双曲积的渐近上曲率.pdf 12页

'˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn双曲积的渐近上曲率谢桂玲，肖映青湖南大学数学与计量经济学院，长沙　410082摘要：M.Bonk和T.Foertsch提出了Gromov双曲空间的渐近上曲率的概念，并提议研究双曲积的渐近上曲率。本文研究了上述问题，证明了Ku(Y∆;o)maxfKu(X1);Ku(X2)g，其中(X1;o1);(X2;o2)是两个给定基点的Gromov双曲空间，Y∆;o是它们的双曲积，Ku(X)是双曲空间X的渐近上曲率。此外，在一定的条件下我们得到Ku(Y∆;o)大于等于Ku(X2)。关键词：Gromov双曲空间；渐近上曲率；双曲积中图分类号：O181OntheAsymptoticuppercurvatureofhyperbolicproductsXIEGui-Ling,XIAOYing-QingCollegeofMathematicsandEconometrics,HunanUniversity,Changsha,410082Abstract:M.BonkandT.FoertschintroducedthenotionofasymptoticuppercurvatureforGromovhyperbolicspacesandsuggestedtostudytheasymptoticuppercurvatureofhyperbolicproducts.Inthispaper,westudytheseproblemsandprovethatKu(Y∆;o)maxfKu(X1);Ku(X2)g;where(X1;o1);(X2;o2)aretwopointGromovhyperbolicspaces,Y∆;oistheirhyperbolicproductandKu(X)istheasymptoticuppercurvatureofahyperbolicspaceX.Moreover,weobtainsomeextraconditionstosurethatKu(Y∆;o)isnosmallerthanKu(X2).Keywords:Gromovhyperbolicspace;Asymptoticuppercurvaturebound;Hyperbolicproduct.0IntroductionGiventwoGromovhyperbolicmetricspaces(seeDefinition1),theirCartesianproductwilltypicallyfailtobehyperbolicitself.Forexample,RisaGromovhyperbolicspacebutR2isnot.Butin[1],T.FoertschandV.SchroederintroducedthenotionofhyperbolicproductfortwoGromovhyperbolicmetricspacesandshowedthatthehyperbolicproductisstillaFoundations:ThisworkwassupportedbytheNationalNaturalScienceFoundationofChina(No.11301165).AuthorIntroduction:XieGuiling(1991-),female,graduatestudents,majorresearchdirection:complexanalysis.Cor-respondenceauthor:XiaoYingqing(1979-),male,associateprofessor,majorresearchdirection:complexanalysis,E-mailaddress:ouxyq@hnu.edu.cn.-1- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnGromovhyperbolicmetricspace.Accordingtotherequirements,wegiveabriefreviewaboutit,andsomebasicnotionsonecanfindinpreliminaries.Weusethenotationjxyjforthedistancebetweenx;yinagivenmetricspaceX.Fora;b;c2Randc0,onedefine:a=cb()jabjc:LetX1,X2betwohyperbolicmetricspacesandY:=X1X2betheCartesianproductofX1andX2.OnY,T.FoertschandV.Schroederconsideredthemaximummetric,i.e.,jxyj:=maxfjx1y1j;jx2y2jg;8x=(x1;x2);y=(y1;y2)2Y:Takeapointoi2Xi,wesaythat(Xi;oi)isapointmetricspacefori=1;2.Suppose∆0,wewriteo:=(o1;o2)2Yanddefine:Y∆;o=X1∆;oX2:=f(x1;x2)2Yjjo1x1j=∆jo2x2jg:ThespaceY∆;oYisendowedwiththerestrictionofthemaximummetriconYandiscalledthehyperbolicproductof(X1;o1)and(X2;o2)byT.FoertschandV.Schroederin[1].Itwas′′′provedin[1]thatifX1;X2are-hyperbolic,thenY∆;ois-hyperbolicforsome=(∆;).ThenotionofasymptoticuppercurvatureKu(X)foraGromovhyperbolicspaceX(seeDefinition3)wasintroducedin[2].ItistheanalogofsectionalcurvatureonRiemannianmanifolds.TherearecloselyconnectionbetweentheasymptoticuppercurvatureKu(X)andthepinchedsectionalcurvatureofaRiemannianmanifoldX.WhenXisacompletesimply-connectedRiemannianmanifoldwithpinchedsectionalcurvatureb2Ka2<0,itwasshownthatb2K(X)a2<0in[2].ThisimpliesinparticularthattheasymptoticuuppercurvatureofrealhyperbolicspaceHnis1.Inthepaper[2],therearevariousotherquestionsinrelationwiththeasymptoticup-percurvaturewereproposed.Forexampleitsbehaviorunderconstructionssuchasgluingsalongquasi-convexsetsorhyperbolicproducts.Inthispaper,westudytheasymptoticuppercurvatureofthehyperbolicproduct.Firstly,weobtainthefollowingtheorem.Theorem1.Supposethat(X1;o1);(X2;o2)aretwopointGromovhyperbolicspaces,∆0andY∆;oisthehyperbolicproductof(X1;o1)and(X2;o2),whereo=(o1;o2)2X1X2.ThenKu(Y∆;o)maxfKu(X1);Ku(X2)g:Nextly,weestimatethelowerboundofKu(Y∆;o)undersomeextraconditionsandobtainthefollowingtheorem.Theorem2.Supposethat(X1;o1);(X2;o2)aretwopointGromovhyperbolicspaces,∆0andY∆;oisthehyperbolicproductof(X1;o1)and(X2;o2),whereo=(o1;o2)2X1X2.-2- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnAssumethatKisanonnegativeconstantandforallx;x′2X,thereexisty;y′2Xsuchthat12z=(x;y);z′=(x′;y′)2Yand∆;o′:′(xjx)o1=K(yjy)o2:ThenKu(Y∆;o)Ku(X1):Thepaperisorganizedasfollows.Insection2,somebasicdefinitionsandpropertiesarepresented.Insection3,wegiveproofsofTheorems1and2.1PreliminariesLetXbeametricspace,fixabasepointo2X,theGromovproductofx;x′2Xwithrespecttooisdefinedas′1′′(xjx)o:=(joxj+joxjjxxj):2Notethat(xjx′)0bythetriangleinequality.oDefinition1.ThemetricspacesXis-hyperbolicforsome0,ifitsatisfiesthe-inequality(xjy)ominf(xjz)o;(zjy)ogforallx;y;z2Xandeverybasepointo2X.ThemetricspaceXiscalled(Gromov)hyperbolic,ifitis-hyperbolicforsome0.Obviously,ifametricspaceXsatisfiesthe-inequalityforabasepointo2X,thenforotherbasepointo′2X,the2-inequalityisfulfilled.ForpropertiesofGromovhyperbolicspaces,wereferto[3].ForaGromovhyperbolicspace,thefollowingclaimonceoccurredin[4],buttherearenoevidence.Forcompleteness,wegiveaproofhere.Lemma1.IfXisa-hyperbolicspaceforbasepointo2X.Thenthereexistsaconstanta0onlydependingonsuchthat′(xjx)omin(xi−1jxi)oalogn1≤i≤nforallx;x′2Xandalln-chainsx=x;x;


;x=x′2X.01nProof.Letx;x′2Xandx=x;x;


;x=x′2Xisan-chain.01nWefirstconsiderthecasewhenn=2k;k2N+,andclaimthat′(xjx)omin(xi−1jxi)ok:(1)1≤i≤n-3- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnWeprovetheinequality(1)byinductivemethod.BytheconditionthatXisa-hyperbolicspace,theinequality(1)holdwhenn=2.Nowassumethatitistruewhenn=2k−1.Thatis,forthechainx0;x1;


;x2k−1,wehave(xjx2k−1)omin(xi−1jxi)o(k1):(2)1≤i≤2k−1Bytheinductionhypothesis,forthechainx2k−1;x2k−1+1;


;xn=x2k−1+2k−1,wehave(x2k−1jxn)omin(xi−1jxi)o(k1):(3)2k−1+1≤i≤n′kByinequalities(2)and(3),forthechainx=x0;x1;


;x2k−1;


;xn=x;n=2,wehave′(xjx)ominf(xjx2k−1)o;(x2k−1jxn)ogmin(xi−1jxi)ok:1≤i≤nThereforetheinequality(1)iscorrect.Nextly,weconsiderthecasewhen2k0,wehaveKu(Y∆;o)=maxf1+";2+"g=maxfKu(X1);Ku(X2)g+";whichimpliesthatKu(Y∆;o)maxfKu(X1);Ku(X2)g:Inthefollowing,weestimatethelowerboundofKu(Y∆;o)undersomeextraconditions.ProofofTheorem2.Assumethatx=x;x;


;x=x′isachaininX,thenthere01n1existachainy=y;y;


;y=y′suchthat01n′z=z0=(x0;y0);z1=(x1;y1);


;zn=(xn;yn)=zisachaininY∆;o,andforeachi21;2;


;n,wehave:(xi−1jxi)o1=K(yi−1jyi)o2:AccordingtotheLemma3,foreachi21;2;


;n,wehave(zi−1jzi)ominf(xi−1jxi)o1;(yi−1jyi)o2g(xi−1jxi)o1K:SinceY∆;oisaGromovhyperbolicspace,weknowthatY∆;oisanACu()-spaceforsome2[1;0).Thenthereexistc0suchthat′1(zjz)omin(zi−1jzi)oplognc1≤i≤n1min(xi−1jxi)o1plogn(K+c):1≤i≤nBytheLemma3again,wehave′′′(xjx)o1minf(xjx)o1;(yjy)o2g′(zjz)o∆1min(xi−1jxi)o1plogn(∆+K+c);1≤i≤nwhichimpliesthatX1isanACu()-space,moreover,Ku(X1)Ku(Y∆;o):Wesaythatamapf:(X;dX)!(Y;dY)isarough-isometricembeddingifthereisak>0suchthat′′′dX(x;x)kdY(f(x);f(x))dX(x;x)+k-8- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnforx;x′2X.Ifinaddition,foreachy2Ythereexistsx2XsuchthatdY(f(x);y)k;thenfiscalledarough-isometry.Wesaythattwometricspacesareroughisometricifthereexistsaroughisometrybetweenthem.Lemma4.Let(X;dX);(Y;dY)betwoGromovhyperbolicspacesandf:X!Yisarough-isometry.ThenKu(X)=Ku(Y):Proof.Bytheassumption,thereexistak0suchthat′:′jyyj=kjxxj:forallx;x′2X,wherey=f(x);y′=f(x′)2Y.Bycalculating,wehave′1′′(xjx)o=(joxj+joxjjxxj)2:1′′=3k(juyj+juyjjyyj)22′=(yjy)u;whereo2Xandu=f(o)2Y.Assumethatx=x;x;:::;x=x′isachaininX,then01nthereexistachainy=y;y;:::;y=y′inYand01n:(xi−1jxi)o=3k(yi−1jyi)u2foralli21;2;:::;n.NotethatYisanACu()-spaceforsome2[1;0),sinceYisaGromovhyperbolicspace.Thenthereexistc0suchthat′′3(xjx)o(yjy)uk213min(yi−1jyi)uplognck1≤i≤n21min(xi−1jxi)oplognc3k;1≤i≤nwhichimpliesthatXisalsoanACu()-spaceandKu(X)Ku(Y):Ontheotherhand,lety2Y,thereexistx2Xsuchthatjyf(x)jksincef:X!Yisarough-isometry.Leto2Xandu=f(o)2Y,thenjuyjjf(o)f(x)j+jf(x)yjjoxj+2k;-9- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnandjoxjjf(o)f(x)j+kjuyj+jyf(x)j+kjuyj+2k:Inotherwords,wehave:juyj=2kjoxj:Nowlety;y′2Y,thereexistx;x′2Xsatisfy′′′′′jyyjjyf(x)j+jf(x)f(x)j+jf(x)yjjxxj+3k;and′′′′′′′jxxjjf(x)f(x)j+kjf(x)yj+jyyj+jyf(x)j+kjyyj+3k:Then′:′jyyj=3kjxxj:So,wehave′1′′(yjy)u=(juyj+juyjjyyj)2:1′′=7k(joxj+joxjjxxj)22′=(xjx)o;Similarly,weobtainKu(Y)Ku(X):Fromabove,weobtainKu(X)=Ku(Y):Lemma5.Let(X;dX);(Y;dY)betwoGromovhyperbolicspacesandf:X!Yisarough-isometricembedding.ThenKu(X)Ku(Y):Proof.Notethatf:X!f(X)isarough-isometry.AccordingtoLemma4,wehaveKu(X)=Ku(f(X)):NotethatKu(f(X))Ku(Y)sincef(X)Y.ThusKu(X)Ku(Y):Asanapplicationoftheabovetheorem,weobtainthefollowingconclusion.-10- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cnCorollary1.Supposethat(X1;o1);(X2;o2)aretwopointGromovhyperbolicspaces,∆0andY∆;oisthehyperbolicproductof(X1;o1)and(X2;o2),whereo=(o1;o2)2X1X2.Iff:X1!X2isarough-isometricembeddedwith(x;f(x))2Y∆;oforallx2X1,thenKu(X1)Ku(Y∆;o)Ku(X2):Proof.Bytheassumption,forallx;x′2X,thereexisty=f(x);y′=f(x′)2Ysuchthatz=(x;y);z′=(x′;y′)2Yand∆;o′:′jyyj=kjxxj:Notethatjjoxjjoyjj∆andjjox′jjoy′jj∆.Bycalculating,wehave1212′1′′(xjx)o1=(jo1xj+jo1xjjxxj)2:1′′=∆+1k(jo2yj+jo2yjjyyj)22′=(yjy)o2:FromTheorem2,wehaveKu(X1)Ku(Y∆;o):AccordingtoLemma5,wehaveKu(X1)Ku(X2).ThenbyTheorem1,weobtainKu(Y∆;o)maxfKu(X1);Ku(X2)g=Ku(X2):Inconclusion,weobtainKu(X1)Ku(Y∆;o)Ku(X2):Corollary2.LetXbeaGromovhyperbolicspace,∆0andY∆;oisthehyperbolicproductof(X;o1)and(X;o2),whereo=(o1;o2)2XX.Ifjo1o2j∆.ThenKu(Y∆;o)=Ku(X):Proof.Supposethatx2X.Sincejxo1jjxo2jjo1o2j∆,wehavef(x;x):x2XgY∆;o:DefineiX:X!XbyiX(x)=x.Obviously,jiX(x)iX(y)j=jxyj;whichimpliesthatiXisanisometricembeddingmap.BytheaboveCorollary,wehaveKu(Y∆;o)=Ku(X):-11- ˖ڍመ੾᝶஠ڙጲhttp://www.paper.edu.cn参考文献（References）[1]T.Foertsch,V.Schroeder.Aproductconstructionforhyperbolicmetricspaces,IllinoisJournalofMathematics,49(2005),793-810.[2]M.Bonk,T.Foertsch.Asymptoticuppercurvatureboundsincoarsegeometry,Mathema-tischeZeitschrift,253.4(2006),753-785.[3]S.Buyalo,V.Schroeder,ElementsofAsymptoticGeometry,EMSMonographsinMath-ematics.EuropeanMathematicalSociety,Zürich(2007).[4]M.Bonk,B.Kleiner.Rigidityforquasi-Mobiusactionsonfractalmetricspaces,JournalofDifferentialGeometry,61.1(2002),81-106.[5]Yin,Q.Thurstonmapsandasymptoticuppercurvature,GeometriaeDedicata,176.1(2011),1-23.-12-'