关于完备收缩的Ricci-harmonic孤子的研究.pdf

数 学 杂 志 Vo1．36(2016) J．ofMath．(PRC) NO．3 0N C0M LETE SHRINKING RICCI—HARM 0N IC S0LIToNS YANG Fei，ZHANG Liang—di fsc。。lo]M。em。ticsn佗dPhysics，Chi。UniversityofGeoscie扎ces，WuhanJ30063，Chiha) Abstract：In thispaper，westudy thegeometry ofshrinkingRicci—harmonicsolitons．By utilizingthemethodofManola，GabrieleandCarlo[4】undertheRiccisoliton，weprovetheresult thatevervcompactshrinkingRicci—harmonicsolitonisagradientone，whichextendstheresultin thecaseofRiccisolition．Moreover，byutilizingthemethodofZhang[14]，weproveamoreprecise volumegrowth estimate than thatofatmostEuclidean growth forthecompletenon—compact gradientshrinkingRicci—harmonicsoliton，whichextendstheresultofZhang[14]inthecaseof Riccisolition． Keywords： shrinkingRicci-harmonicsoliton；gradient；volumegrowth 2010MR SubjectClassification： 53C25；53C43 D0cumentcode： A ArticleID： 0255—7797(2016)03—0494-07 1 Introduction Let(M ，g)beacompletesmoothRiemannianmanifold，themetricgiscalledaRicci— harn1onicsoliton ifthereexistsavectorfieldX andaconstant ，suchthat __ 1知 where ：(Mn，g)__÷(Ⅳ ，h)isamapbetweentheRemannianmanifolds(M ，g)and (Ⅳm，)，RcistheRiccicurvatureof(M，9)， =traeeVd~and isanonnegativeconstant· WeealltheRicci—harmonicsoliton(1．1)ashrinking，steady，expandingRicci—harmonic solitonif 0 ：0，or入0．IfX isagradientofsomefunctionf，then~xg= 。f，we ca11theRicci．harmonicsolitonagradientRicci—harmonicsolitonwithpotentialfunctionf． SimilartotheRiccisoliton，theRicci—harmonicsolitonisaself-similarsolutiontothe Ricci—harmonicflow， J，0g．一2Rc+2(t)ve~ve， f1．21 、【0