Regularity scales and convergence of the Calabi flow

0.35 MB·English

by  Haozhao Li

Regularity scales and convergence of the Calabi flow
Haozhao Li , BingWang , and Kai Zheng
∗ †
5
1
0 InmemoryofProfessorWeiyueDing
2

n
Abstract
a
J 1
Wedefineregularityscalesasalternativequantitiesof max Rm − tostudythebehavior
2
(cid:18) M | |(cid:19)
1 oftheCalabiflow. Basedonestimatesoftheregularityscales, weobtainconvergencetheo-

remsoftheCalabiflowonextremalKa¨hlersurfaces,undertheassumptionofglobalexistence
]
G oftheCalabiflowsolutions.OurresultspartiallyconfirmDonaldson’sconjecturalpicturefor
theCalabiflowincomplexdimension2. Similarresultsholdinhighdimensionwithanextra
D
assumptionthatthescalarcurvatureisuniformlybounded.
.
h
t
a
m Contents
[

2 1 Introduction 2
v
1
5 2 Regularityscales 7
8
1 2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
0
2.2 Estimatesbasedoncurvaturebound . . . . . . . . . . . . . . . . . . . . . . . . 8
.
1
0 2.3 Frommetricequivalence tocurvaturebound . . . . . . . . . . . . . . . . . . . . 14
5
1 2.4 Curvaturescale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
:
v 2.5 Harmonicscale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
i
X
2.6 Backwardregularity improvement . . . . . . . . . . . . . . . . . . . . . . . . . 27
r
a
3 AsymptoticbehavioroftheCalabiflow 28
3.1 DeformationofthemodifiedCalabiflowaroundextKmetrics . . . . . . . . . . 28
3.2 Convergence ofKa¨hlerpotentials . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3 Convergence ofcomplexstructures . . . . . . . . . . . . . . . . . . . . . . . . . 35
SupportedbyNSFCgrantNo.11131007.
∗
SupportedbyNSFgrantDMS-1312836.
†
1
4 Furtherdiscussion 40
4.1 Exampleswithglobalexistence . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2 BehavioroftheCalabiflowatpossible finitesingularities . . . . . . . . . . . . . 41
4.3 Furtherstudy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
1 Introduction
In the seminal work [5], E. Calabi studied the variational problem of the functional (S S)2,
M −
theCalabienergy,amongKa¨hlermetricsinafixedcohomologyclass. ThevanishingpRointsofthe
Calabi energy are called the constant scalar curvature Ka¨hler (cscK) metrics. The critical points
oftheCalabienergyarecalledtheextremalKa¨hler(extK)metrics. Tosearchsuchmetrics,Calabi
introduced a geometric flow, which is now well known as the Calabi flow. Actually, on a Ka¨hler
manifold(Mn,ω,J),theCalabiflowdeformsthemetricby
∂
g = S , (1.1)
∂t i¯j ,i¯j
where g is the metric determined by ω(t) and J, and S is the scalar curvature of g. Note that in
theclass[ω],everymetricformcanbewrittenasω+ √ 1∂∂¯ϕforsomesmoothKa¨hlerpotential
−
functionϕ. Therefore, ontheKa¨hlerpotential level,theaboveequation reducesto
∂
ϕ = S S = gi¯j logdet g +ϕ S, (1.2)
∂t − − kl¯ kl¯ ,i¯j−
(cid:8) (cid:0) (cid:1)(cid:9)
where S is the average of scalar curvature, which is a constant depending only on the class [ω].
Notethatequation(1.2)isafourthorderfullynonlinearPDE.Thisorderincursextremetechnical
difficulty. Inspiteofthisdifficulty, theshort timeexistence ofequation (1.2)wasproved byX.X.
Chen and W.Y.He in [15]. Furthermore, they also proved the global existence of (1.2) under the
assumption thattheRiccicurvature isuniformly bounded.
About twodecades after the birth of the Calabi flow, in [30], S.K.Donaldson (See [33] also)
pointed out that the Calabi flow fits into a general frame of moment map picture. In fact, by
fixingtheunderlying symplectic manifold (M,ω)anddeforming thealmostcomplexstructures J
alongHamiltonianvectorfields,C (M)hasaninfinitesimalactiononthemodulispaceofalmost
∞
complexstructures. Thefunction S S canberegardedasthemomentmapofthisaction, where
−
S is the Hermitian scalar curvature in general. Therefore, (S S)2 isthe moment map square
M −
function, definedonthemodulispaceofalmostcomplexstRructures. Thenthedownwardgradient
flowofthemomentmapnormsquarecanbewrittenas
d 1
J = J ∂¯ X , (1.3)
J S
dt −2 ◦
where X is the symplectic dual vector field of dS. When the flow path of (1.3) locates in the
S
integrable almost complex structures, the Hermitian scalar coincides with the Riemannian scalar
curvature. Therefore, theflow(1.3)isnothing buttheclassical Calabiflow(1.1)uptodiffeomor-
phisms. Basedonthismomentmappicture,Donaldsonthendescribedsomeconjecturalbehaviors
oftheCalabiflow.
2
Conjecture1.1(S.K.Donaldson[30]). SupposetheCalabiflowshaveglobalexistence. Thenthe
asymptotic behavior oftheCalabi flow starting from (M,ω,J)falls into one ofthe fourpossibili-
ties.
1. Theflowconverges toacscKmetriconthesamecomplexmanifold(M,J).
2. The flow is asymptotic to a one-parameter family of extK metrics on the same complex
manifold(M,J),evolvingbydiffeomorphisms.
3. The manifold does not admit an extK metric but the transformed flow J on converges
t
J
to J . Furthermore, one canconstruct adestabilizing testconfiguration of (M,J)suchthat
′
(M,J )isthecentralfiber.
′
4. The transformed flow J on does not converge in smooth topology and singularities de-
t
J
velop. However,onecanstillmakesufficientsenseofthelimitof J toextractaschemefrom
t
it,andthisschemecanbefittedinasthecentralfiberofadestabilizing testconfiguration.
Conjecture 1.1 has attracted a lotof attentions for the study of the Calabi flow. Onthe wayto
understand it, there are many important works. For example, R. Berman [3], W.Y. He [36] and
J. Streets [45] proved the convergence of the Calabi flow in various topologies, under different
geometric conditions. G.Sze´kelyhidi [48]constructed examplesofglobal solutions oftheCalabi
flow which collapse at time infinity. A finite-dimension-approximation approach to study the
Calabiflowwasdeveloped in[32]byJ.Fine.
Note that the global existence of the Calabi flows is a fundamental assumption in Conjec-
ture 1.1. OnRiemann surfaces, the global existence and the convergence of the Calabi flowhave
been proved by P.T. Chrusical [26], X.X. Chen [12] and M. Struwe [47]. However, much less is
knowninhighdimension. Itwasconjectured byX.X.Chen[13]thateveryCalabiflowhasglobal
existence. This conjecture sounds to be too optimistic at the beginning. However, there are posi-
tive evidences for it. In [44], J. Streets proved the global existence of the minimizing movement
flow,whichcanberegardedasweakCalabiflowsolutions. Therefore, theglobalexistence ofthe
Calabi flow can be proved if one can fully improve the regularity of the minimizing movement
flow,althoughthereexistterrificanalyticdifficultiestoachievethis. Ingeneral,Chen’sconjecture
was only confirmed in particular cases. For example, if the underlying manifold is an Abelian
surfaceandtheinitialmetricisT-invariant,H.N.HuangandR.J.Fengprovedtheglobalexistence
in[37].
Inshort,theCalabiflowcanbeunderstood fromtwopointsofview: eitherasaflowofmetric
forms(Calabi’spointofview)withinagivencohomologousclassonafixedcomplexmanifold,or
as a flow of complex structures (Donaldson’s point of view) on a fixed symplectic manifold. Let
(Mn,ω,J)beareference Ka¨hler manifold, gbethe reference metric determined byωand J. If J
isfixed,thentheCalabiflowevolvesinthespace
, ω ϕ C (M),ω = ω+ √ 1∂∂¯ϕ > 0 . (1.4)
ϕ ∞ ϕ
H (cid:26) (cid:12) ∈ − (cid:27)
(cid:12)
(cid:12)
Ifωisfixed,thentheCalabiflowe(cid:12)volvesinthespace
, J J isanintegrable almostcomplexstructure compatible with ω . (1.5)
′ ′
J |
(cid:8) (cid:9)
3
Weequip both and withCk,21 topology for some sufficiently large k = k(n), with respect to
H J
thereferencemetricg. EachpointofviewoftheCalabiflowhasitsownadvantage. Weshalltake
bothpointsofviewandmayjumpfromonetotheotherwithoutmentioning thisexplicitly.
Theorem1.2. Suppose (M2,ω,J)isanextKsurface. Define
, ω TheCalabiflowinitiating fromω hasglobalexistence ,
ϕ ϕ
LH ∈ H|
n o
g , Thepath-connected component of containing ω.
LH LH
g
Then the modified Calabi flow (c.f. Definition 3.1) starting from any ω converges to ̺ ω
ϕ ∗
∈ LH
forsome̺ Aut (M,J),inthesmoothtopology ofKa¨hlerpotentials.
0
∈
In the setup of Conjecture 1.1, we have = = automatically. Therefore, Theo-
LH LH H
rem1.2confirmsthefirsttwopossibilities ofConjecture1.1incomplexdimension2.
g
Theorem 1.3. Suppose (M2,ω,J ),s D is a smooth family of Ka¨hler surfaces parametrized
s
∈
bythedisk D = zz C,nz < 2 ,withthefoollowingconditions satisfied.
{ | ∈ | | }
Thereisasmoothfamilyofdiffeomorphisms ψ : s D 0 suchthat
s
• { ∈ \{ }}
[ψ ω]= [ω], ψ J = J , ψ = Id.
∗s ∗s s 1 1
[ω]isintegral.
•
(M2,ω,J )isacscKsurface.
0
•
Denote
, J s D,theCalabiflowinitiating from(M,ω,J )hasglobalexistence ,
s s
LJ { | ∈ }
g , Thepath-connected component of containing J0.
LJ LJ
g
Then the Calabi flow starting from any J converges to ψ (J ) in the smooth topology of
s ∗ 0
∈ LJ
sectionsofTM T M,whereψ Symp(M,ω)depends on J .
∗ s
⊗ ∈
Theorem 1.3 partially confirms the third possibility of Conjecture 1.1 in complex dimension
2, in the case that the C -closure of the C-leaf of J contains a cscK complex structure, for a
∞ 1
G
polarized Ka¨hler surface. Note that by the integral condition of [ω] and reductivity of the auto-
morphismgroupsofcscKcomplexmanifolds, theconstruction ofdestablizing testconfigurations
followsfrom[29]directly.
Theorem 1.2 and Theorem 1.3 have high dimensional counterparts. However, in high dimen-
sion, due to the loss of scaling invariant property of the Calabi energy, we need some extra as-
sumptionsofscalarcurvature toguarantee theconvergence.
4
Theorem1.4. Suppose (Mn,ω,J)isanextremalKa¨hlermanifold.
Foreachbigconstant A,weset
, ω TheCalabiflowinitiating fromω hasglobalexistence and S A ,
A ϕ ϕ
LH ∈ H| | |≤
n o
g , Thepath-connected component of containing ω,
A A
LH LH
′ , A. g
LH LH
[A>0
ThenthemodifiedCalabiflowstartingfromeachω convergesto̺ ωforsome̺= ̺(ϕ)
ϕ ′ ∗
∈ LH ∈
Aut (M,J),inthesmoothtopology ofKa¨hlerpotentials.
0
Note that by Chen-He’s stability theorem(c.f. [15]), the set is non-empty if A is large
A
LH
enough. Therefore, isanon-empty subsetof . Wehavetherelationships
′
LH LH
. (1.6)
′
LH ⊂ LH ⊂ H
Therefore, inorder to understand the global behavior of theCalabi flow,itis crucial tosetup the
equalities.
= , (1.7)
LH H
= . (1.8)
′
LH LH
Equality(1.7)isnothing buttherestatement ofChen’sconjecture. Equality(1.8)ismoreorlessa
globalscalarcurvature boundestimate.
Theorem 1.5. Suppose (Mn,ω,J ),s D isasmooth family ofKa¨hler manifolds parametrized
s
{ ∈ }
bythedisk D = zz C, z < 2 ,withthefollowingconditions satisfied.
{ | ∈ | | }
Thereisasmoothfamilyofdiffeomorphisms ψ : s D 0 suchthat
s
• { ∈ \{ }}
[ψ ω]= [ω], ψ J = J , ψ = Id.
∗s ∗s s 1 1
[ω]isintegral.
•
(Mn,ω,J )isacscKmanifold.
0
•
Denote
, J s D,theCalabiflowinitiating from J hasglobalexistence and S A ,
A s s
LJ { | ∈ | |≤ }
gA , Thepath-connected component of A containing J0,
LJ LJ
′ , A. g
LJ LJ
[A>0
Then the Calabi flow starting from any J converges to ψ (J ), in the smooth topology of
s ′ ∗ 0
∈ LJ
sectionsofTM T M,whereψ Symp(M,ω)depends on J .
∗ s
⊗ ∈
5
It is interesting to compare the Calabi flow and the Ka¨hler Ricci flow on Fano manifolds at
the current stage. For simplicity, we fix [ω] = 2πc (M,J). Modulo the pioneering work of
1
H.D.Cao([9], global existence) andG.Perelman([43], scalar curvature bound), Theorem 1.4and
Theorem 1.5basically says that theconvergence ofthe Calabi flowcan beasgood asthatfor the
Ka¨hler Ricci flow on Fano manifolds, whenever some critical metrics are assumed to exist, in a
broadersense. TheKa¨hlerRicciflowversionofTheorem1.4andTheorem1.5hasbeenstudiedby
G.TianandX.H.Zhuin[51]and [52],basedonPerelman’sfundamentalestimate. Amoregeneral
apporach was developed by G. Sze´kelyhidi and T.C. Collins in [27]. Our proof of Theorem 1.4
and Theorem 1.5 uses a similar strategy to that in Tian-Zhu’s work [52]. However, our proof is
basedonbackwardregularityimprovementtheorems,Theorem2.22andTheorem2.23,whichare
themaintechnical newingredients ofthispaper.
If the flows develop singularity at time infinity, then the behavior of the Calabi flow and the
Ka¨hler Ricci flow seems much different. Based on the fundamental work of Perelman, we know
collapsing does not happen along the Ka¨hler Ricci flow. In [23] and [24], it was proved by X.X.
Chen and the second author that the Ka¨hler Ricci flow will converge to a Ka¨hler Ricci soliton
flow on a Q-Fano variety. A different approach was proposed in complex dimension 3 in [50],
by G.Tian and Z.L. Zhang. However, under the Calabi flow, G. Sze´kelyhidi [48] has shown that
collapsingmayhappenattimeinfinity,byconstructing examplesofglobalsolutionsoftheCalabi
flow on ruled surfaces. In this sense, the Calabi flow is much more complicated. Of course, this
is not surprising since we do not specify the underlying Ka¨hler class. A more fair comparison
should be between the Calabi flow and the Ka¨hler Ricci flow, in the same class 2πc (M,J), of a
1
given Fano manifold. However, few is known about the Calabi flow in this respect, except the
underlying manifoldisatoricFanosurface (c.f.[17]).
Theorem1.2andTheorem1.3pushthedifficultyoftheCalabiflowstudyonKa¨hlersurfacesto
theproof ofglobal existence, i.e.,Chen’s conjecture. Theorem 1.4and Theorem 1.5indicate that
thestudyoftheCalabiflowwithboundedscalarcurvatureisimportant. Itisnotclearwhetherthe
global existence always holds. If global existence fails, what will happen? In other words, what
is the best condition for the global existence of the Calabi flow? Whether the scalar curvature
bound is enough to guarantee the global existence? In order to answer these questions, we can
borrowideasfromthestudyoftheRicciflow. In[42],N.SesumshowedthattheRicciflowexists
as long as the Ricci curvature stays bounded. Same conclusion holds for the Calabi flow, due to
thework[15]ofX.X.Chen andW.Y.He. However, wecanalso translate Sesum’s result into the
Calabiflowalonganother route. NotethattheCalabiflowsatisfiesequation (1.1). Sothemetrics
evolve by ¯S, the complex Hessian of the scalar curvature. Correspondingly, under the Ricci
flow, the m∇e∇trics evolve by 2R . Modulo constants, we can regard ¯S as the counterpart of
ij
− ∇∇
Ricci curvature in the Calabi flow. Consequently, one can expect that the Calabi flow has global
existence whenever ¯S is bounded. This is exactly the case. Tostate our results precisely, we
|∇∇ |
introduce thenotations
O (t)= sup S , P (t) = sup ¯ S , Q (t) = sup Rm . (1.9)
g | |g(t) g ∇∇ g(t) g | |g(t)
M M (cid:12) (cid:12) M
(cid:12) (cid:12)
(cid:12) (cid:12)
Weshallomitgandtiftheyareclearinthecontext.
Theorem 1.6. Suppose that (Mn,g(t)), T t < 0 is a Calabi flow solution and t = 0 is the
{ − ≤ }
6
singulartime. Thenwehave
limsupPt δ , (1.10)
0
| | ≥
t 0
→
whereδ = δ (n). Furthermore,foreachα (0,1),wehave
0 0
∈
limsupOαQ2 αt C , (1.11)
− 0
| | ≥
t 0
→
whereC =C (n,α). Inparticular, ift = 0isasingular timeoftype-I, thenwehave
0 0
limsupO2t > 0. (1.12)
| |
t 0
→
Theorem 1.6isnothing buttheCalabiflowcounterpart ofthemaintheorems in[53]and[22].
ThetoolsweusedintheproofofTheorem1.6aremotivatedbythestudyoftheanaloguequestion
of the Ricci flow by the second author in [22] and [53]. Actually, the methods in [53] and [22]
werebuiltinaquitegeneralframe. Itwasexpectedtohaveitsadvantageinthestudyofthegeneral
geometricflows.
Thepaperisorganizedasfollows. Insection2,wedeveloptwoconcepts—curvature scaleand
harmonic scale— to study geometric flows. Based on the analysis of these two scales under the
Calabi flow, we show global backward regularity improvement estimates. In section 3, we com-
bine the regularity improvement estimates, the excellent behavior of the Calabi functional along
the Calabi flow and the deformation techniques to prove Theorem 1.2-Theorem 1.5. In section
4, we first show some examples where Theorem 1.2-Theorem 1.5 can be applied and then prove
Theorem1.6.
Acknowledgements: Theauthors would like tothank Professor Xiuxiong Chen, SimonDon-
aldson, Weiyong He, Claude LeBrun and Song Sun for insightful discussions. Haozhao Li and
KaiZhengwouldliketoexpresstheirdeepestgratitudetoProfessorWeiyueDingforhissupport,
guidanceandencouragementduringtheproject. PartofthisworkwasdonewhileHaozhaoLiwas
visitingMITandhewishestothankMITfortheirgenerous hospitality.
2 Regularity scales
2.1 Preliminaries
Let Mn beacompact Ka¨hler manifold of complex dimension nand g aKa¨hler metric on M with
the Ka¨hler form ω. The Ka¨hler class corresponding to ω is denoted by Ω = [ω]. The space of
Ka¨hlerpotentials isdefinedby
(M,ω)= ϕ C (M) ω+ √ 1∂∂¯ϕ > 0 .
∞
H (cid:26) ∈ (cid:12) − (cid:27)
(cid:12)
(cid:12)
(cid:12)
7
In[5]and[6],Calabiintroduced theCalabifunctional
2
Ca(ω ) = S(ω ) S ωn,
ϕ Z ϕ − ϕ
M (cid:16) (cid:17)
where S(ω ) denotes the scalar curvature of the metric ω = ω+ √ 1∂∂¯ϕ and S is the average
ϕ ϕ
−
ofthescalarcurvatureS(ω ). ThegradientflowoftheCalabifunctional iscalledtheCalabiflow,
ϕ
whichcanbewrittenbyaparabolic equation ofKa¨hlerpotentials:
∂ϕ
= S(ω ) S.
ϕ
∂t −
Themetricsevolvebytheequation:
∂
g = S .
∂t i¯j ,i¯j
Consequently, theevolution equations ofmetric-related quantities aredetermined bydirectcalcu-
lation:
∂
Γk = gkl¯ S , (2.1)
∂t ij ϕ∇j il¯
∂
+∆2 R = R (R ) (R R R R )
∂t ϕ! i¯jkl¯ − kl¯r¯s i¯j rs¯− pq¯i¯j qp¯ − ip¯ p¯j kl¯
R R R +(R ) (R ) +(R ) (R ) , (2.2)
− m¯n in¯pl¯ m¯jkp¯ in¯ p n¯jkl¯ p¯ in¯kl¯ p n¯j p¯
n o
∂
+∆2 R = R S ∆ (R R R R ), (2.3)
∂t ϕ! i¯j − i¯jk¯l kl¯− ϕ pq¯i¯j qp¯ − ip¯ p¯j
∂
+∆2 S = Si¯jR . (2.4)
∂t ϕ! − i¯j
Theevolution equation ofcurvature tensor(2.2)canbesimplifiedasfollows:
∂
Rm = ¯ ¯S = ∆2Rm+ 2Rm Rm+ Rm Rm, (2.5)
∂t −∇∇∇∇ − ∇ ∗ ∇ ∗∇
wheretheoperator denotes somecontractions oftensors. Thus,wehavetheinequality
∗
∂
Rm 4S +c(n)Rm 2S . (2.6)
∂t| | ≤ ∇ | | ∇
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
2.2 Estimates basedoncurvature bound
The global high order regularity estimate of the Calabi flow was studied in [17], when Rieman-
niancurvature andSobolevconstant arebounded uniformly. Takingadvantage ofthelocalization
technique developed in[34]and[46],onecanlocalizetheestimatein[17].
Lemma2.1. Suppose (Ωn,g(t)),0 t T isaCalabiflowsolutiononanopenKa¨hlermanifold
{ ≤ ≤ }
Ω, B (x,r)isageodesic completeballinΩ. Suppose
g(T)
C (B (x,r),g(T)) K , (2.7)
S g(T) 1
≤
∂ ∂
sup Rm + g + g K . (2.8)
2
Bg(T)(x,r) [0,T](| | (cid:12)(cid:12)∂t (cid:12)(cid:12) (cid:12)(cid:12)∇∂t (cid:12)(cid:12))≤
× (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
8
Thenforeverypositive integer j,thereexistsC =C(j, 1,K ,K )suchthat
r 1 2
sup jRm(,T) C.
|∇ | · ≤
Bg(T)(x,0.5r)
Proof. Thisfollowsfrom thesameargument asTheorem 4.4of[46]andtheSobolev embedding
theorem. (cid:3)
Lemma 2.2. Let (Mn,g,J) be a complete Ka¨hler manifold with Rm + Rm C . Then there
1
| | |∇ | ≤
exist positive constants r ,r depending only onC ,nsuch that for each p M there is amapΦ
1 2 1
∈
fromtheEuclideanball Bˆ(0,r )inCn to M satisfying thefollowingproperties.
1
(1) Φisalocalbiholomorphic mapfrom Bˆ(0,r )toitsimage.
1
(2) Φ(0) = p.
(3) Φ (g)(0) = g ,whereg isthestandard metriconCn.
∗ E E
(4) r 1g Φ g r g in Bˆ(0,r ).
2− E ≤ ∗ ≤ 2 E 1
Proof. Thisisonlyanapplication ofProposition 1.2ofTian-Yau[49]. Similarapplication canbe
foundin[10]. (cid:3)
Theorem2.3(J.Streets[46]). Suppose (Mn,g(t)),0 t T isaCalabiflowsolution satisfying
{ ≤ ≤ }
sup Rm K.
| |≤
M [0,T]
×
Thenwehave
1+l
1 2
sup lRm (x,t) C K + , (2.9)
x M(cid:12)∇ (cid:12) ≤ √t!
∈ (cid:12)(cid:12)∂l (cid:12)(cid:12) 1 1+2l
sup Rm (x,t) C K + , (2.10)
x M(cid:12)(cid:12)∂tl (cid:12)(cid:12) ≤ √t!
∈ (cid:12)(cid:12) (cid:12)(cid:12)
foreveryt (0,T]andpositiveint(cid:12)(cid:12)eger l. H(cid:12)(cid:12) ereC =C(l,n). Inparticular, wehave
∈
∂ 1
sup Rm (x,T) C K3+ .
x M(cid:12)(cid:12)∂t| |(cid:12)(cid:12) ≤ T32!
∈ (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. Byequation(2.2),weseethat(2.10)followsfrom(2.9). Weshallonlyprove(2.9).
Wearguebycontradiction. Asin[46],wedefinefunction
l
2
f(x,t,g) = jRm 2+j (x).
l ∇ g(t)
Xj=1(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
9
Suppose(2.9)doesnotholduniformlyforsomepositiveintegerl. Thenthereexistsasequenceof
Calabi flowsolutions (Mn,g(t)),0 t T satisfying the assumptions ofthe theorem and there
{ i i ≤ ≤ }
arepoints(x,t) M [0,T]suchthat
i i i
∈ ×
f(x,t,g)
lim l i i i = .
i→+∞ K +t−21 ∞
i
Suppose that the maximum of fl(x,t,gi) on M (0,T] is achieved at (x,t). We can rescale the
K+t−21 × i i
i
metricsby
g˜ (x,t) , λg x,t +λ 2t , λ , f(x,t,g).
i i i −i i l i i i
(cid:16) (cid:17)
Byconstruction, tλ2 1forilargeandtheflowg˜ (t)existsonthetimeperiod[ 1,0]. Moreover,
i i ≥ i −
itsatisfiesthefollowingproperties.
lim sup Rm = 0.
• i→+∞Mi×[−1,0]| |g˜i
sup f(, ,g˜ (t)) 1.
l i
• · · ≤
Mi [ 1,0]
×−
f(x,0,g˜ )= 1.
l i i
•
By Lemma 2.2, we can construct local biholomorphic map Φ from a ball Bˆ(0,r) Cn to M,
i i
⊂
with respect to the metric g˜ (0) and base point x. Note that the radius r is independent of i. Let
i i
h˜ (t) = Φ g˜ (0). Thenweobtain asequence ofCalabiflows (Bˆ(0,r),h˜ (t)), 1 t 0 satisfying
i ∗i i { i − ≤ ≤ }
(2.7)and(2.8),uptoshifting oftime. Furthermore, wehave
l
2
lim Rm (0) = 0, jRm 2+j (0) = 1. (2.11)
i→∞| |h˜i(0) Xj=1(cid:12)∇ (cid:12)h˜i(0)
(cid:12) (cid:12)
(cid:12) (cid:12)
ByLemma2.1,wecantakeconvergence inthesmoothCheeger-Gromovtopology.
Cheeger Gromov C
Bˆ(0,0.5r),h˜ (0) − − ∞ B˜,h˜ (0) .
i
−−−−−−−−−−−−−−−−−→ ∞
(cid:16) (cid:17) (cid:16) (cid:17)
Onone hand, Rm 0on B˜, whichinturn implies that jRm 0on B˜ for each positive
h˜ (0) ≡ ∇ h˜ (0) ≡
integer j. Ontheot∞herhand, takingsmoothlimitof(2.11),weobta∞in
l
2
jRm 2+j (0) = 1.
∇ h˜ (0)
Xj=1(cid:12) (cid:12) ∞
(cid:12) (cid:12)
(cid:12) (cid:12)
Contradiction. (cid:3)
TheproofofTheorem2.3followsthesamelineasthatin[46]byJ.Streets,wedonotclaimthe
originality oftheresult. Weinclude theproofherefortheconvenience ofthereaders andtoshow
the application of the local biholomorphic map Φ, which will be repeated used in the remainder
part of this subsection. Actually, by delicately using interpolation inequalities, the constants in
10