Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

0.11 MB·English

by  Tobias Holck Colding

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