Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications TobiasHolckColdingand Aaron Naber ∗ 2 1 January 9, 2012 0 2 n a J Abstract 6 ] Consideralimitspace(Mα,gα,pα) G→H (Y,dY,p),wherethe Mαn havealowerRiccicurvaturebound G and are volumenoncollapsed. The tangentconesof Y at a point p Y are knownto be metric cones ∈ D C(X),howevertheyneednotbeunique. LetΩ M betheclosedsubsetofcompactmetricspaces Y,p GH ⊆ . X whichariseascrosssectionsforthetangentsconesofY at p. Inthispaperwestudythepropertiesof h t ΩY,p. Inparticular,we givenecessaryandsufficientconditionsforanopensmoothfamilyΩ (X,gs) a ≡ m of closed manifolds to satisfy Ω = ΩY,p for some limit Y and point p Y as above, where Ω is the ∈ closure of Ω in the set of metric spaces equipped with the Gromov-Hausdorfftopology. We use this [ characterizationtoconstructexampleswhichexhibitfundamentallynewbehaviors.Thefirstapplication 3 is toconstructlimitspaces(Yn,d ,p)withn 3suchthatat p thereexistsforevery0 k n 2 a v Y ≥ ≤ ≤ − 4 tangentconeat p ofthe formRk C(Xn k 1), where Xn k 1 isasmoothmanifoldnotisometrictothe − − − − 4 × standardsphere. Inparticular,thisisthefirstexamplewhichshowsthatastratificationofalimitspace 2 3 Y basedonthe Euclideanbehavioroftangentconesisnotpossibleor evenwelldefined. Itisalso the . firstexampleofathreedimensionallimitspacewithnonuniquetangentcones. Thesecondapplication 8 0 is to construct a limit space (Y5,d ,p), such that at p the tangent cones are not only not unique, but Y 1 nothomeomorphic. Specifically,sometangentconesarehomeomorphictoconesoverCP2♯CP2 while 1 : othersarehomeomorphictoconesoverS4. v i X 1 Introduction r a GH In this paper weare interested in pointed Gromov-Hausdorff limits (M ,g ,p ) (Y,d ,p) such that the α α α Y → M ’saren-dimensional andsatisfythelowerRiccibound α Ric(M ) (n 1)g, (1) α ≥ − − andthenoncollapsing assumption Vol(B (p )) v> 0. (2) 1 α ≥ DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA02139. Emails: [email protected] ∗ [email protected]. ThefirstauthorwaspartiallysupportedbyNSFGrantsDMS0606629,DMS1104392,andNSFFRG grantDMS0854774andthesecondauthorbyanNSFPostdoctoralFellowship. 1 For any such limit Y, by Gromov’s compactness theorem [GLP, G], any sequence r 0 contains a i → GH subsequence r such that (Y,r 1d ,p) (Y ,d,p), where Y is a length space. Any such limit Y is said j −j Y → p p p to be a tangent cone of Y at p. By the noncollapsing assumption (2) it follows from [ChC1], [ChC2] that any tangent cone must be a metric cone Y C(X ) over a compact metric space X with diamX π p p p p ≡ ≤ andHausdorff dimension equalton 11. However,by[ChC2]tangent conesofY at pneednotbeunique; − GH cf. [P2]. Moreprecisely, itmay happen that there isa different sequence r˜ 0such that (Y,r˜ 1d ,p) j → −j Y → (C(X˜ ),d,p)convergestoatangentconeC(X˜ )whereX˜ andX arenotisometric. Wearethereforejustified p p p p indefiningfor p Y thefamilyΩ X ofmetricspacessuchthatC(X )arisesasatangentconeofY at Y,p s s ∈ ≡ { } p. It is known that the family Ω M , viewed as a subset of the space of all compact metric spaces Y,p GH ⊆ endowedwiththeGromov-Hausdorfftopology,iscompactandpathconnected. Itfollowsfrom[ChC2]that thevolumeVol(),ormorepreciselythe(n 1)-dimensional Hausdorffmeasure,isindependent ofthecross · − section X Ω andisbounded fromabovebythatoftheroundunitsphereofdimension n 1. Thatis, s Y,p ∈ − Vol(X ) = V Vol(Sn 1(1)). (3) s − ≤ Further, ifX Ω isasmoothcrosssection, e.g. asmoothclosedmanifold, thenbecauseRic(C(X )) 0 s Y,p s ∈ ≥ wehavethat Ric(X ) n 2. (4) s ≥ − In fact, it is fairly clear that (4) holds in the more general sense of [LV], [S] even for singular X . Tofully s understand thefamilyΩ weintroduce onemoreconcept, thatofRicciclosability. Y,p Definition1.1. Let(Mn 1,g)beasmooth closed Riemannian manifold. Wesay that M isRicciclosable if − foreveryǫ > 0,thereexistsasmooth(open)pointed Riemannianmanifold(Nn,h ,q )suchthat: ǫ ǫ ǫ 1. Ric(N ) 0. ǫ ≥ 2. Theannulus A (q ) N isisometrictoA (C(M,(1 ǫ)g)). 1, ǫ ǫ 1, ∞ ⊆ ∞ − Remark 1.1. Note that if the stronger condition that there exists N with Ric(N) 0 and A (q) 1, ≥ ∞ ≡ A (C(M,g)) holds, then (M,g) is certainly Ricci closable. Ricci closability acts as a form of geomet- 1, ∞ rictrivialcobordism condition. Nowweaskthequestion: What subsets Ω M can arise as Ω for some limit space Y coming from a sequence M Y GH Y,p α ⊆ → whichsatisfiesconditions (1)and(2)? Wehave written down somebasic necessary conditions on Ω , and our main theorem is that these condi- Y,p tionsaresufficentaswell. 1Withoutthenoncollapsingassumptiontangentconesneednotbemetricconesby[ChC2]andneednotevenbepolarspaces by[M4]. 2 Theorem1.1. LetΩbeanopenconnectedmanifold,ourparameterspace. Let (Xn−1,gs) s Ω MGH,with { } ∈ ⊆ n 3,beasmoothfamilyofclosed manifolds suchthat(3)and(4)holdandsuchthatforsome s wehave 0 ≥ GH that X is Ricci closable. Then there exists a sequence of complete manifolds (Mn,g ,p ) (Y,d ,p) s0 α α α → Y which satisfy (1) and (2) for which X = Ω , where X is the closure of the set X in the Gromov- s Y,p s s { } { } { } Hausdorff topology. Remark1.2. Infact,intheconstruction wewillbuildthe M tosatisfyRic(M ) 0. NoteherethatΩ,asa α α ≥ parameterspace,isasmoothmanifoldwhichweareviewingasbeingembeddedΩ M insidethespace GH ⊆ ofmetricspaces. In the applications we will be interested not so much in the smooth cones C(X ) which arise as tangent s conesat p Y,butintheconesC(X)where X liesintheboundary oftheclosure X X X . Thereare s s ∈ ∈{ }\{ } two primary examples we will be interested in constructing through Theorem 1.1. First, we will construct anexampleofalimitspace(Y,d ,p)suchthatat p Y tangentconesarehighlynonunique, andinfact,for Y ∈ every0 k n 2wecanfindatangent conethatsplits offprecisely anRk factor. Notethis isindistinct ≤ ≤ − contrast totheRn case, whereifonetangentconeatapointisRn,thensoarealltheothertangentconesat that point, see[C]2. Notethat ifatangent cone splits offanRn 1 factor, then by[ChC2]itisactually aRn − factor, sothatthenonunique splitting ofRk factorsforevery0 k n 2isthemostdegenerate behavior ≤ ≤ − onecangetatasinglepoint. Morepreciselywehavethefollowing: GH Theorem 1.2. For every n 3, there exists a limit space (Mn,g ,p ) (Y,d ,p) where each M satisfy α α α Y α ≥ → (1) and (2), and such that for each 0 k n 2, there exists a tangent cone at p which is isometric to ≤ ≤ − Rk C(X),where X isasmoothclosedmanifoldnotisometrictothestandardsphere. × Thisexamplehasthe,potentiallyunfortunate,consequencethatatopologicalstratificationofalimitspace Y inthe context of lowerRicci curvature can’t bedone based on tangent cone behavior alone. This should becontrasted tothecaseofAlexandrovspaces,see[P3]. Thisalsogivesanexampleofathreedimensional limitspacewithnonunique tangentcones. Ournextexampleisofalimitspace(Y,d ,p),suchthatat p Y thereexistdistincttangentconeswhich Y ∈ arenotonlynotisometric, buttheyarenotevenhomeomorphic. Moreprecisely wehave: GH Theorem1.3. Thereexistsalimitspace(M5,g ,p ) (Y5,d ,p)ofasequence M satisfying(1)and(2), α α α Y α → 2 andsuchthatthereexistsdistincttangentconesC(X ),C(X )at p Y withX homeomorphictoCP2♯CP 0 1 0 ∈ and X homeomorphic toS4. 1 Both of the last two theorems have analogues for tangent cones at infinity of open manifolds with non- negative Ricci curvature and Euclidean volume growth. Wesay that an open n-dimensional manifold with nonnegative Riccicurvature hasEuclideanvolumegrowthifforsome p M (henceall p M)thereexists ∈ ∈ somev> 0suchthatforallr > 0wehavethatVol(B (p)) vrn. r ≥ Theorem1.4. Wehavethefollowing: 2Foralimitofasequencethatcollapsesthesituationisquitedifferent,see[M2]. 3 1. For n 3, there exists a smooth open Riemannian manifold (Mn,g) with Ric 0 and Euclidean ≥ ≥ volume growth such that for each 0 k n 2 one tangent cone at infinity of M is isometric to ≤ ≤ − Rk C(X),whereX isasmoothclosedmanifoldnotisometrictothestandard sphere. × 2. ThereexistsasmoothopenRiemannianmanifold(M5,g)withRic 0andEuclidean volumegrowth ≥ 2 that has distinct tangent cones at infinity C(X ) and C(X ) with X homeomorphic to CP2♯CP and 0 1 0 X homeomorphic toS4. 1 Relatedtotheaboveexamplesweconjecture thefollowing: Conjecture 1.1. Let Yn be a noncollapsed limit of Riemannian manifolds with lower Ricci bounds. Let NU Y bethesetofpointswherethetangentconesatthegivenpointarenotunique,thendim (NU) Haus ⊆ ≤ n 3. − Conjecture 1.2. Let Yn be a noncollapsed limit of Riemannian manifolds with lower Ricci bounds. Let NH Y bethesetofpointswherethetangentconesatthegivenpointarenotofthesamehomeomorphism ⊆ type,thendim (NH) n 5. Haus ≤ − In particular, webelieve that for a four dimensional limit at each point tangent cones should be homeo- morphic. Finally, wemention that[CN1]and [CN2]contains somerelated results. Inparticular, in[CN2]wewill usesomeoftheconstructions ofthispaper. 2 Proof of Theorem 1.1 Themaintechnical lemmaintheproofofTheorem1.1isthefollowing. Lemma 2.1. Let Xn 1 be a smooth compact manifold with g(s), s ( , ), a family of metrics with − ∈ −∞ ∞ h < 1suchthat: ∞ 1. Ric[g(s)] (n 2)g(s). ≥ − 2. d dv(g(s)) = 0,wheredvistheassociated volumeform. ds 3. ∂ g(s), ∂ ∂ g(s) 1and ∂ g(s) 1,wherethenormsaretakenwithrespecttog(s). s s s s | | | | ≤ |∇ | ≤ Thenthereexistfunctionsh :R+ (0,1)and f :R+ ( , )withlim h(r) = 1,lim h(r) = h , r 0 r → → −∞ ∞ → →∞ ∞ lim f(r) = ,lim f(r) = andlim rf (r) = 0suchthatthemetricg¯ = dr2+r2h2(r)g(f(r))on r 0 r r 0, ′ → −∞ →∞ ∞ → ∞ (0, ) X satisfies Ric[g¯] 0. ∞ × ≥ Further if for some T ( , )we have that g(s) = g(T) for s T then we can pick h such that for r ∈ −∞ ∞ ≤ sufficiently smallh(r) 1. ≡ Proof. Weonlyconcernourselveswiththeconstructionof f andhforr (0,1). Extendingtheconstruction ∈ forlargeristhesame. 4 Nowfirstwenotethatifg¯ = dr2+r2h2(r)g(f(r))asabovethenthefollowingequationsholdfortheRicci tensor, wheretheprimesrepresent rderivatives. (rh) 1 (rh) 1 Ric = (n 1) ′′ + gabgpqg g ′gabg gabg . (5) rr − − rh 4 ′ap ′bq − rh ′ab − 2 ′a′b 1 1 Ric = [∂ (gabg ) ∂(gabg )+ (gab) (∂g g gpq∂ g )] (6) ir 2 a ′bi − i ′ab 2 ′ i ab − ib a pq (rh) (rh) 1 Ric = Ric +r2h2[( (n 2)( ′)2 ′′ gabg )g ij ij − − rh − rh − 2 ′ab ij n(rh) 1 1 +( ′ gabg )g + gabg g ] (7) −2 rh − 4 ′ab ′ij 2 ′ai ′bj In the estimates it will turn out that terms involving either second derivatives of g or products of first derivativesofhandgcannotbecontrolled ingeneral. Luckilytheconstantvolumeformtellsusthat gabg = 0, ′ab andbytakingther derivativewegetthat gabg = gabgpqg g . ′a′b ′ap ′bq Whenwesubstitute theseinto(5)aboveweget (rh) 1 Ric = (n 1) ′′ gabgpqg g , (8) rr − − rh − 4 ′ap ′bq similarsubstitutions maybemadefortheotherequations. Nowforpositivenumbers E,F 1tobechosendefinethefunctions ≤ E h(r) = 1 ǫ(r) = 1 (9) − − log( log(r r)) 0 − and f(r) = Flog(log( log(r r))), (10) 0 − − forr r tobechosen. Thefollowingcomputations arestraightforward: 0 ≤ E E ǫ(r) = ,ǫ (r) = , ′ log( log(r r)) (log( log(r r)))2( log(r r))r 0 0 0 − − − E( 1+ 1 + 2 ) ǫ′′(r) = − (−log(r0r)) log(−log(r0r))(−log(r0r)) (11) (log( log(r r)))2( log(r r))r2 0 0 − − andso (rh) 1 ǫ 1 E 1 ′ = ( ′ )= (1 ) , (12) rh r − 1 ǫ r − (1 ǫ)(log( log(r r)))2( log(r r)) ≤ r 0 0 − − − − 5 E(1+ 1 + 2 ) (rh) ǫ 2ǫ ′′ = ( ′′ ′ ) = − (−log(r0r)) log(−log(r0r))(−log(r0r)) rh −1 ǫ − r(1 ǫ) (log( log(r r)))2( log(r r))r2(1 ǫ) 0 0 − − − − − E = , (13) −2(log( log(r r)))2( log(r r))r2 0 0 − − wherethelastinequality holdsforr 1andr sufficientlysmall. Alsobyourassumptions ong(s)wehave 0 ≤ that g f F . Finally, if weplug all of this into our equations for the Ricci tensor | ′| ≤ | ′| ≤ log( log(r0r))( log(r0r))r weget,where D = D−(n)isa−dimensional constant: E DF2 Ric rr ≥ (log( log(r r)))2( log(r r))r2 − (log( log(r r)))2( log(r r))2r2 0 0 0 0 − − − − E , (14) ≥ 2(log( log(r r)))2( log(r r))r2 0 0 − − DF Ric − , (15) ir ≥ log( log(r r))( log(r r))r 0 0 − − (n 2)ǫ E DF Ric r2h2[ − + ii ≥ r2h2 2(log( log(r r)))2( log(r r))r2 − log( log(r r))( log(r r))r2 0 0 0 0 − − − − DF2 E ] r2h2 , (16) −(log( log(r r)))2( log(r r))2r2 ≥ log( log(r r))r2 0 0 0 − − − where the last inequalities on (14) and (16) require E E(n,F) and r sufficiently small. Now it is clear 0 ≥ from the above that weget positive Ricci in the r and M directions. Thedifficulty is that we have a mixed term(15)whichcancertainlybenegativeandinfactdominatesthepositivityof(14). Toseepositivity fixa point(r,x) (0,1) M andassumeatthispointg (f(r)) = δ . Theneveryunitdirection atthispointisof ij ij ∈ × theformδrˆ+ √1 δ2ˆiforδ [0,1]andwecancompute: r−h ∈ 1 Eδ2 Ric [ (δr+√1rh−δ2i)(δr+√1rh−δ2i) ≥ log(−log(r0r))r2 2log(−log(r0r))(−log(r0r)) 2DFδ√1 δ2 − +E(1 δ2)] (17) − ( log(r r))h − 0 − 1 Eδ2 DFδ√1 δ2 [ − +E(1 δ2), (18) ≥ 2log( log(r r))r2 log( log(r r))( log(r r)) − ( log(r r)) − 0 0 0 0 − − − − wherethelastinequality isforr 1andafterpossibly changing D. Toseethisispositiveforanyδ [0,1] ≤ ∈ we break it into two cases, when √1 δ2 1 and √1 δ2 1 . For the first case we see − ≥ ( log(r0r)) − ≤ ( log(r0r)) − − that √1 δ2 DF E Ric − [ − + ] 0, (19) (δr+√1rh−δ2i)(δr+√1rh−δ2i) ≥ log(−log(r0r))r2 (−log(r0r)) (−log(r0r)) ≥ forE DF. Forthecase √1 δ2 1 wefirstnotethatδ 1 forr 1andthengroupthefirsttwo ≥ − ≤ ( log(r0r)) ≥ 2 ≤ − termstoget: δ E DF Ric [ ] 0 (20) (δr+√1rh−δ2i)(δr+√1rh−δ2i) ≥ log(−log(r0r))(−log(r0r))r2 2log(−log(r0r)) − (−log(r0r)) ≥ 6 for E DF andr 1,andr sufficiently smallasclaimed. 0 ≥ ≤ Now extending f and h to the rest of r can be done in the same manner, and handling the case when g(s) = g(T)stabilizesiscomparativelysimpleandcanbedonewithacutofffunctionsothath(r)isconcave inthisregion. Noteforanyh wecanpickF,andhenceE,sufficientlysmallastomakethevolumelossas ∞ smallaswewish. (cid:3) WiththeaboveinhanditiseasytofinishTheorem1.1. ProofofTheorem1.1. Webeginbyconstructing whatwillbethelimitspaceY =C(X)ofthetheorem. Let c : ( , ) Ω be a smooth map such that for every open neighborhood U Ω there are t such a −∞ ∞ → ⊆ → ∞ thatc( t ) = c(t ) U. a a − ∈ Inthecasewhencondition (3)isassumedwecanapplyatheoremofMoser[Mo],whichtellsusthatfor acompactmanifold X ifw ,w arevolumeformswiththesamevolumethenthereexistsadiffeomorphism 0 1 φ : X X such thatw = φ w . Withthisinmindthereisnolossinassuming thatforeach s,t ( , ) 1 ∗ 0 → ∈ −∞ ∞ wehavedv = dv ,sincetheotherconditions ofthetheorem arediffeomorphism invariant. g(c(s)) g(c(t)) Because g(x) is smooth for x Ω we can be sure, after possibly reparametrizing c, that g(t) g(c(t)) ∈ ≡ satisfiesLemma2.1. Wetake g¯ = dr2+r2h2(r)g(f(r)) fromthislemma. Theconditions onhguarantee thatthemetricextendstoacompletemetricontheconeY. Now we argue that Y satisfies the conditions of the theorem, hence for each s Ω¯ that the metric cone ∈ C(X ) is realized as a tangent cone of Y. So let r 0 such that c(f(r )) s, which we can do by the s a a → → conditions on f andtheconstruction ofc. Ifweconsider therescaled metric r 2g¯ dr2 +r2h2(r r)g(f(r r)), a− ≈ a a thenbythecondition lim rf (r) = 0weseethatthisconverges tothedesiredtangent coneasclaimed. r 0 ′ → Finally, we wish to show that if for some s Ω that if X is Ricci closable, then (Y,d) can be realized 0 ∈ s0 asalimit(M ,g ,p )ofRiemannianmanifolds withnonnegative Riccicurvature. Foreachαletc (t)bea α α α α smoothcurvesuchthat c(t) ift α c (t) = ≥ − . α  s ift 2α  0 ≤ − Foreachαlet(C(X),d )bethemetricspaceassociated withthecurve α g (t) (1 α 1)g(c (t)), α − α ≡ − as by Lemma 2.1 (again, if need be we can reparametrize c (t) for t < α to force g (t) to satisfy the α α − requirementsoftheLemma). Neartheconepointwehavethat(C(X),d )isisometrictoC(X,(1 1)g(s )). α − α 0 Bytheassumption ofRicciclosability thereexistsacompleteRiemannianmanifold(N ,h ,p )suchthat α α α Ric(N ) 0, α ≥ and A (p ) A (C(M,(1 α i)g(s ))). 1, α 1, − 0 ∞ ≡ ∞ − 7 ThuswecangluethesetogethertoconstructsmoothRiemannianmanifolds(M ,g ,p ). Thisisourdesired α α α sequence. (cid:3) 3 Example I Ourfirstapplication ofTheorem1.1istoprovide,forn 3,examplesoflimitspaces ≥ (Mn,g ,p )GH (Yn,d ,p), (21) α α α Y → where each M has nonnegative Ricci curvature with Vol(B (p )) > v > 0, and such that at p Y the α 1 α ∈ tangentconesarenotonlynonunique, butforeach0 k n 2wecanfindasequence rk 0suchthat ≤ ≤ − a → (Y,(rk) 1d ,p)GH Rk C(Xn k 1), (22) a − Y → × − − where the Xn k 1 are smooth manifolds with Vol(Xn k 1) < Vol(Sn k 1). That is, for each 0 k n 2 − − − − − − ≤ ≤ − wecan findatangent cone which splits offprecisely anRk factor. Aswasremarked earlier this isoptimal, in that if any tangent cone were to split a Rn 1-factor, then by [ChC2] we would have that p is actually a − regularpointofY,andinparticular by[C]everytangentconewouldbeRn. Toconstructourexamplewewillbuildafamilyofsmoothmanifolds(Sn 1,g¯ ),andapplyTheorem1.1. − s Todescribe this family let us first define for 0 < t 1 the t-suspension, S (X), over a smooth manifold X. t ≤ That is, for 0 < t 1 and a smooth manifold X, the metric space S (X) is homemorphic to the suspension t ≤ over X anditsgeometryisdefinedbythemetric 1 dr2 +sin2( r)d2 , t X for r (0,tπ). Notice then that S (X) is the standard metric suspension of X. Now for any~t D ~t 1 ∈ ∈ ≡ { ∈ Rn 1 : 0< t t ... t 1 wecandefinethemetric − n 1 n 2 1 − ≤ − ≤ ≤ ≤ } g S (...S (S1(t ))), ~t ≡ t1 tn−2 n−1 whereS1(t )isthecircleofradiust . Noteinparticular thatg isthen 1sphereofradiust. More n 1 n 1 (t,...,t) − − − generally, wehavethatg ,wherethefirstkentriesare1,isisometrictothek-foldsuspensionofthe (1,...,1,t,...,t) n k 1sphereofradiust. Thistellsusinparticular that − − C((Sn 1,g )) Rk C(Sn k 1(t)). − (1,...,1,t,...,t) − − ≡ × LetusdefinethesubsetΩ Rn 1 bythecondition − ⊆ Ω ~t Rn 1 : 0 < t t ... t < 1andVol(g)= Vol(g ) . ≡ { ∈ − n−1 ≤ n−2 ≤ ≤ 1 ~t 12,...,21 } WehavethatΩsatisfiesthefollowingbasicproperties: 1. Ωisasmooth,connected, opensubmanifold ofdimension n 2. − 2. (1,..., 1) Ω. 2 2 ∈ 8 3. For each 0 k n 2 0 < t < 1 and ~t Ω (1,...,1,t ,...,t ) such that (Sn 1,g ) GH ≤ ≤ − ∃ k i ∈ → k k − ~ti → (Sn 1,g ),wherethefirstkentriesare1. − (1,...,1,tk,...,tk) Nowthecollection g with s Ωalmostdefinesourfamily. Noticeinparticular thatsince g isthe s ∈ (21,...,21) n 1sphere ofradius 1 itiscertainly Ricciclosable, andthatforevery0 k n 2wehavebythethird − 2 ≤ ≤ − condition above that Rk C(Sn k 1(t )) g(Ω), where the closure is in the Gromov-Hausdorff sense. The − − k × ∈ remainingissueissimplythatourmetricsg onSn 1 arenotsmooth. However,for~t Ωtheydosatisfy s − ∈ sec[g]> 1+ǫ(~t), ~t both on the smooth part and in the Alexandrov sense on the whole, where ǫ(~t) 0 as~t ∂Ω. Although → → not smooth, the singularities are isometric spheres and may be easily smoothed in a canonical fashion by writing in normal coordinates with respect to the singular spheres, see [P1], [M1], [M3] for instance. We let g¯ be such a smoothing, where for each~t we can then easily arrange, by smoothing a sufficiently small ~t amount,that 1 sec[g¯ ] > 1+ ǫ(~t) (23) ~t 2 while Vol(g¯ ) Vol(g) < δ(~t), (24) | ~t − ~t | where δ(~t) << ǫ(~t). Thus, after a slight rescaling of each g¯ , we can guarantee that the volumes continue ~t to coincide and that sec 1 for s Ω. This family thus satisfies Theorem 1.1, and we can construct the ~t ≥ ∈ desiredlimitspace(Mn,g ,p ) (Yn,d ,p)asintheTheorem. α α α → Y 4 Example II Inthissectionwepresentonefurtherexampleofinterest. Wewishtoconstruct acompletelimitspace (M5,g ,p ) (Y5,d ,p), (25) α α α → Y whereeach M satisfyRic 0,Vol(B (p )) v > 0,andsuchthatat pthetangentconesofY arenotonly α α 1 α ≥ ≥ not unique, but there exist distinct tangent cones which are not even homeomorphic. Specifically there are sequences r 0andr 0with a → a′ → (Y,r 1d ,p) (C(X ),d ,p), a− Y → p Yp (Y,r 1d ,p) (C(X ),d ,p), (26) a′− Y → ′p Yp′ andsuchthathomeomorphically wehave X CP2♯CP2, p ≈ X S4. (27) ′p ≈ 9 To construct our example we wish to again use Theorem 1.1. We will construct a family of metrics (CP2♯CP2,g)witht (0,2]whichsatisfythehypothesis ofthetheorem andsuchthat t ∈ lim(CP2♯CP2,g) = (S4,g ). t 0 t 0 → Geometrically, (S4,g ) will contain two singular points and will look roughly like a football. On the other 0 hand,(CP2♯CP2,g )willhaveasufficiently niceformthatwewillbeabletoshowthatitisRicciclosable. 2 Oncethisfamilyisconstructed wecanimmediatelyapplyTheorem1.1toproduce ourexample. The construction of the family will be done in several steps. We begin by introducing our basic ansatz. LetS3 bethethreesphere, viewedastheLieGroupSU(2),withthestandard frame X,Y,Z suchthat [X,Y]= 2Z, [Y,Z]= 2X, [Z,X]= 2Y. Eachpieceofthevariousconstructions willbeametricon(r ,r ) S3 whichtakestheform 0 1 × dr2+A(r)2dX2+B2(r) dY2+dZ2 , (28) (cid:16) (cid:17) where 0 r < r π. Notice that by employing various boundary data on A and B we can get these ≤ 0 1 ≤ 2 metrics to close up to smooth metrics on CP2, CP2♯CP2 or CP2 D4, where D4 is the closed 4-ball. The \ Riccicurvatureofthesemetricssatisfytheequations A B Ric(r,r) = ′′ 2 ′′ , (29) − A − B 1 A A B A2 Ric(X,X)= ′′ 2 ′ ′ +2 , (30) X2 − A − AB B4 | | 1 B A B B 2 2B2 A2 Ric(Y,Y)= ′′ ′ ′ ′ +2 − , (31) Y 2 − B − AB − B! B4 | | 1 B A B B 2 2B2 A2 Ric(Z,Z)= ′′ ′ ′ ′ +2 − , (32) Z2 − B − AB − B! B4 | | withallotherRiccitermsvanishing. 4.1 Bubble Construction Ourbubbles mimicthose of[P1],see also[M1],[M3]. Let0 < b 1beaconstant whichwillbefixedat 0 ≤ theendoftheconstruction. Foreach0 ǫ 1letusconsiderthemetricspacesBǫ definedby ≤ ≤ 1 Aǫ (r) b sin(2r), (33) B 0 ≡ 2 1 1 1 ǫ Bǫ (r) b ( )ǫ cosh( r), (34) B 0 ≡ 100 − 2 − 100 ! 100 forr (0,r ], wherer issuchthat Aǫ (r ) = Bǫ (r ). Ourbubbles Bǫ aresmoothmanifolds withboundary ǫ ǫ B ǫ B ǫ ∈ which are homeomorphic to CP2 D4. Notice that 0 < r r r π, and that for each such ǫ the \ 1 ≤ ǫ ≤ 0 ≡ 4 10