Global existence and convergence of $Q$-curvature flow on manifolds of even dimension

0.33 MB·

by  Quôc-Anh Ngô

GLOBAL EXISTENCE AND CONVERGENCE OF Q-CURVATURE FLOW ON
MANIFOLDS OF EVEN DIMENSION
QUO´ˆCANHNGOˆ ANDHONGZHANG
Abstract. Usinganegativegradientflowapproach,wegeneralizeandunifysomeexis-
7 tencetheorems fortheproblem ofprescribing Q-curvature firstbyBaird, Fardoun, and
1 Regbaoui(Calc. Var.2775-104)for4-manifoldswithapossiblesign-changingcurvature
0 candidatethenbyBrendle(Ann. Math. 158323-343)forn-manifoldswithevennwith
2 positivecurvaturecandidate tothecaseofn-manifoldsofallevendimensionwithsign-
changingcurvaturecandidates. MakinguseoftheŁojasiewicz–Simoninequality,wealso
n
addresstherateoftheconvergence.
a
J

9

]
P 1. Introduction
A
On a closed manifold (Mn,g) of dimension n > 3, a formally self-adjoint geometric
.
h differentialoperatorA beingcalculatedwiththemetricgiscalledconformallycovariant
g
t
a ofbidegree(a,b)if
m A (ϕ)=e bwA (eawϕ) (1.1)
gw − g
[ for all ϕ C∞(Mn) where gw := e2wg is a conformal metric conformally to g, usually
∈
1 denotedbyg [g].
w
∈
v
Ofimportanceinconformalgeometryisthesecond-orderconformalLaplaciandefined
7
4 onclosedmanifoldsofdimensionn>3. Tobeprecise,ifwedenote
2 n 2
2 L := ∆ + − R
g g g
− 4(n 1)
0
−
. where R is the scalar curvatureof g, then L is conformallycovariantof bidegree((n
1 g g
0 2)/2,(n+2)/2)because −
7
L (ϕ)=e (n+2)w/2L (e(n 2)w/2ϕ) (1.2)
1 gw − g −
v: forallϕ C (Mn). Uponusingthechangeofvariableu4/(n 2) =e2w,Eq. (1.2)leadsusto
∞ −
∈
i
X L (ϕ)=u (n+2)/(n 2)L (uϕ)
gw − − g
r
a forallϕ C∞(Mn).Bysettingϕ 1,thescalarcurvatureRgobeystheso-calledconformal
∈ ≡
change
4(n 1)
− n −2 ∆gu+Rgu=Rguu(n+2)/(n−2) (1.3)
−
whereg =u4/(n 2)g. Intheliterature,Eq. (1.3)forunknownuisknownastheprescribed
u −
scalarcurvatureproblemwiththeprescribedfunctionR . Thischallengingproblemhas
gu
already been captured much attention by many mathematicians in the last few decades.
NoticethataspecialcaseofEq.(1.3)whenR isconstantisthefamousYamabeproblem.
gu
Sincethesetwoproblemswereinthecoreofconformalgeometry,wedonotmentionany
particularreferencesinceitcanbeeasilyfoundintheliterature.
Date:10thJanuary,2017at01:24.
Keywordsandphrases. Łojasiewicz–Simon inequality, Q-curvature, negative gradient flow, closed mani-
folds,evendimension.
1
2 Q.A.NGOˆ ANDH.ZHANG
Thefirsthigh-orderexampleofconformaloperatorsisthefourth-orderPaneitzoperator
P4for(M4,g),discoveredby[Pan83]. Thisoperatorisgivenasfollows
2
P4 =( ∆ )2 div R g 2Ric d , (1.4)
g − g ·− g 3 g − ·
(cid:16)(cid:0) (cid:1) (cid:17)
where d is the differential and Ric denotes the Ricci tensor. (Note that the subscript 4
denotestheorderofthePaneitzoperatorbeingused,notthedimensionoftheunderlying
manifold.)Clearly,thePaneitzoperatorP4isconformallycovariantofbidegree(0,4)since
P4 (ϕ)=e 4wP4(ϕ) (1.5)
gw − g
forallϕ C (M4)whereg =e2wg. Uponlettingϕ 1,wededucethat
∞ w
∈ ≡
Q =e 4u(P4u+Q ),
gw − g g
wherethequantityQisgivenby
1
Q = (∆R R2+3 Ric 2).
g −6 g− g | g|
The quantity Q is called the Q-curvature associated with the Paneitz operator P4 on
g
(M4,g), which was discovered by Branson [Bra85]. One of interesting features of the
Q-curvature,bytheGauss–Bonnet–Cherntheorem,isthatitobeysthefollowingidentity
1
Q dµ + W 2dµ =8π2χ(M),
Z g g 4Z | g|g g
M M
whereχ(M) isthe Euler-characteristicof M4 andW is the Weyltensor. Since the Weyl
g
tensor W is conformally invariant, the total Q-curvature Q dµ is also conformally
g M g g
invariant;thatisthevalue R
k := Q dµ
4 g g
Z
M
is independent of the metric g but the conformal class [g] represented by g. (Note that
the subscript 4 in k indicates the order of the Paneitz operator which then defines Q .)
4 g
SimilartoAubin’sthresholdintheYamabeproblem,thereisathreshold16π2fork when
4
workingwithQ-curvaturesonM4. (Thenumber16π2,whichis3!vol(S4),isexactlyequal
to S4QgS4dµgS4 beingcalculatedwithrespecttothestandardmetricgS4 onS4.)
RFor general even dimension n > 4, Graham, Jenne, Mason and Sparling [GJMS92]
discovered a similar operator Pn which is a conformally invariant self-adjoint operator
g
of order n with leading term ( ∆)n/2. This operator is commonly known as the GJMS
operator. The work [GJMS92]−was based on an earlier work by Fefferman and Graham
[FG85]. Moreprecisely,theGJMSoperatorisconformallycovariantofbidegree(0,n)in
thesensethat
Pn (ϕ)=e nwPn(ϕ) (1.6)
gw − g
forallϕ C (Mn)whereg =e2wg.Inaddition,thereiscurvaturequantityQnassociated
∈ ∞ w g
with the operator Pn, analogue with the Q-curvature Q4, which also satisfies a similar
g g
conformalchange
Q =e nu(Pnu+Q ). (1.7)
gu − g g
Inthispaper,weconsidertheproblemofprescribingQ-curvatureonaclosedmanifold
of even dimension. Let us first describe the problem in details. Let (Mn,g ) be closed
0
manifoldsofevendimensionnwithbackgroundmetricg .Thisproblemcanbeformulated
0
as follows: Let f be a smooth function on M. The problem is to ask if there exists a
pointwise conformal metric g [g ], that is g = e2ug for some u, such that f can be
0 0
∈
realizedastheQ-curvatureofg.Thankstotheruleofconformalchange(1.7),thisproblem
isequivalenttosolvingthehighordernonlinearequation
P u+Q = fenu, (1.8)
0 0
Q-CURVATUREFLOWONMANIFOLDSOFEVENDIMENSION 3
where,forsimplicity,wesetP :=Pn andQ = Qn .
0 g0 0 g0
For f > 0 keeping sign, Brendle [Bre03] used a flow method to show that Eq.(1.8)
hasasolutionprovidedthatP is(weakly)positivewithkernelconsistingoftheconstant
0
functionsand
Q dµ =:k <(n 1)!ω ,
Z 0 g0 n − n
M
whereω isthevolumeofunitsphereinRn+1. HerebypositivityofP wemean
n 0
u P u dµ >λ (P ) (u u)2dµ >0 (1.9)
Z · 0 g0 1 0 Z − g0
M M
forallsmoothfunctionuwhereλ (P )isthefirstnontrivialeigenvalueoftheoperatorP
1 0 0
andu = (vol(M,g )) 1 udµ . (Whenn = 4,thethreshold(n 1)!ω fork coincides
0 − M g0 − n n
with 16π2, the thresholRdfork .) More precisely,givenanyinitialdata u , he considered
4 0
theevolutionofmetricsasfollow
∂g(t) = Q MQ0 dµg0 f g(t) fort>0,
Then,Brendleshowedg∂th(t0a)tg=iev−2eun(cid:18)0ga0gn.(yt)−u0RtRhMeflfodwµg(01.1(cid:19)0)hasasolutionwhichexistsf(o1r.1a0l)l
timeandconvergesatinfinitytoametricg with
∞
Q dµ
Q = M 0 g0 f.
g R
∞ M f dµg0
R
Later on, Baird et al. [BFR06] consideredthe problemof prescribing Q-curvatureon 4-
manifolds, where they allow f to change sign. They adoptan abstract negativegradient
flowwhichisdifferentfromBrendle’storealizeit. Preciselyspeaking,theyconsideredthe
functional
J[u]= u P u dµ +2 Q u dµ ,
Z · 0 g0 Z o g0
M M
ontheSobolevspaceH = H2(M,g )undertheconstraint
0
u X := u H : e4uf dµ = Q dµ .
∈ (cid:26) ∈ ZM g0 ZM 0 g0(cid:27)
Then,theythoughtofXasahypersurfaceoftheSobolevspaceHandstudiedthenegative
gradientflowforconformalfactorswithrespectivetothishypersurfacebelow
∂u= XJ(u)
t
−∇ (1.11)
u(0)=u X.
 0 ∈
Clearly,iftheflow(1.11)existsforalltimeandconvergesatinfinity,thenthelimitfunction
u yieldsasolutionof(1.8)withn=4.
∞
Forgeneralmanifoldsof even dimension,there were two waysto improveBrendle’s.
First,byusingavariationalapproach,Bairdetal.[BFR09]improvedBrendle’sbyrelaxing
thepositivitypropertyofP andallowing f >0.However,theykeptusingtheassumption
0
that P has kernel consisting of constant functions. Note that positivity property of the
0
operator P0 on (Sn,gSn) is no longer needed as it automatically holds; see Appendix C.
In another work of the same research group, Fardoun et al. [FR12] improvedBrendle’s
byassumingthepositiveoperatorP hasnon-trivialkernelinthesensethatitskernelhas
0
dimensionatleast2.
Inspiredby the work [BFR06] for sign-changingcandidatesin low dimensionand by
[Bre03] for positive candidatesin high dimensions, the aim of this paper is to study the
prescribing Q-curvatureproblemonall evendimensionalmanifoldswherethe candidate
curvature f is allowed to change sign. The method here we used is again the negative
4 Q.A.NGOˆ ANDH.ZHANG
gradient flow; see also [NX15]. However, instead of the abstract negativegradient flow
method(1.11),weusingaflowwhichisananalogueof (1.10). Inthefollowing,wewill
describeourflowmethodandstatethemainresultofthepaper.
Asin[BFR06],weconsiderthefollowingenergyfunctionalforconformalfactors
n
E[u]= u P u dµ +n Q u dµ (1.12)
2Z · 0 g0 Z 0 g0
M M
onasubsetoftheSobolevspaceHn/2(M,g )givenbythefollowingconstraint
0
u u Hn/2(M,g ): fenu dµ = Q dµ =:Y. (1.13)
∈ ∈ 0 Z g0 Z 0 g0
n M M o
Because P has the leading term ( ∆)n/2, it is not hardto see thatthere exists a positive
0
−
constantCdependingonlyonM,suchthatforanyu Hn/2(M,g )wehave
0
∈
C 1 u ( ∆ )n/2udµ 6 u P u dµ 6C u ( ∆ )n/2udµ . (1.14)
− Z · − 0 g0 Z · 0 g0 Z · − 0 g0
M M M
NoticethatanusualnormforHp(M,g )isgivenasfollows
0
u 2 = ( ∆ )p/2u2dµ + u2dµ . (1.15)
k kHp(M,g0) Z | − 0 | g0 Z g0
M M
However,inviewof(1.14)andsincetheGJMSoperatorP ispositivewithkernelconsist-
0
ingofconstantfunctionswecandefineanequivalentnormonHn/2(M,g )asthefollowing
0
u 2 = u P u dµ + u2 dµ . (1.16)
k kHn/2(M,g0) ZM · 0 g0 ZM g0
Now, we define our negativegradient flow for the functional(1.12) under the constraint
(1.13)asfollows
∂
g(t)= 2(Q λ(t)f)g(t), (1.17)
∂t − g(t)−
wheretheparameterλischosen,insteadofpreservingthevolumeofmetricg(t)in(1.10),
tofixthequantity f dµ foralltime. Adirectcalculationfor d f dµ =0shows
M g(t) dt M g(t)
R R
f(λ(t)f Q )dµ =0. (1.18)
g(t) g(t)
Z −
M
Hence,thenaturalchoiceofλis
fQ dµ
λ(t)= M g(t) g(t). (1.19)
R
f2 dµ
M g(t)
R
Consequently,the constraint(1.13) is preservedalongthe flow. Up to thispoint, we can
stateourmainresultasfollows.
Theorem 1.1. Let (M,g ) be a compact, oriented n-dimensional Riemannian manifold
0
withneven. AssumethattheGJMSoperatorP ispositivewithkernelconsistingofcon-
0
stantfunctions.Moreover,assumethatthemetricg satisfies
0
Q dµ <(n 1)!ω .
Z 0 g0 − n
M
Also,wechoosetheinitialmetricg(0)=e2u0g0withu0 Y andlet f beasmoothfunction
∈
onMsuchthat
(i) sup f(x)>0 if Q dµ >0,
x∈M ZM 0 g0 
(ii) supx∈M f(x)>0andinfx∈M f(x)<0 ifZMQ0dµg0 =0,  (1.20)
(iii) supx∈M f(x)6C0and infx∈M f(x)<0 ifZMQ0dµg0 <0, 
Q-CURVATUREFLOWONMANIFOLDSOFEVENDIMENSION 5
where C is a positive constant depending only on f = max( f(x),0),u ,g and M.
0 − 0 0
−
Then, the flow (1.17) has a solution which is defined for all time and converges in C -
∞
topologytoametricg with Q = λ f astimetendstoinfinity,whereλ isaconstant
and, in addition, equa∞lto 1 if ∞ Q d∞µ , 0. Moreover, if we write g(t)∞= e2u(t)g and
M 0 g0 0
g = e2u g , thentheconvergeRnceg(t) g isnotworsethanapolynomialrateinthe
se∞nsetha∞tth0ereexisttwopositiveconsta→ntsB∞andβsuchthat
u(t) u 6 B(1+t) β
C (M) −
k − ∞k ∞
for allt > 0. Finally, if Q dµ < 0 and f 6 0, then the convergenceg(t) g is
M 0 g0 → ∞
exponentiallyfast. R
WenowconsiderthecaseM =Sn,thestandardn-sphere.LetGbeagroupofisometries
ofSn. Recallthatafunction f issaidtobeG-invriantifitsatisfies
f(σx)= f(x)
for all σ G and x Sn. Furthermore, we say that a conformal metric g to gSn is G-
invarianti∈fuisaG-in∈variantfunctionwhengiswrittenasg = e2ugSn. LetΣbethesetof
fixedpointsofG,thatis,
Σ= x Sn :σx= xforallσ G .
{ ∈ ∈ }
Ournextresultisasfollows.
Theorem 1.2. Let f be a smooth and G-invariant function on Sn with sup f > 0.
x Sn
Choosetheinitialmetricg(0)=e2u0gSn withu0 Y beingG-invariant.Ifeither∈
∈
(a) Σ= or
∅
(b)
E[u ]
sup f(x)6(n 1)!exp 0 ,
x∈Σ − n− (n−1)!ωno
then the flow (1.17) has a solution which is defined for all time and converges in C -
∞
topology to a G-invariant conformal metric g with Q = f as time tends to infinity.
Moreover,ifwewriteg(t)=e2u(t)gSn andg∞ =e∞2u∞gSn,th∞entheconvergenceg(t)→g∞is
notworsethanapolynomialrateinthesensethatthereexisttwopositiveconstantsBand
βsuchthat
u(t) u 6 B(1+t) β
C (M) −
k − ∞k ∞
forallt>0.
RecallthatonSn, theGJMSoperatorP ispositivewithkernelconsistingofconstant
0
functions.AsadirectconsequenceofTheorems1.1and1.2,wehave
Corollary 1.3. Let (M,g ) be a compact, oriented n-dimensional Riemannian manifold
0
withneven. AssumethattheGJMSoperatorP ispositivewithkernelconsistingofcon-
0
stantfunctions. Let f beanon-constant,smoothfunctionon M. Then,thefollowinghold
true.
(i) If Q dµ < 0, then there exists a conformal metric to g with associated
M 0 g0 0
Q-Rcurvature f given that f satisfies inf f(x) < 0 and there exists a positive
x M
constantCdependingonlyon f ,g and∈Msuchthatsup f(x)6C. Inpartic-
− 0 x M
ular,if f 60,thereexistsaconformalmetrictog suchth∈at f canberealizedits
0
Q-curvature.
(ii) If Q dµ = 0, then there exists a conformal metric to g with associated
M 0 g0 0
Q-Rcurvature αf for α 1,1 providedsup f(x) > 0, inf f(x) < 0, and
f dµ ,0. ∈ {− } x∈M x∈M
M g0
(iii) IRf 0 < Q dµ < (n 1)!ω , thenthere exists a conformalmetric to g with
M 0 g0 − n 0
associaRtedQ-curvature f ifandonlyifsup f(x)>0.
x M
∈
6 Q.A.NGOˆ ANDH.ZHANG
(iv) LetM =Snand f beaG-invariantfunctiononSnwithsup f >0. Ifeither
x Sn
(a) Σ= or ∈
(b) there∅existy Σandr >0suchthat
0 0
∈
sxuΣp f(x)6max(cid:18)Z−Sn f ◦φy0,r0 dµSn,0(cid:19), (1.21)
∈
thenthereexistsaG-invariantconformalmetrictog withassociatedQ-curvature
0
f. Inparticular,if
sup f(x)6max f dµSn,0 ,
x Σ Z−Sn
∈ (cid:0) (cid:1)
thenthereexistsaG-invariantconformalmetrictog withassociatedQ-curvature
0
f.
InviewofCorollary1.3(ii)above,weareunabletogetridofthemultipleαin(ii),due
tothe factthatwe cannotconfirmthe signofλ . However,notethatifsup f(x) > 0
x M
and f dµ <0,thenGeandXu[GX08]succ∞essfullyremovedthemultiple∈αbyusing
M g0
avarRiationalapproach.Note,however,thattheauthorsin[BFR06,Corollary2.4]wasable
toconfirmthesignofλ insomecases. Theresultin[BFR06]roughlysaysthatifthere
exists some u C2(M)∞with small ∆u2dµ , then there exists a conformal metric g
∈ M| | g0
withitsQ-curvatureequalto sgn( R f dµ )f. Theproofmakesuseof(n/2)2=n;hence
e − M eg0
it is not easy to generalize such aRresult for the general operator P because the precise
0
formulaforP isnotclear.
0
Beforeclosingthissection, we notethat, asin [BFR06] and[Bre03], the metricg in
0
Theorem1.1islimitedtotheso-called“subcritical”inthesense k < (n 1)!ω . When
n n
−
k >(n 1)!ω ,theanalysisisdelicatesinceblow-upphenomenamayoccurs.Thiscanbe
n n
easilyse−enbyconsideringthefamilyofconformaldiffeomorphismofSn. Onthestandard
model Sn, hence k = (n 1)!ω , a lot of attempts by different methods have already
n n
−
beenperformed,forinterested readers, we referto [WX98, Bre03, Bre06, MS06, CX11,
Ho12]andthereferencestherein.Anothersourceofusefulinformationfortheproblemof
prescribingQ-curvaturecanalsobefound,forexample,in[DM08,LLL12,GM15].
2. Theflowequationanditsenergyfunctional
In this section, we first derive the flow equations for the conformal factor u and the
curvatureQandthenshowthattheenergyfunctionalE isnon-increasing.
Since the evolution equation (1.17) preserves the conformal structure, we may write
g(t) = e2u(t)g forsomereal-valuedfunctionu. Then,weclearlyhavethefollowingequa-
0
tionfortheconformalfactoru
∂
u(t)=λ(t)f Q (F)
∂t − g(t)
forallt>0. Thanksto(1.7),wealsodeducethat
∂
u= P u e nuQ +λ(t)f
∂t − g(t) − − 0
forallt>0. Byeasycalculations,wecanobtainevolutionequationsfor∂Q and∂λ(t)
t g(t) t
alongtheflow. Thisisthecontentofthefollowinglemma.
Lemma2.1. Supposethatu(x,t)isasolutionoftheflow(F). Then,wehavethefollowing
claims:
∂
Q = P (Q λ(t)f)+nQ (Q λ(t)f)
∂t g(t) − g(t) g(t)− g(t) g(t)−
and
λ(t)= f2 dµ −1 P f nλ(t)f2 λ(t)f Q dµ
′ g(t) g(t) g(t) g(t)
Z Z − −
(cid:16) M (cid:17) M(cid:0) (cid:1)(cid:0) (cid:1)
Q-CURVATUREFLOWONMANIFOLDSOFEVENDIMENSION 7
where, as always, P = e nuP is the GJMS operatorwith respective to the conformal
g(t) − 0
metricg(t).
Proof. Sincethisisjustaroutinecalculation,weleavetheprooftothereader. (cid:3)
Lemma2.2. Supposethatu(x,t)isasolutionoftheflow(F). Then,onehas
dE[u]
= n (Q λ(t)f)2dµ .
g(t) g(t)
dt − Z −
M
Inparticular,theenergyfunctionalE[u]isnon-increasingalongtheflow.
Proof. Itfollowsfromtheflowequationforu,(1.7),and(1.18)that
dE[u] n
= u P u+u P u dµ +n Q u dµ
dt 2Z t· 0 · 0 t g0 Z 0 t g0
M M
= n (Q λ(t)f)Q dµ
g(t) g(t) g(t)
− Z −
M
= n (Q λ(t)f)2dµ +nλ(t) f(λ(t)f Q )dµ
g(t) g(t) g(t) g(t)
− Z − Z −
M M
= n (Q λ(t)f)2dµ .
g(t) g(t)
− Z −
M
Theproofiscomplete. (cid:3)
Before going further, let us recall some important inequalities that will also be used
in the present paper. First, we recall the Gagliardo–Nirenberg interpolation inequality,
appliedtoanyLs-integrablefunctionϕ,
p(1 α)/r pα/q
jϕpdµ 6Cp mϕrdµ − ϕqdµ (2.1)
ZM|∇ | g0 (cid:18)ZM|∇ | g0(cid:19) (cid:18)ZM| | g0(cid:19)
forsomeC >0with0<α<16q,r6+ satisfying1/p= j/n+(1/r m/n)α+(1 α)/q.
∞ − −
SincethehighorderGJMSoperatorP isself-adjointandpositivewithkernelconsist-
0
ingofconstantfunctions,wecanapplyAdam’sinequality[Ada88,Theorem2]toget
n!ω (u u)2
exp n − dµ 6C (2.2)
ZM (cid:16) 2 Mu·P0u dµg0(cid:17) g0 A
R
forsomeconstantC >0. Hereandthroughoutthepaper,bywwemeantheaverageofw
A
calculatedwithrespecttog ,thatisw=(vol(M,g )) 1 wdµ . (Intherestofourpaper,
0 0 − M g0
byvol(K)wemeanthevolumeofK MbeingcalculaRtedwithrespecttothebackground
⊂
metricg . IfwewanttoemphasizethemetricgbeingusedtocalculatethevolumeofK,
0
we shall write vol(K,g).) A detailed explanation for the validity of (2.2) can be found,
for example, in [Bre03, page 330]. As a consequence of (2.2), we obtain the following
Trudinger-typeinequality
α2
exp α(u u) dµ 6C exp u P u dµ (2.3)
ZM (cid:0) − (cid:1) g0 A (cid:16)2n!ωn ZM · 0 g0(cid:17)
forallrealnumberα.
3. BoundednessofuinHn/2for06t6T
Inthissection,wetrytoshowthatuisuniformlyboundedinHn/2(M,g ). Toseethis,
0
wewillsplitourargumentintofourcasesdependingonthesignandvalueofQ .
0
8 Q.A.NGOˆ ANDH.ZHANG
3.1. Thecase Q dµ = 0. Itfollowsfromourchoiceoftheinitialdatau in(1.13)
M 0 g0 0
and(1.18)thatR f dµ =0forallt>0,whichimpliesthefollowinglemma.
M g(t)
R
Lemma3.1. Alongtheflow(F),thevolumeofthemetricgispreserved. Hence,ifweset
V0 = Menu0 dµg0 then Menu dµg0 =V0forallt.
R R
Proof. Clearly,wehave
d
enu dµ =n u dµ =n λ(t)f Q dµ
dtZ g0 Z t g(t) Z − g(t) g(t)
M M M
(cid:0) (cid:1)
=n λ(t) f dµ Q dµ =0.
Z g(t)−Z 0 g0
(cid:16) M M (cid:17)
Hence enu dµ isconstantalongtheflow;thisprovestheassertion. (cid:3)
M g0
R
Lemma3.2. ThereexistsauniformconstantC >0suchthat
u 6C
k kHn/2(M,g0)
foralltimetinthemaximalintervalofexistence.
Proof. FromLemma2.2,itfollowsthat
n
u P u dµ +n Q (u u) dµ 6E[u ]. (3.1)
2Z · 0 g0 Z 0 − g0 0
M M
UsingthePoincare´-typeinequality(1.9),weget
1
(u u)2 dµ 6 u P u dµ ,
ZM − g0 λ1(P0)ZM · 0 g0
whichtogetherwithYoung’sinequalityimplythat
n n
n Q (u u) dµ 6 u P u dµ + Q2 dµ . (3.2)
(cid:12)(cid:12) ZM 0 − g0(cid:12)(cid:12) 4ZM · 0 g0 λ1(P0)ZM 0 g0
(cid:12) (cid:12)
Bysubstituti(cid:12)ng(3.2)into(3.1),w(cid:12)ehave
(cid:12) (cid:12)
4 4
u P u dµ 6 E[u ]+ Q2 dµ , (3.3)
ZM · 0 g0 n 0 λ1(P0)ZM 0 g0
Hence,inviewof (1.16),tobound u ,itsufficestobound udµ . ByJensen’s
k kHn/2(M,g0) M g0
inequalityandLemma3.1,wehavethat R
enu 6 enu dµ =V0/vol(M).
Z− g0
M
Therefore,
1
u6 log(V0/vol(M)).
n
Toboundufrombelow,weapplytheTrudinger-typeinequality(2.3)and(3.3)toget
2E[u ] 2n
exp n(u u) dµ 6C exp 0 + Q2 dµ .
ZM (cid:0) − (cid:1) g0 A (cid:16)(n−1)!ωn λ1(n−1)!ωn ZM 0 g0(cid:17)
Since enu dµ =V0,weconcludethat
M g0
R C 2E[u ] 2n
exp nu 6 A exp 0 + Q2 dµ .
(cid:0)− (cid:1) V0 (cid:16)(n−1)!ωn λ1(n−1)!ωn ZM 0 g0(cid:17)
Hence,weget
1 C 2 2E[u ]
u> log A + Q2 dµ + 0 .
−(cid:16)n (cid:16)V0(cid:17) λ1(n−1)!ωn ZM 0 g0 n!ωn (cid:17)
Bycombiningalltheestimatesabove,wethuscompletetheproof. (cid:3)
Q-CURVATUREFLOWONMANIFOLDSOFEVENDIMENSION 9
3.2. Thecase Q dµ > 0. Inthiscase,itfirstfollowsfromourchoiceofinitialdata
M 0 g0
u and(1.18)thRat fenu dµ = Q dµ forallt>0,whichimpliesthat
0 M g0 M 0 g0
R R
Q dµ = fenu dµ 6(sup f) enu dµ .
ZM 0 g0 ZM g0 M ZM g0
Hence,
1
enu dµ > Q dµ . (3.4)
Z g0 sup f Z 0 g0
M M M
Itfollowsfrom(2.3)and(3.4)that
n/2
u P u dµ >log exp n(u u) dµ logC
(n−1)!ωn ZM · 0 g0 ZM (cid:0) − (cid:1) g0 − A
1
>log Q dµ nu.
(cid:18)CAsupM f ZM 0 g0(cid:19)−
Thus,wehave
n/2 1 1
u> u P u dµ + log Q dµ . (3.5)
−n!ωn ZM · 0 g0 n (cid:18)CAsupM f ZM 0 g0(cid:19)
Using(3.5)andLemma2.2,weget
E[u ]>E[u]
0
n
= u P u dµ +n Q (u u) dµ +nu Q dµ
2Z · 0 g0 Z 0 − g0 Z 0 g0
M M M
>n 1 MQ0 dµg0 u P u dµ +n Q (u u) dµ (3.6)
2(cid:18) − R(n−1)!ωn (cid:19)ZM · 0 g0 ZM 0 − g0
1
+ Q dµ log Q dµ .
ZM 0 g0 (cid:18)CAsupM f ZM 0 g0(cid:19)
ByYoung’sinequalityand(1.9),wehave
n Q (u u) dµ 6 n 1 MQ0 dµg0 u P u dµ
(cid:12)(cid:12)(cid:12) ZM 0 − g0(cid:12)(cid:12)(cid:12) 4(cid:18) − R(n−1)!ωn (cid:19)ZM · 0 g0 (3.7)
(cid:12) (cid:12) n Q2 dµ
(cid:12) (cid:12) + M 0 g0
R
λ (P ) 1 ( Q dµ /(n 1)!ω
1 0 − M 0 g0 − n
(cid:0) R (cid:1) (cid:1)
Since Q dµ < (n 1)!ω , by substituting (3.7) into (3.6), we conclude that there
M 0 g0 − n
existsaRnuniformconstantC >0suchthat
2
u P u dµ 6C . (3.8)
Z · 0 g0 2
M
Substituting(3.8)into(3.5)givesthatthereexistsanuniformconstantC suchthat
3
u>C . (3.9)
3
To get an upper bound for u, we notice that (3.2) still holds in this case. Hence by
substituting (3.2) into the first inequality in (3.6), the positivity of P and the inequality
0
Q dµ >0,weget
M 0 g0
R
(n/λ (P )) Q2 dµ +E[u ]
u6 1 0 M 0 g0 0 . (3.10)
R
n Q dµ
M 0 g0
R
By plugging (3.8), (3.9) and (3.10) into (1.9), we conclude that there exists a uniform
constantC >0suchthat
4
u2 dµ 6C . (3.11)
Z g0 4
M
Theassertionisthenprovedbycombining(3.8)and(3.11).
10 Q.A.NGOˆ ANDH.ZHANG
3.3. Thecase Q dµ <0. Toachieveourgoal,wefirstneedananalogueof[BFR06,
M 0 g0
Lemma4.1]whRoseproofisprovidedinAppendixBforcompleteness.
Lemma 3.3. Let K be a measurablesubsetof M with vol(K) > 0. Then there exist two
constantsα>1dependingonMandg andC >1dependingonM,g ,andvol(K)such
0 K 0
that
α
enu dµ 6C exp α u 2 max enu dµ ,1 .
ZM g0 K (cid:0) k 0kHn/2(M,g0)(cid:1) (cid:26)(cid:16)ZK g0(cid:17) (cid:27)
NotethatLemma3.3holdsregardlessofthesignof Q dµ < 0. Withthehelpof
M 0 g0
thislemma,weareabletoshowtheuniformboundofthRevolumeofthetimemetricg(t).
Lemma3.4. Supposetheflow(F)existson[0,T)forsomeT >0. Also,assumethatthere
existsaconstantC dependingon f ,u , g , and M suchthatsup f 6 C . Then,there
0 − 0 0 M 0
existsauniformconstantγ>0suchthat
enu dµ 6γ
Z g0
M
forallt [0,T).
∈
Proof. Let f+ =max(f,0)anddefine
1
K = x M : f(x)6 inf f .
∈ 2 M
n o
Notice that K , due to the fact that inf f < 0. Hence, by the continuity of f, we
M
∅
concludevol(K)>0. Sinceu Y,wehave
0
∈
Z Q0 dµg0 =−Z fenu0 dµg0 =Z f−enu0 dµg0 −Z f+enu0 dµg0,
M M M M
whichimpliesthat
1
(−infM f)ZMQ0 dµg0 6ZMenu0 dµg0.
From(2.3)andYoung’sinequality,itfollowsthat
n n
ZMenu0 dµg0 6CAexp(cid:18)2(n−1)!ωn ZMu0·P0u0 dµg0 + vol(M)ZMu0 dµg0(cid:19)
n!ω n
6C exp n exp u P u +u2 dµ (3.12)
A (cid:16)2vol(M)(cid:17) (cid:16)2(n−1)!ωn ZM(cid:2) 0· 0 0 0(cid:3) g0(cid:17)
6C exp C u 2 ,
2 2k 0kHn/2(M,g0)
(cid:16) (cid:17)
whereC isaconstantdependingon M andg ,whichwemayassumetobegreaterthan
2 0
1. Therefore,wecanget
1
Q dµ 6C exp C u 2 . (3.13)
−infM f ZM 0 g0 2 (cid:0) 2k 0kHn/2(M,g0)(cid:1)
NowchoosetheconstantC inthelemmasuchthat
0
inf f
C = − M exp (α(C +1) C ) u 2 ,
0 8αCKC2α−1 · (cid:0)− 2 − 2 k 0kHn/2(M,g0)(cid:1)
whereC >1andα>1areconstantsofLemma3.3. Moreover,set
K
γ=2C (8C )αexp (C +1)α u 2 .
k 2 2 k 0kHn/2(M,g0)
Then,weclaimthat (cid:0) (cid:1)
enu dµ 6γ, (3.14)
Z g0
M
forallt [0,T).Toseethis,welet
∈
I = t [0,T): enu(s) dµ 6γforalls [0,t] .
∈ Z g0 ∈
n M o