manifolds Generalized Ricci ﬂow II: Existence for complete noncompact Diﬀerential Geometry and its Applications

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Contents lists available atScienceDirect

Diﬀerential Geometry and its Applications

www.elsevier.com/locate/difgeo

Generalized Ricci ﬂow II: Existence for complete noncompact manifolds

Yi Li¹

SchoolofMathematicsandShing-TungYauCenterofSoutheast University,SoutheastUniversity, Nanjing 211189,China

a r t i cl e i n f o a b s t r a c t

Articlehistory:

Received11January2018

Receivedinrevisedform25January 2019

Accepted30May2019 Availableonlinexxxx CommunicatedbyF.Fang MSC:

primary54C40,14E20 secondary46E25,20C20 Keywords:

SteadygradientRiccisolitons Geometricﬂow

Ricci-ﬂatmetrics

Inthispaper,wecontinuetostudythegeneralizedRicciflow.Wegiveacriterionon steadygradientRiccisolitononcompleteandnoncompactRiemannianmanifolds thatisRicci-flat,andthenintroduceanaturalflowwhosestablepointsareRicci-flat metrics. Modifyingtheargument usedby Shi andList,weprove the short time existenceandhigherorderderivativesestimates.

1. Introduction

Ricci-flat metrics play an important role in geometry and physics. For compact Kähler manifold with trivialfirstChernclass,theexistenceofa(Kähler)Ricci-flatmetricwasprovedbyYauinhisfamouspaper [28] on theCalabiconjecture. IntheRiemanniansetting, Ricci-flatmetricsare stationarysolutionsof the RicciflowintroducedbyHamilton[13] asapowerfultool,together withPerelman’sbreakthrough [22–24], tostudy thePoincaréconjecture.

InthestudyofthesingularitiesoftheRicciﬂow,Riccisolitonsnaturallyarisesastheself-similarsolutions.

Fromthedeﬁnition,a Ricci-ﬂatmetricisindeedaRiccisoliton.

E-mailaddress:[email protected].

1 YiLiispartiallysupportedby“theFundamentalResearchFundsfortheCentralUniversities”(China)No.3207019208.

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1.1. Compactsteadygradient Riccisolitons

Inparticular,weconsiderasteadygradientRiccisoliton whichisatriple(M,g,f),whereM isasmooth manifold, gisaRiemannianmetriconM andf isasmoothfunction,suchthat

Ricg+∇²gf = 0 or Rij+∇i∇jf = 0. (1.1) Hamilton[15] showedthatonacompactmanifoldanysteadygradientRiccisolitonmustbeRicci-ﬂat;this, together with Perelman’sresult[22] thatany compact Riccisoliton isnecessarilyagradientRiccisoliton, impliesthatanycompactsteadyRiccisoliton mustbeRicci-ﬂat (cf.[5,22])

1.2. Completenoncompact steadyRiccisolitons

Nowwesuppose(M,g,f) isacompletenoncompactsteadygradientRiccisoliton.Thesimplestexample is Hamilton’scigarsoliton orWitten’sblack hole([9,11]),whichisthecompleteRiemannsurface(R²,gcs) wheregcs:= (dx⊗dx+dy⊗dy)/(1+x²+y²).Ifwedeﬁnef(x,y):=−ln(1+x²+y²),thenRicgcs+∇²gcsf = 0.

Thecigarsolitonisrotationallysymmetric,haspositiveGaussiancurvature,andisasymptotictoacylinder near inﬁnity;moreover,up to homothety,the cigarsoliton is theuniques rotationally symmetricgradient RiccisolitonofpositivecurvatureonR²(cf.[9,11]).Theclassiﬁcationoftwo-dimensionalcompletecompact steady gradientRiccisolitons wasachieved byHamilton[15],whichstatesthatAny completenoncompact steady gradientRiccisoliton withpositiveGaussian curvatureisindeedthecigarsoliton.

The cigarsolitoncanbe generalizedtoarotationally symmetricsteadygradient Riccisolitoninhigher dimensionsonRⁿ.TheresultingsolitonsarereferredtobeBryant’ssolitons(see[11] fortheconstruction), which is rotationally symmetric and has positiveRiemann curvature operator. Other examples of steady gradientRiccisolitons wereconstructedbyCao[4] and Ivey[16].

For three-dimensional case, Perelman [22] conjectured a classification of complete noncompact steady gradientRiccisolitonwithpositivesectionalcurvaturewhichsatisfiesanon-collapsingassumptionatinfinity.

Namely, a three-dimensionalcomplete and noncompact steadygradient Riccisoliton whichis nonﬂatand κ-noncollapsed, is isometric to the Bryant soliton up to scaling. Under some extra assumptions, it was provedin[1,6,7].AcompleteproofwasrecentlyachievedbyBrendle[2] anditsgeneralizationcanbefound in[3].

AnotherimportantresultisChen’sresult[8] sayingthatanycompletenoncompactsteadygradientRicci soliton hasnonnegativescalarcurvature.Forcertaincases,thelowerbounded forthescalarcurvature can be improved [10,12]. When the scalar curvature of a complete steady gradient Ricci soliton achieves its minimum, Petersen andWilliam [25] provedthat suchasoliton must be Ricci-flat.On theother hand,if acomplete noncompact steadygradientRicci solitonhaspositiveRiccicurvature and itsscalarcurvature achieves its maximum, then it must be diffeomorphic to the Euclidean space with the standard metric ([15,5]);inparticular,inthiscase,suchasoliton isRicci-flat.

To removethecurvaturecondition,wecanprovethefollowing

Proposition 1.1. Suppose M is a compact or complete noncompact manifold of dimension n. Then the following conditionsare equivalent:

(i) there existsaRicci-ﬂatRiemannianmetricon M;

(ii) thereexistreal numbers α,β,asmoothfunctionφonM,andaRiemannianmetric gonM suchthat 0 =−R_ij+α∇i∇jφ, 0 = Δ_gφ+β|∇gφ|²g. (1.2) Theproofis giveninsubsection2.1.

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Remark1.2.Inthecompactcase,thesecondconditionin(1.2) canbe removed.However,inthecomplete noncompact case, the second condition in (1.2) is necessarily. For example, the cigar soliton is a steady gradientRiccisolitonwithnonzeroscalarcurvature4/(1+x²+y²).

Theequation(1.2) suggestsus tostudytheparabolic ﬂow

∂_tg(t) =−2Ric_g(t)+ 2α∇²g(t)φ(t), ∂_tφ_t= Δ_g(t)φ(t) +β∇g(t)φ(t)²

g(t). (1.3)

The system(1.3) is similar to the gradientﬂow of Perelman’s entropyfunctional W [22]. Let Met(M) denote the space of smooth Riemannian metrics on acompact smooth manifold M of dimension m. We deﬁnePerelman’sentropyfunctionalW:Met(M)×C^∞(M)×R⁺ −→Rby

W(g, f, τ) :=

M

τ

R_g+|∇gf|²g

+f −m e^−f

(4πτ)^m/2dV_g, (1.4)

wheredV_g standsforthevolumeform ofg.Perelman’sshowedthatthegradientﬂow of(1.4) is

∂_tg(t) =−2Ric_g(t)−2∇²g(t)f(t),

∂_tf(t) =−Δ_g(t)f(t)−R_g(t)+ m

2τ(t), (1.5)

d

dtτ(t) =−1;

moreover,theentropyWisnondecreasingalong(1.5).SinceWisdiﬀeomorphicinvariant,i.e.,W(Φ^∗g,Φ^∗f, τ) =W(g,f,τ) forany diﬀeomorphismsΦ onM,itfollowsthatthesystem(1.5) isequivalentto

∂_tg(t) =−2Ric_g(t),

∂_tf(t) =−Δ_g(t)f(t) +∇g(t)f(t)²

g(t)−R_g(t)+ m

2τ(t), (1.6)

d

dtτ(t) =−1;

Thus, (1.3) is a mixture of (1.5) and (1.6). There also are lots of interesting generalized Ricci ﬂows, for example,see[14,17–21].

1.3. A parabolicﬂow

Inthispaper,weconsider aclass ofRiccﬂowtypeparabolic diﬀerentialequation:

∂tg(t) =−2Ric_g(t)+ 2α1∇g(t)φ(t)⊗ ∇g(t)φ(t) + 2α2∇²g(t)φ(t), (1.7)

∂tφ(t) = Δ_g(t)φ(t) +β1∇g(t)φ(t)²

g(t)+β2φ(t), (1.8)

where α1,α2,β1,β2 are given constants. Whenα1 = α2 =β1 =β2 =φ(t)= 0, the system(1.7)–(1.8) is exactlytheRicciﬂowintroduced byHamilton[13]. Whenα2 =β1=β2= 0,it reducesto List’sﬂow [19].

Recently,Huand Shi[27] introducedastaticﬂowoncomplete noncompactmanifoldthatissimilar toour ﬂow.Themain resultis

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Theorem 1.3. Let (M,g) be an m-dimensional complete and noncompact Riemannian manifold with

|Rm_g|² ≤k₀ on M and φ a smooth functionon M satisfying |φ|²+|∇gφ|²g ≤k₁ and |∇²gφ|²g ≤k₂. Then there exists a positive constant T, depending only on m,k0,k1,k2,α1,α2,β1,β2, such that the -regular (α1,α2,β1,β2)-ﬂow (1.7)–(1.8) withtheinitial data(g,φ) has asmoothsolution (g(t),φ(t))on M×[0,T] andsatisﬁes the followingcurvature estimate. For any nonnegative integer n, there existuniform positive constantsC_k,depending only onm,n,k₀,k₁,k₂,α₁,α₂,β₁,β₂,such that

∇ⁿg(t)Rm_g(t)²

g(t)≤C_n

tⁿ , ∇ⁿ⁺²_g(t)φ(t)²

g(t)≤ C_n tⁿ on M×[0,T].

Forthedefinitionofregularflowand -regularflow,see Definition2.11andSection3.

1.4. Notionsandconvenience

Manifolds are denote by M,N,· · ·. If g is a Riemannian metric on M, we write Rmg,Ricg,Rg,∇g, and dVg the Riemann curvature, Ricci curvature, scalar curvature, Levi-Civita connection, and volume form of g, respectively. We always omit the time variable t in concrete computations. For a family of Riemannian metrics, we denote by Δ_t and dV_t the corresponding Beltrami-Laplace operatorand volume form respectively.

If P and Q are two quantities (may depend on time) satisfying P ≤ CQ for some positive uniform constantC,thenweset PQ.Similarly,wecandeﬁneP ≈QifP QandQP.

WealsousetheEinsteinsummationfortensorﬁelds;forexample, a, bg=aijb^ij :=

1≤i,j≤m

aijb^ij =

1≤i,j,k,≤m

g^ikg^jaijbk

forany two2-tensorﬁeldsa= (aij) andb= (bij) onaRiemannianmanifold (M,g) ofdimensionm.

If A and B are two tensor ﬁelds on a Riemannian manifold (M,g) we denote by A∗B any quantity obtainedfrom A⊗B byoneormoreoftheseoperations(a slightlydiﬀerentfrom thatin[9]):

(1) summation overpairsofmatchingupperandlower indices,

(2) multiplicationbyconstants dependingonlyonthedimensionofM andtheranksofAand B.

WealsodenotebyA^kanyk-foldproductA· · ·A.Theaboveproducta,bgcanbewrittenasa,bg=a∗b;

inorderto stressthemetricg,wealsowrite itasa,bg=g⁻¹∗g⁻¹∗a∗b.

2. Aparabolicgeometricﬂow

Inthissectionweintroduceaparabolicgeometric ﬂowmotivatedby(1.2).AtﬁrstwewillprovePropo- sition1.1

2.1. Acharacterizationof Ricci-ﬂatmetrics

Recall thatasteadygradientRiccisoliton isatriple(M,g,f) satisfying(1.1).

Proposition 2.1.(See also Proposition 1.1) Suppose M is a compact or complete noncompact manifold of dimension m.Thenthefollowingconditionsare equivalent:

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(i) there existsaRicci-ﬂatRiemannianmetriconM;

(ii) there existreal numbersα,β,asmoothfunctionφon M,andaRiemannianmetricg on M suchthat 0 =−Rij+α∇i∇jφ, 0 = Δgφ+β|∇gφ|²g. (2.1) Proof. Onedirection(i)⇒(ii)istrivial,sincewecantakeα=β =φ= 0.Inthefollowing weassumethat theequation(2.1) holdsforsomeα,β,andφ,g.WhenMiscompact,aresultofHamilton[15] tellsusthat g mustbeRicci-ﬂat.NowweassumethatM iscompletenoncompact.

Takingthetraceoftheﬁrstequation in(2.1),weget

Rg=αΔgφ. (2.2)

Inparticular,

Rg=−αβ|∇gφ|²g. (2.3)

Hence, if αβ ≥ 0, then Rg ≤ 0; on the other hand, by a result of Chen [8], we know that any com- pletenoncompactsteady gradientRiccisolitonhasnonnegative scalarcurvature.Together withthosetwo inequalities,wemusthaveR_g= 0 and∇gφ= 0 by (2.3).Consequently,from(2.1),weseethatR_ij = 0.

To dealwiththe caseαβ <0, wetakethederivative ∇ⁱ ontheﬁrst equation of(2.1): 0=−¹₂∇jR_g+ αΔg∇jφ since ∇ⁱRij = ¹₂∇jRg. According to the identity Δg∇jφ = ∇jΔgφ+Rjk∇^kφ, we arrive at 0=−¹₂∇jRg+α(∇jΔgφ+Rjk∇^kφ).Using (2.1) and(2.3),weobtain

0 =∇j

α² 2 −αβ

|∇gφ|²g−1 2Rg

= α(α−β)

2 ∇j|∇gφ|²g.

Inthecaseαβ <0,wemusthaveα= 0,β = 0,andα=β,sotheaboveidentityyields|∇gφ|²g =cforsome constantc, and hence R_g =−αβc using again (2.1). From the proved identity 0=−¹₂∇jR_g+αΔ_g∇jφ, we obtain Δg∇jφ = 0. Consequently 0 = 2∇^jφΔg∇jφ = Δg|∇gφ|²g−2|∇²gφ|²g = 0−2|∇²gφ|²g and then

|∇²gφ|²g= 0.Inparticular, ∇i∇jφ= 0 andhenceRij= 0. Ineachcase,wegetaRicci-ﬂatmetric. 2 2.2. Evolution equations

MotivatedbyProposition2.1, weconsideraclassofRiccﬂow typeparabolic diﬀerentialequation:

∂_tg(t) =−2Ric_g(t)+ 2α₁∇g(t)φ(t)⊗ ∇g(t)φ(t) + 2α₂∇²g(t)φ(t), (2.4)

∂_tφ(t) = Δ_g(t)φ(t) +β₁∇g(t)φ(t)²

g(t)+β₂φ(t), (2.5)

where α₁,α₂,β₁,β₂ are given constants. Whenα₁ = α₂ =β₁ =β₂ =φ(t)= 0, the system(1.7)–(1.8) is exactlytheRicciﬂowintroduced byHamilton [13].Whenα₂=β₁=β₂= 0,itreduces toList’sﬂow [19].

Tocomputeevolutionequationsfor(2.4)–(2.5),werecallvariationformulasstatedin[9].Consideraﬂow

∂tgij =hij where hisafamilyofsymmetric2-tensorﬁelds.Then

∂tg^ij =−g^ikg^jhk, ∂tΓ^k_ij = 1

2g^k(∇ihj+∇jhi− ∇hij),

∂_tR_ijk= 1

2g^p(∇i∇jh_kp+∇i∇kh_jp− ∇i∇ph_jk

− ∇j∇ih_kp− ∇j∇kh_ip+∇j∇ph_ik),

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∂_tR_jk =1

2g^pq(∇q∇jh_kp+∇q∇kh_jp− ∇q∇ph_jk− ∇j∇kh_qp),

∂_tR=−Δ_ttrh+ div (divh)− h,Ric, ∂_tdV_t = 1 2trh dV_t. Wenow takehij:=−2Rij+ 2α1∇iφ∇jφ+ 2α2∇i∇jφ.

Lemma 2.2.Under (2.4)–(2.5),we have

∂_tΓ^k_ij=−∇iR_j^k− ∇jR_i^k+∇^kR_ij+ 2α₁∇i∇jφ· ∇^kφ +α₂∇^k∇i∇jφ−α₂

R_i^k_j^p∇pφ+R_j^k_i^p∇pφ . Proof. Compute

∂tΓ^k_ij =−∇iRjk− ∇jRik+∇^kRij+ 2α1∇i∇jφ· ∇^kφ +α2g^k(∇i∇j∇φ+∇j∇i∇φ− ∇∇i∇jφ). Accordingto theRicciidentity, weobtainthedesiredresult. 2

∂tRij = ΔtRij−2RikR^kj+ 2RpijqR^pq−2α1Rpijq∇^pφ∇^qφ+ 2α1Δ_g(t)φ· ∇i∇jφ

−2α1∇i∇kφ∇^k∇jφ+α2(Rip∇p∇jφ+Rjp∇p∇iφ+∇pRij∇^pφ). Proof. Notethat

∂tRij =−1

2Δthij−1

2∇i∇j(g^pqhpq) +1

2g^pq(∇p∇jhiq+∇p∇ihjq).

I2=∇i∇jR−α1(∇i∇j∇kφ+∇j∇i∇kφ)∇^kφ−2α1∇i∇kφ∇j∇^kφ−α2∇i∇j(Δtφ).

The symmetryofI3 andI4 allowsusto consideronly oneterm, sayingforexampleI3. Since∇p∇j∇iφ=

∇p∇i∇jφ=∇i∇p∇jφ−R^k_pij∇kφ,wehave I3=−1

2∇i∇jR+RpijqR^pq−RikR^kj+α1[∇i∇j∇pφ∇^pφ−Rpijq∇^pφ∇^qφ]

+α1[Δtφ∇i∇jφ+∇i∇pφ(t)∇j∇^pφ+∇iφΔt∇jφ] +α2∇^q∇j(∇i∇qφ). Consequently,we arriveat

∂_tR_ij = Δ_tR_ij−2R_ikR^k_j+ 2R_pijqR^pq−2α₁R_pijq∇^pφ∇^qφ + 2α₁Δ_tφ· ∇i∇jφ−2α₁∇i∇kφ∇^k∇jφ+ Λ, where

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Λ :=−α₂Δ_t(∇i∇jφ)−α₂∇i∇j(Δ_tφ) +α₂∇^q∇j(∇i∇qφ) +α₂∇^q∇i(∇j∇qφ)

=α2 Δt∇i∇jφ− ∇i∇jΔtφ−2Ripjq∇^p∇^qφ− ∇iRjp∇^pφ− ∇jRip∇^pφ+ 2∇pRij∇^pφ

where we used thecontractBianchi identity ∇^pR_jkp =∇jR_k− ∇kR_j inthe last line.Theﬁnal stepis to simplifythe diﬀerence [Δ_g(t),∇i∇j]φ(t). According to the Ricci identity, we get Λ = α₂(R_i∇∇jφ+ Rj∇∇iφ+∇Rij∇φ). 2

Lemma2.4. Under (2.4)–(2.5),wehave

∂_tR_g(t) = Δ_g(t)R_g(t)+ 2Ric_g(t)²

g(t)+ 2α₁|Δ_g(t)φ(t)|²g(t)−2α₁∇²g(t)φ(t)²

g(t)

−4α1

Ric_g(t),∇g(t)φ(t)⊗ ∇g(t)φ(t)

g(t)+α2

∇g(t)R_g(t),∇g(t)φ(t)

g(t). Proof. Bytheaboveformulafor∂tR_g(t),weobtain

∂_tR_g(t)= Δ_g(t)R_g(t)+ 2|Ric_g(t)|²g(t)+ 2α₁|Δ_g(t)φ(t)|²g(t)−2α₁∇²g(t)φ(t)²

g(t)

−2α1R^ij∇iφ(t)∇jφ(t)−2α1

Δ_g(t)∇iφ(t)− ∇iΔ_g(t)φ(t)

∇ⁱφ(t) +I

whereI:=−2α2Δt(Δtφ)+2α2∇ⁱ∇^j(∇i∇jφ)−2α2R^ij∇i∇jφ.BytheRicciidentitywegetI=α2∇tR,∇φ andthedesiredformula. 2

FollowingHamilton, weintroducethetensor ﬁeld

B_ijk :=−g^prg^qsR_ipjqR_krs. (2.6) NotethatBjik =Bijk andBijk=Bkij.

Lemma2.5. Under (2.4)–(2.5),wehave

∂tRijk = ΔtRijk+ 2(Bijk−Bijk+Bikj−Bijk)−(RipRpjk+RjpRipk

+ RkpRijp+RpRijkp) + 2α1 ∇i∇φ∇j∇kφ− ∇i∇kφ∇j∇φ

+ α₂ ∇^pR_ijk∇pφ−R_ijk^p∇p∇φ+R^p_jk∇i∇pφ+R_i^p_k∇j∇pφ+R_ij^p∇k∇pφ

.

Proof. Recall theevolutionequation

∂tR_ijk= 1 2g^p

∇i∇khjp+∇j∇phik− ∇i∇phjk− ∇j∇khip−R^q_ijkhqp−R_ijp^q hkq

where∂tgij =hij.Applyingtheaboveformulatohij =−2Rij+ 2α1∇iφ∇jφ+ 2α2∇i∇jφimplies∂tR_ijk = I1+I2+I3,where

I₁:=g^p

∇i∇pR_jk+∇j∇kR_ip− ∇i∇kR_jp− ∇j∇pR_ik+R^q_ijkR_qp+R^q_ijpR_kq

, I₃:=g^p R^q_ijk

−α₁∇qφ∇pφ−α₂∇q∇pφ

+R^q_ijp

−α₁∇kφ∇qφ−α₂∇k∇qφ

,

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and I₂:= the rest terms.Accordingto[9,13] we have I1= ΔtR_ijk+g^pq

R^r_ijpR_rqk−2R_pik^r R_jqr+ 2R_pirR_jqk^r

− Rir

R_rjk−Rjr

R_irk−Rkr

R_ijr+Rr

R^r_ijk. It canbe showedthat

I₂=−α₁R^q_ijk∇qφ∇φ+α₁R_ij^q∇qφ∇kφ+ 2α₁

∇i∇φ∇k∇jφ− ∇i∇kφ∇j∇φ

+α₂

∇i∇k∇j∇φ+∇j∇∇i∇kφ− ∇j∇k∇i∇φ− ∇i∇∇j∇kφ

; together withI3,wearriveat

I2+I3=−2α1R^q_ijk∇qφ∇φ+ 2α1

∇i∇φ∇k∇jφ− ∇i∇kφ∇j∇φ

+α2

∇^qRijk−Rk

jq∇i∇qφ−Rkiq∇j∇qφ−Rijkq∇q∇φ−Rijq∇k∇qφ

where we used the Ricci identity and the formula ∇iRk

jq+∇jRkiq =−∇^qRijk. Replacing by s in I1,I2,I3,wehave∂tR^s_ijk = ˜I1+ ˜I2+ ˜I3,where wedenotebyI˜i thecorresponding terms;hence

∂tRijk= (−2RsR_ijk^s +gsI˜1) + 2α1R^s_ijk∇φ∇sφ+ 2α2R^s_ijk∇∇sφ+gs( ˜I2+ ˜I3).

Theﬁrst bracketontheright-handside followsfrom Hamilton’scomputation[9,13];theresttermscanbe computed fromtheexpressionsforI˜2and I˜3. 2

Nextwecomputeevolutionequationsforφ(t).

g(t)

− 2α₁|∇g(t)φ(t)|⁴g(t)+ (4β₁−2α₂)

∇g(t)φ(t)⊗ ∇g(t)φ(t),∇²g(t)φ(t)

g(t).

Proof. Using(2.5) wehave∂t∇iφ= Δt∇iφ−Rij∇^jφ+2β1∇^jφ∇i∇jφ+β2∇iφ(t),whereweusetheidentity

∇iΔtφ= Δt∇iφ−Rij∇^jφ.Using(2.4) wethenget

∂t|∇g(t)φ(t)|²g(t) = −2α1|∇g(t)φ(t)|⁴g(t)−2α2∇ⁱ∇^jφ(t)∇iφ(t)∇jφ(t) + 2∇ⁱφ(t)Δ_g(t)∇iφ(t) + 4β1∇i∇kφ(t)∇ⁱφ(t)∇^kφ(t) + 2β2|∇g(t)φ(t)|²g(t)

whichimpliesthedesiredequation. 2 Lemma 2.7.Under (2.4)–(2.5),we have

∂t(∇i∇jφ) = Δt(∇i∇jφ) + 2Rpijq∇^p∇^qφ+β2∇i∇jφ−Rip∇^p∇jφ−Rjp∇^p∇iφ

− 2α1|∇g(t)φ(t)|²g(t)∇i∇jφ+ (2β1−α2)∇kφ∇k∇i∇jφ + 2β₁∇i∇^kφ∇j∇kφ+ 2(β₁−α₂)R_pijq∇^pφ∇^qφ.

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Proof. Compute∂_t(∇i∇jφ) =∇i∇j(∂_tφ)−∂_tΓ^k_ij·∂_kφ.Using (2.5) wehave

∇i∇j(∂_tφ) = ∇k∇i∇j∇^kφ−R_ikj∇∇^kφ+R^k_ik∇j∇φ− ∇iR_j∇φ

−R_j∇i∇φ+β₂∇i∇jφ+ 2β₁

∇i∇^kφ∇j∇kφ+∇^kφ∇i∇j∇kφ . Since∇k∇i∇j∇^kφ= Δt(∇i∇jφ)− ∇^kR_ikj∇φ−R_ikj∇^k∇φ,wehave

∇i∇j(∂tφ) = Δt(∇i∇jφ(t))−Ri∇∇jφ−Rj∇∇iφ+β2∇i∇jφ−(∇iRj

+∇jRi− ∇Rij)∇φ−2R_ikj∇∇^kφ+ 2β1

∇i∇^kφ∇j∇kφ+∇^kφ∇i∇j∇kφ . UsingLemma2.2implies

∂t(∇i∇jφ) = Δt(∇i∇jφ)−2Rikj∇^k∇φ−Ri∇∇jφ−Rj∇∇iφ+β2∇i∇jφ + 2β1∇i∇^kφ∇j∇kφ+ 2β1∇^kφ∇i∇j∇kφ

−2α1|∇g(t)φ(t)|²g(t)∇i∇jφ−α2∇^kφ∇k∇i∇jφ+ 2α2Rikj∇φ∇^kφ.

NowLemma2.7followsfrom ∇i∇j∇kφ=∇k∇i∇jφ−R_ikj∇φ. 2 Lemma2.8. Under (2.4)–(2.5),wehave

∂_t(∇iφ∇jφ) = Δ_t(∇iφ(t)∇jφ)− ∇^kφ(R_ik∇jφ+R_jk∇iφ)

−2∇i∇^kφ∇j∇kφ+ 2β₂∇iφ∇jφ+ 2β₁∇^kφ(∇iφ∇j∇kφ(t) +∇jφ∇i∇kφ). Proof. From theevolutionequationfor∇iφ(t) obtainedintheproof ofLemma2.6,we get

∂_t(∇iφ∇jφ) = ∇jφ

Δ_t∇iφ−R_ik∇^kφ+ 2β₁∇^kφ∇i∇kφ+β₂∇iφ

+∇iφ

Δ_t∇j−R_jk∇^kφ+ 2β₁∇^kφ∇j∇kφ+β₂∇jφ

whichimpliestheequation. 2

2.3. Regular ﬂowsoncompactmanifolds

Let (g(t),φ(t))_t_∈_[0,T) be the solution of (2.4)–(2.5) on acompact m-manifold M with theinitial value (˜g,φ).˜ Deﬁne

˜ c:= max

M |∇˜gφ˜|²˜g, D:= 1

4|2β1−α2|²−α1. (2.7) Proposition 2.9.Suppose (g(t),φ(t))_t∈[0,T) is thesolution of (2.4)–(2.5) onacompact m-manifoldM with theinitialvalue (˜g,φ).˜ Thenwehave

(1) Case1:4β1−2α2= 0.

(1.1) If α1>0and β2>0,then

|∇g(t)φ(t)|²g(t)≤ cβ˜ ₂e^2β²^t

˜

cα1e^2β²^t+ (β2−˜cα1).

(10)

(1.2) If α₁>0andβ₂≤0,then|∇g(t)φ(t)|²g(t)≤˜c.

(1.3) If α1= 0,then |∇g(t)φ(t)|²_g(t)≤ce˜^2β²^t.

(1.4) If α1<0andβ2≤0,then|∇g(t)φ(t)|²_g(t)≤˜c/(1+ 2α1˜ct).

(1.5) If α₁<0andβ₂>0,then

∇g(t)φ(t)²

g(t)≤ ˜cβ₂e^2β²^t β2+ ˜cα1(e^2β²^t−1). (2) Case2:4β₁−2α₂= 0.

(2.1) If D <0and β₂>0,then

|∇g(t)φ(t)|²g(t)≤ ˜cβ2e^2β²^t

−cDe˜ ^2β²^t+ (β₂+ ˜cD). (2.2) If D <0and β₂≤0,then |∇g(t)φ(t)|²g(t)≤c.˜

(2.3) If D= 0,then|∇g(t)φ(t)|²_g(t)≤˜ce^2β²^t.

(2.4) If D >0and β₂≤0,then |∇g(t)φ(t)|²g(t)≤c/(1˜ −2Dct).˜ (2.5) If D >andβ₂>0,then

∇g(t)φ(t)²

g(t)≤ ˜cβ2e^2β²^t β2−˜cD(e^2β²^t−1).

Proof. For any time t, we have ∇φ⊗ ∇φ,∇²φ ≤ |∇φ|²|∇²φ|. By Lemma 2.6 we have ∂t|∇φ|² ≤ Δ|∇φ|²+ 2β2|∇φ|²−2|∇²φ|²−2α1|∇φ|⁴+|4β1−2α2||∇φ|²|∇²φ|. For convenience, set u:= |∇φ|² and v:=|∇²φ|.Then

∂tu≤Δtu+ 2β2u−2v²−2α1u²+|4β1−2α2|uv.

(1) Case1:4β1−2α2= 0.Inthis case,theaboveinequalitybecomes

∂_tu≤Δ_tu+ 2β₂u−2α₁u². If α₁ ≥0,then∂_tu≤Δ_tu+ 2β₂uand ∂_t

e^−2β²^tu

=e^−2β²^t

−2β₂u+∂_tu

≤Δ

e^−2β²^tu

from whichwe obtainu≤˜ce^2β²^tbythemaximumprinciple.

If α₁ <0 and β₂ ≤0,then ∂_tu≤Δ_tu−2α₁u² and u_t ≤ _1+2α^u(0)₁_u(0)t ≤ _1+2α^c^˜₁_ct_˜. On the other hand,if β₂>0,then u≤˜cβ₂e^2β²^t/[β₂+ ˜cα₁(e^2β²^t−1)] sincethesolutiontotheordinarydiﬀerentialequation

U(t) = 2β₂U(t)−2α₁U²(t), U(0) = ˜c is oftheform

U(t) = cβ˜ 2e^2β²^t β₂+ ˜cα₁(e^2β²^t−1).

(2)Case2:4β1−2α2= 0.Usingtheinequalityab≤a²+₄¹b² foranya,b≥0 andanypositivenumber ,weobtain

∂tu≤Δtu+ 2β2u−(|4β1−2α2| −2)v²+

|4β₁−2α₂| 4 −2α1

u².

Choosing:= 2/|4β₁−2α₂|implies∂_tu≤Δ_tu+ 2β₂u+ 2Du², where D isgivenin(2.7).This isjustthe case(1)ifwereplaceα₁by−D.Thefollowingdiscussioncanbeobtained. 2