071IntroductionLet+fc)bean(几+1)一dimensionalconnectedRiemannianspaceformwithconstantsecti0nalcurvaturec．AccordingtoC>0，C=0orc<0，itiscalledspherespace，Euclideanspaceorhyperb

4、olicspace，respectively，anditdenotedbySn+1(c)，RorH¨(c)．LetMnbeann—dimensionalhypersurfaceinMn+l(c)．Asitiswellknowntheremanyrigidityresultswithconstantmeancurvature，constantscalarcurvatureorconstantk—thmeancur—vatureinMn+l(c)．forexample，see[1-61inSn+1(c)orR叶and[7-9]inhyperbolicsp

5、ace日n+(c)．In[9]，Wuprovedthefollowingtheorem．Theorem1．1LetM”(n3)beacompletehypersurfacein日”+(一1)withconstantmeancurvatureH(IHl>1)andtwodistinctprincipalcurvatureswithmultiplicitiesn一1，1．SetS=一n+佗0H22(咒一1)千IfthesquarelengthofthesecondfundamentalformsatisfiesS≤S+S：S0rS：S一，andMnisi

6、sometrictohyperboliccylinderSn-1(Receiveddate：2012—0305Accepteddate：2012-11-05Foundationitem：SupportedbyNationalNaturalScienceFoundationofChina(10971029；11201400：11026062)：ProjectofHenanProvincialDepartmentofEducation(2011Al10015)；TalentYouthTeacherFundofXinyangNormalUniversity

7、．Biography：HanYingbo(1978一)，male，bornatHeze，Shandong，PH．D．，associateprofessor，majoringindiferentialgeometry．634JournalofMathematicsV01．34or日一f三s(一1)，herenlHI土、士=2(n一11Inthisnote．weshallalsoinvestigaten—dimensionalhypersurfaceswithconstantcurva—ture日(IHl>1)inHn+(一1)andobtainthef

8、ollowingresult：Theorem1．2LetM(n3)beacompletehypersurfa