manifolds with quadratic curvature decay and slow volume growth

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1、ManifoldswithQuadraticCurvatureDecayandSlowVolumeGrowthJohnLott∗DepartmentofMathematicsUniversityofMichiganAnnArbor,MI48109-1109ZhongminShenDepartmentofMathematicsIndianaUniversity-PurdueUniversityatIndianapolisIndianapolis,IN46202-3216September17,1998ToDetlefGromollonhis60thbirthdayAb

2、stractWeshowthattherearetopologicalobstructionsforanoncompactmanifoldtoadmitaRiemannianmetricwithquadraticcurvaturede-cayandavolumegrowthwhichisslowerthanthatoftheEuclideanarXiv:math/9809097v1[math.DG]17Sep1998spaceofthesamedimension.∗SupportedbyNSFGrantDMS-9704633.11IntroductionAmajor

3、themeisRiemanniangeometryistherelationshipbetweencurvatureandtopology.Forcompactmanifolds,onecanconstrainthecurvatureanddiameterandaskwhetheroneobtainstopologicalrestrictionsonthemani-fold.Ifthemanifoldisnoncompactthenareplacementforadiameterboundisaconstraintonhowthecurvaturebehavesin

4、termsofthedistancefromabasepoint.Moreprecisely,letMbeacompleteconnectedn-dimensionalRiemannianmanifold.Fixabasepointm0∈M.Definition1.1Mhasquadraticcurvaturedecay(withconstantC>0)ifforallm∈Mandall2-planesPinTmM,thesectionalcurvatureK(P)ofPsatisfies2

5、K(P)

6、≤C/d(m0,m).(1)Notethatcondition(1)

7、isscale-invariantinthatitisunchangedbyaconstantrescalingoftheRiemannianmetric.Manyinterestingresultshavebeenobtainedwhenthesectionalcurvaturehassomefaster-than-quadraticcurvaturedecay[1,8].Inthispaperweconcentrateinsteadonthecaseofquadraticcurvaturedecay.Initself,condition(1)doesnotput

8、anyre-strictionsonthetopologyofamanifold.OnecanshowthatanyconnectedsmoothparacompactmanifoldhasaRiemannianmetricwithquadraticcur-vaturedecay;see[9,p.96]orLemma2.1below.Ontheotherhand,wewillshowthatifinadditiononerestrictsthevolumegrowthofthemetric,thenonedoesobtaintopologicalrestrictio

9、nsonM.ThefirstquestioniswhetherMhasfinitetopologicaltype.Definition1.2MhasfinitetopologicaltypeifMishomotopy-equivalenttoafiniteCW-complex.Definition1.3Mhaslowerquadraticcurvaturedecay(withconstantC>0)ifforallm∈Mandall2-planesPinTmM,thesectionalcurvatureK(P)ofPsatisfies2K(P)≥−C/d(m0,m).(2)L