Morse-Novikov cohomology of almost nonnegatively curved manifolds

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Advances in Mathematics

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Morse-Novikov cohomology of almost nonnegatively curved manifolds

Xiaoyang Chen

SchoolofMathematicalSciences,InstituteforAdvancedStudy,TongjiUniversity, Shanghai,China

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

Articlehistory:

Received9July2019

Receivedinrevisedform26April 2020

Accepted21May2020 Availableonline9June2020 CommunicatedbytheManaging Editors

Keywords:

Morse-Novikovcohomology Almostnonnegativecurvature

LetMⁿbeaclosedmanifoldofalmostnonnegativesectional curvature and nonzero first de Rham cohomology group.

Using a topological argument, we show that the Morse- NovikovcohomologygroupH^p(Mⁿ,θ) vanishesforanypand [θ]∈ H_dR¹ (Mⁿ),[θ] = 0. Based on a new integral formula, wealsoshowthatasimilarresultholdsforaclosedmanifold of almostnonnegative Riccicurvature underthe additional assumptionthatitscurvatureoperatorisuniformlybounded frombelow.

©2020ElsevierInc.Allrightsreserved.

1. Introduction

LetMⁿbeasmoothmanifoldandθarealvaluedclosedoneformonMⁿ.SetΩ^p(Mⁿ) thespaceofrealsmoothp-formsanddefined_θ: Ω^p(Mⁿ)→Ω^p+1(Mⁿ) asd_θα=dα+θ∧α forα∈Ω^p(Mⁿ).Thenwehaveacomplex

· · · →Ω^p−1(Mⁿ)−→^dθ Ω^p(Mⁿ)−→^dθ Ω^p+1(Mⁿ)→ · · ·

E-mailaddress:xychen100@tongji.edu.cn.

https://doi.org/10.1016/j.aim.2020.107249 0001-8708/©2020ElsevierInc. Allrightsreserved.

whose cohomologyH^p(M,θ)=H^p(Ω^∗(Mⁿ),dθ) iscalled thep-th Morse-Novikovcoho- mologygroupofMⁿwithrespecttoθ.Ifθ1,θ2aretworepresentativesinthecohomology class [θ],then H^p(M,θ1)H^p(M,θ2).HenceH^p(M,θ) only dependsonthe deRham cohomology class of θ. This cohomology shares many propertieswith the ordinary de Rham cohomology.See[11,18,19] andsection2fordetails.

If [θ] = 0,the Novikov cohomology group H^p(M,θ) is isomorphic to the de Rham cohomology group H_dR^p (Mⁿ).There arelots of workrelating de Rham cohomology to curvature properties of Riemannian manifolds. See for example [20]. In particular, a celebrated theorem of Gromov says that the Betti number of a closed manifold with almost nonnegativesectionalcurvatureis boundedabovebyaconstantdependingonly the dimension of the manifold [10]. Here we saythat a Riemannianmanifold Mⁿ has almostnonnegativesectionalcurvatureifitadmitsasequenceofRiemannianmetricsgi

suchthat

sec(g_i)≥ −1 i D(g_i)≤1,

where sec(gi) isthesectionalcurvatureofgi andD(gi) isthediameterofgi.

However,therearequitefewworkdiscussingtherelationshipbetweenMorse-Novikov cohomology H^p(M,θ) and curvature when [θ] = 0. This paper is trying to make an attempttowardsthisdirection.Ourfirstresultisthefollowingtheorem.

Theorem 1.1.LetMⁿ beaclosed Riemannianmanifoldof almostnonnegativesectional curvature andnonzero first de Rham cohomology group, then theMorse-Novikov coho- mology H^p(M,θ)= 0forany p(includingp= 0)and any[θ]∈H_dR¹ (Mⁿ),[θ]= 0.

From the workin[8,15],we knowthataclosed Riemannianmanifold Mⁿ of almost nonnegative sectional curvature is an almost nilpotent space. Namely, there is afinite cover of Mⁿ, denoted by Mˆⁿ, such that π₁( ˆMⁿ) is a nilpotent group that operates nilpotently on π_k( ˆMⁿ) for every k ≥ 2. Recall that an action by automorphisms of a group G on an abelian group V is called nilpotent if V admits a finite sequence of G-invariantsubgroups

V =V₀⊃V₁⊃V₂⊃ · · · ⊃V_k = 0

suchthattheinducedactionofGonVj/Vj+1istrivialforanyj.NowTheorem1.1isa consequenceofthefollowingtopological result.

Theorem 1.2. Let Mⁿ be a smooth manifold with nonzero first de Rham cohomol- ogy group. If Mⁿ is an almost nilpotent space, then the Morse-Novikov cohomology H^p(M,θ)= 0 forany pandany [θ]∈H_dR¹ (Mⁿ),[θ]= 0.

ByTheorem 2.1 in section2, we see thatn

p=0(−1)^pdimH^p(Mⁿ,θ) isequal to the EulercharacteristicnumberofMⁿ. Hencewegetthefollowing

Corollary 1.3. Let Mⁿ be a smooth manifold with nonzero first de Rham cohomology group.IfMⁿ isanalmostnilpotentspace,thenitsEulercharacteristicnumbervanishes.

Corollary 1.3implies thataclosed Riemannianmanifold of almost nonnegative sec- tional curvature and nonzero first de Rham cohomology group has vanishing Euler characteristic number. This result has previously been proved by Yamaguchi in [23]

usingcollapsingtheory.

Theorem 1.1 fails for closed manifolds of almost nonnegative Ricci curvature. Re- call thataRiemannianmanifold hasalmost nonnegative Riccicurvature ifit admitsa sequenceofRiemannianmetricsgi suchthat

Ric(g_i)≥ −n−1 i D(g_i)≤1,

where Ric(gi) is the Riccicurvature of gi and D(gi) is the diameterof gi. LetM⁴ be themanifoldperformingsurgery alongameridiancurve inT⁴,i.e., removingatubular neighborhood of the curve and attaching acopy of D²×S². In [1], Anderson showed thatM⁴admits asequenceofRiemannianmetricsgi suchthat

|Ric(g_i)| ≤ n−1 i D(gi)≤1.

Moreover,itsfundamentalgroupisisomorphictoZ³anditsEulercharacteristicnumber isnonzero.Forany[θ]∈H_dR¹ (M⁴),[θ]= 0,byTheorem2.1andTheorem2.3insection2, we getH^p(M⁴,θ)= 0 forp= 2 and H²(M⁴,θ)= 0.However, thesectional curvature of g_i constructed by Anderson can nothave a uniform lower bound. Otherwise, there will be also an upper bound of the sectional curvature and by Theorem 1 in[22], M⁴ will fiberover S¹ which is impossible bythe construction. Inparticular, the curvature operatorofgicannothaveauniformlowerbound.BythefollowingTheorem1.4andits Corollary1.5, M⁴ infactcannotadmit asequenceof Riemannianmetricsg_i ofalmost nonnegativeRiccicurvaturewithcurvatureoperatoruniformlybounded frombelow.

Theorem 1.4. Let Mⁿ be a closed Riemannian manifold with nonzero first de Rham cohomologygroupand admitsasequenceof Riemannianmetricsg_i suchthat

Ric(gi)≥ −n−1 i D(gi)≤1.

If the curvature operator of gi is uniformly bounded from below by −Id, then for any [θ]∈H_dR¹ (Mⁿ),[θ]= 0,there existssome t∈R,t= 0 suchthat H^p(M,tθ)= 0 forany p,whereH^p(M,tθ)istheMorse-Novikovcohomologygroup withrespecttotθ.

Corollary 1.5.Let Mⁿ be a closed Riemannian manifold with nonzero first de Rham cohomologygroup.IfMⁿadmitsasequenceofRiemannianmetricsofalmostnonnegative Ricci curvature with curvature operator uniformly bounded from below, then the Euler characteristic numberof Mⁿ vanishes.

ForaclosedRiemannianmanifold(Mⁿ,gi) withalmost nonnegativeRiccicurvature and nonzerofirst deRham cohomologygroup,Theorem 1 in [22] alsoimpliesthatMⁿ hasvanishingEuler numberifthesectionalcurvature ofgi hasauniformupperbound.

Theorem1 in[22] wasprovedbycollapsingtheoryandisquitedifferentfromourmethod inthispaper.

It has been known that the fundamental group of a closed manifold M of almost nonnegativeRiccicurvatureisalmostnilpotent[3,16].ByTheorem2.3,H¹(M,θ)= 0 for any[θ]= 0 withoutanyadditionalassumption.See[14] forrelatedworkonnoncollapsed almost Ricciflatmanifolds.

Finally, wepointoutthatforaclosed RiemannianmanifoldM ofnonnegative Ricci curvature andnonzerofirstdeRhamcohomologygroup,thentheMorse-Novikovcoho- mology H^p(M,θ)= 0 foranypand[θ]∈H_dR¹ (M),[θ]= 0.Infact,byCheeger-Gromoll splitting theorem [5], afinite coverof M is diffeomorphicto aproduct of a torus and a simplyconnected manifold. ByTheorem 2.1 andExample 1, wesee thatthe Morse- Novikovcohomology H^p(M,θ)= 0 for anypand[θ]∈H_dR¹ (M),[θ]= 0.

The proofof Theorem 1.2is basedonCartan-Lerayspectral sequence onequivalent homology[4].Bypassingtoafinitecover,wecanassumethatMⁿ isanilpotentspace.

The closedoneform θ onMⁿ defines alinearrepresentationof thefundamentalgroup of Mⁿ:

ρ:π1(Mⁿ)→GL(1,C) =C^∗,[γ]→e

γθ

.

TherepresentationρdefinesacomplexrankonelocalsystemC_ρoverMⁿ[6].Wedenote byH^p(Mⁿ,Cρ) thep-thcohomologygroupofMⁿwithcoefficientsinthislocalsystem.

ByTheorem2.2 insection2,foranyp,wehave

H^p(Mⁿ, θ)H^p(Mⁿ,Cρ).

By duality, it suffices to show that H_p(Mⁿ,C_ρ) = 0, where H_p(Mⁿ,C_ρ) is the p-th homology groupof Mⁿ with coefficientsinthis local system.Letπ:Mⁿ →Mⁿ be the universalcoverofMⁿ.BytheCartan-Lerayspectralsequence[4],wehave

E_kl² =Hk(π1(Mⁿ), Hl(Mⁿ,C))⇒Hk+l(Mⁿ,Cρ), (1.1)

whereHk(π1(Mⁿ),Hl(Mⁿ,C)) isthek-th homologygroupofπ1(Mⁿ) with coefficients intheπ1(Mⁿ)-moduleHl(Mⁿ,C).Then weprovebyinductionto getthevanishing of Hp(Mⁿ,Cρ).

Theproof of Theorem 1.4 is basedon Hodgetheory of Morse-Novikov cohomology.

Let d^∗ be the formal L² adjoint of d with respect to the Riemannian metric g_i. We canalso define an operator d^∗_θ as the formal L² adjoint of d_θ with respect to g_i. Fur- ther,Δθ=dθd^∗_θ+d^∗_θdθ isthecorrespondingLaplacian.Theseoperatorsarelower-order perturbations of thecorresponding operators inthe usual Hodge-de Rham theory and therefore havemuch thesameanalytic properties. Forexample, the usualproof of the Hodgedecompositiontheoremgoesthrough,andoneobtainsanorthogonaldecomposi- tion

Ω^p(Mⁿ) =H^p(Mⁿ)⊕dθ(Ω^p−1(Mⁿ))⊕d^∗_θ(Ω^p+1(Mⁿ)),

whereH^p(Mⁿ) isthespaceofΔθharmonicforms,whichisisomorphictoH^p(Mⁿ,θ).

By Hodge theory, for each i we can choose a harmonic form θi inthe cohomology class[θ].LetV(gi) bethevolumeof(Mⁿ,gi),dVithevolumeformofgiandXithedual vectorfield ofθ_i definedby g_i(X_i,Y)=θ(Y).Set t_i = ( ^V(gi)

M n|Xi|²dVi)^1/2>0.Choose a Δtiθi harmonicform αi inH^p(Mⁿ,tiθi).Theideaistoshowthatαi≡0 forsufficiently largei,whichreliesonthefollowingcrucial integralinequalityprovedinCorollary4.3.

Mⁿ

t²_i|Xi|²|αi|²dVi≤Cn

Mⁿ

(ti|∇Xi|+ti|div(Xi)|)|αi|²dVi (1.2)

forsomeconstantCn dependingonlyonn.

AsRic(g_i)≥ −ⁿ⁻_i¹,applyingBochnerformulato X_i,weget

Mⁿ

|∇X_i|²dV_i ≤n−1 i

Mⁿ

|X_i|²dV_i. (1.3)

Combining (1.2) and(1.3),forsufficientlylargeiwewillshow

Mⁿ

|α_i|²dV_i≤ 1 2

Mⁿ

|α_i|²dV_i.

Henceαi≡0.Seesection5fordetails.

Acknowledgments

Theauthor ispartially supportedby NationalNaturalScienceFoundation of China No. 11701427 and the Fundamental Research Funds for the Central Universities. He thanksProfessorBinglongChen,JohnLottandAndreiPajitnovforhelpfuldiscussions.

2. Basicproperties ofMorse-Novikovcohomology

Inthissection wecollectsomebasicpropertiesofMorse-Novikovcohomology.

Theorem2.1. LetMⁿ beacompactn-dimensionalmanifoldandθ aclosedoneformon Mⁿ.Then:

(1) If θ =θ+df,f ∈C^∞(Mⁿ,R),then for any p,we have H^p(Mⁿ,θ) H^p(Mⁿ,θ) and theisomorphismisgivenby themap[α]→[e^fα];

(2) If [θ] = 0 and Mⁿ is connected and orientable, then H⁰(Mⁿ,θ) and Hⁿ(Mⁿ,θ) vanish. Moreover, theintegration

:H^p(Mⁿ,θ)×H^n−p(Mⁿ,−θ),(α,β)→

Mⁿα∧β induces anisomorphism H^p(Mⁿ,θ)(H^n−p(Mⁿ,−θ))^∗.

(3) n

p=0(−1)^pdimH^p(Mⁿ,θ)isequaltotheEuler characteristicnumberof Mⁿ; (4) IfN^d bead-dimensionalmanifoldandγbe aclosed oneformon N^d,thenwehave H^k(Mⁿ×N^d,π₁^∗θ+π^∗₂γ)

p+q=kH^p(Mⁿ,θ)

H^q(N^d,γ),where π1:Mⁿ×N^d→ Mⁿ,π2:Mⁿ×N^d→N^d are theprojection maps.

(5) If π : Mⁿ → Mⁿ is a covering space with finite sheet, then π^∗ : H^p(Mⁿ,θ) → H^p(Mⁿ,π^∗θ) isinjectiveforany p.

Proof. Seepage 476-480 in[11] and Proposition1.2 in[18] for the proof of parts 1-4.

Forpart5,byTheorem 2.2,wehave

H^p(Mⁿ, θ)H^p(Mⁿ,C_ρ),

where C_ρ isthecomplexrankonelocalsystemdefinedbythelinearrepresentation ρ:π₁(Mⁿ)→GL(1,C) =C^∗,[γ]→e

γθ

and H^p(Mⁿ,C_ρ) is the p-th cohomology group of Mⁿ with coefficients in this local system.

As π:Mⁿ →Mⁿ is acoveringspace with finitesheet,onecanconstruct atransfer map (seee.g.[9,12]) h:H^p(Mⁿ,π^∗Cρ)→H^p(Mⁿ,Cρ) such that hπ^∗ =kId, where k is the degreeof π. It follows that π^∗ : H^p(Mⁿ,θ)H^p(Mⁿ,Cρ)→ H^p(Mⁿ,π^∗Cρ) H^p(Mⁿ,π^∗θ) isinjective.

As acorollaryofTheorem 2.1,weget

Example 1.Let Mⁿ be n-dimensionaltorus,thenH^p(Mⁿ,θ)= 0 forany pand[θ]= 0 byTheorem 2.1.

LetθbeaclosedoneformonMⁿ.Considerthefollowinglinearrepresentationofthe fundamentalgroupofMⁿ:

ρ:π1(Mⁿ)→GL(1,C) =C^∗,[γ]→e

γθ

.

TherepresentationρdefinesacomplexrankonelocalsystemCρoverMⁿ[6].Wedenote byH^p(Mⁿ,Cρ) thep-thcohomologygroupofMⁿ withcoefficientsinthislocalsystem.

Theorem2.2. H^p(Mⁿ,θ)H^p(Mⁿ,Cρ)forany p.

Proof. The proof is contained in [19]. For the convenience of the reader, we provide the details here. Let π : Mⁿ → Mⁿ be the universal cover of Mⁿ. The cohomology groupsH^p(Mⁿ,C_ρ) areisomorphicto H_ρ^p(Mⁿ),thecohomologygroupsofthecomplex Ω(Mⁿ,ρ),consisting oftheρ-equivariantdifferentialforms onMⁿ relative tothe usual differential(theproof isanalogous to thesheaf-theoreticproof ofde Rham’s theorem).

Let h be afunction on Mⁿ such that dh = π^∗θ. We give amapping F : Ω^∗(Mⁿ) → Ω^∗(Mⁿ,ρ) by the formula F(w) = e^hπ^∗w. It is easy to see that F is one-to-one and commuteswith thedifferentials.Hence

H^p(Mⁿ, θ)H_ρ^p(Mⁿ)H^p(Mⁿ,C_ρ).

Theorem2.3. LetMⁿ be an-dimensionalmanifold andθ a closed oneform on Mⁿ.If thefundamental groupofMⁿ has afinitelygenerated nilpotentsubgroup offinite index, thenH¹(Mⁿ,θ)=Hⁿ⁻¹(Mⁿ,θ)= 0 forany [θ]= 0.

Proof. Let G⊆π1(Mⁿ) be afinitely generated nilpotent subgroup of finite index and π: Mⁿ →Mⁿ the coveringspace of Mⁿ with π₁(Mⁿ) G. The closed oneform π^∗θ definesalinearrepresentationof G:

ρ:G=π₁(Mⁿ)→GL(1,C) =C^∗,[γ]→e

γπ^∗θ

.

TherepresentationρdefinesacomplexrankonelocalsystemCρoverMⁿ.Wedenoteby H^p(Mⁿ,Cρ) thep-thcohomologygroupofMⁿ withcoefficientsinthelocal systemCρ. LetK(G,1) bethetopologicalspacesuchthatπ1(K(G,1))=G,πi(K(G,1))= 0,i≥2 andLρthecomplexrankonelocalsystemoverK(G,1) definedbyρ.Sincetheclassifying mapMⁿ →K(G,1) inducesoverQacohomologyisomorphismindegreeone,weget

H¹(Mⁿ,C_ρ)H¹(K(G,1),L_ρ).

Asπ:Mⁿ→Mⁿisafinitecover,[θ]= 0 impliesthat[π^∗θ]= 0.ThenL_ρisanontrivial localsystemoverK(G,1).AsGisafinitelygenerated nilpotentgroup,byTheorem2.2 in[17], foranyp,wehave

H^p(K(G,1),Lρ) = 0.

Inparticular,

H¹(Mⁿ,Cρ)H¹(K(G,1),Lρ) = 0.

ByTheorem2.1 andTheorem2.2,we have

H¹(Mⁿ, π^∗θ) = 0 H¹(Mⁿ, θ) = 0

Hⁿ⁻¹(Mⁿ, θ)H¹(Mⁿ,−θ) = 0.

For a smooth manifold which is not an almost nilpotent space, its Morse-Novikov cohomology doesnotnecessarilyvanishas thefollowing exampleshows.

Example 2.[15] Leth:S³×S³→S³×S³ bedefinedby h: (x, y)→(xy, yxy).

This mapisadiffeomorphismwithinversegivenby h⁻¹: (u, v)→(u²v⁻¹, vu⁻¹).

LetM bethemappingtorus ofh.ThenM hasthestructureofafiberbundle:

S³×S³→M→S¹.

Theinducedmap h^∗,3 onH_dR³ (S³×S³) isgivenbythematrix Ah= 1 1

1 2

(2.1) Notice thatthe eigenvaluesof Ah are different from 1 inabsolute value. HenceMⁿ is not an almost nilpotent space. Letλ be aeigenvalue ofAh with λ=e^−t,t= 0,t ∈R and θ ageneratorof H_dR¹ (M). WeclaimthatH³(M,tθ)= 0. Tosee this, observe that tθ defines alinearrepresentationofthefundamentalgroupofM:

ρt:π1(M)→GL(1,C) =C^∗,[γ]→e^t

γθ

.

The representation ρ_t defines a complex rank one local system Cρt over Mⁿ [6]. We denote byH^p(Mⁿ,Cρt) thep-thcohomologygroupofMⁿ withcoefficientsinthislocal system.ByTheorem2.2insection2,foranyp,wehave

H^p(M, tθ)H^p(Mⁿ,C_ρt).

On the other hand,by Wang’sexactsequence in Proposition6.4.8 in[6] page 212,we have

dimCH^p(Mⁿ,Cρt) = dimCker(h^∗,p−e^−tId) + dimCcoker(h^∗,p−1−e^−tId),

where h^∗,p :H^p(S³×S³,C)→H^p(S³×S³,C) isthelinearmap inducedby h. Ase^−t isaneigenvalueof h^∗,3,weseethatdimCker(h^∗,3−e^−tId)>0 andH³(M,tθ)= 0.

3. Cartan-Lerayspectralsequence

In this section we apply Cartan-Leray spectral sequence to proveTheorem 1.2. By passingto a finite cover, wecan assumethat Mⁿ isa nilpotent space. Theclosed one formθ inducesalinearrepresentation ofG=π1(Mⁿ):

ρ:π₁(Mⁿ)→GL(1,C) =C^∗,[γ]→e

γθ

.

ByTheorem 2.2,foranyp,wehave

H^p(Mⁿ, θ)H^p(Mⁿ,Cρ),

where Cρ is the complex rank one local system over Mⁿ defined by ρ. By duality, it suffices to prove the vanishing of Hp(Mⁿ,Cρ), which is the homology group of Mⁿ with coefficients in the local system Cρ. Let Mⁿ be the universal cover of Mⁿ. The representationρtogetherwiththeGactiononMⁿ bydecktransformationinduces the diagonalactiononHl(Mⁿ,C)Hl(Mⁿ,Z)⊗C.BytheCartan-Lerayspectralsequence (Theorem7.9,page173in[4]),wehave

E_kl² =Hk(G, Hl(Mⁿ,C))⇒Hk+l(Mⁿ,Cρ),

where Hk(G,Hl(Mⁿ,C)) is the k-th homology groupof G with coefficients in the G- moduleHl(Mⁿ,C).See[4] formoredetailsofhomologyofgroups.Forus, weonlyneed thefollowinglongexactsequence(Proposition6.1,page71in[4]).

Lemma3.1. Foranyshortexact sequence0→M→M →M→0ofG-modules,there isthefollowinglongexact sequence:

· · · →H_i(G, M)→H_i(G, M)→H_i(G, M)→H_i−1(G, M)→H_i−1(G, M)→ · · ·

→H₁(G, M)→H₁(G, M)→H₁(G, M)→H₀(G, M)→H₀(G, M)

→H₀(G, M)→0.

As Mⁿ is a nilpotent space, then G = π1(Mⁿ) is a nilpotent group that operates nilpotentlyonπm(Mⁿ) foreverym≥2.ByLemma2.18in[13],Goperatesnilpotently onH_l(Mⁿ,Z) foreveryl,thatisV =H_l(Mⁿ,Z) admitsafinitesequenceofG-invariant subgroups

V =V0⊇V1⊇. . . Vk = 0

such thatthe inducedactionofGonVj/Vj+1 istrivialforany j.Therepresentation ρ ofGinducesadiagonalactiononVj⊗C andwehavethefollowingshortexactsequence of Gmodules:

0→Vj+1⊗C→Vj⊗C→Vj/Vj+1⊗C→0.

Wenow proveH_k(G,V_j⊗C)= 0 foranyj byinduction.It isclearthatH_k(G,V_k⊗ C)= H_k(G,0) = 0. As [θ] = 0, we see thatρ is a nontrivial representation of G. By assumption,theinduced actionofGonVj/Vj+1 is trivialforanyj. Thenthediagonal action of GonVj/Vj+1⊗C is nontrivial.As Gisafinitely generated nilpotent group, byTheorem 2.2in[17],weget

Hk(G, Vj/Vj+1⊗C) = 0.

ByLemma3.1andinduction,foranyj,weget H_k(G, V_j⊗C) = 0.

Inparticular,

H_k(G, H_l(Mⁿ,C)) =H_k(G, V₀⊗C) = 0.

BytheCartan-Lerayspectralsequence[4],wehave

E_kl² =H_k(G, H_l(Mⁿ,C))⇒H_k+l(Mⁿ,Cρ).

Henceforanyk,l≥0,wehave

Hk+l(Mⁿ,Cρ) = 0.

Then wegetH^p(Mⁿ,θ)= 0 foranypand[θ]= 0.

4. Anintegral formulaofΔ_θharmonicforms

In sectionwe derive anintegralformula ofΔθ harmonicforms whichwill be crucial intheproofofTheorem1.4.

Let (Mⁿ,g) bea closedRiemannian manifold and θ aclosed real oneform on Mⁿ. Define dθ : Ω^p(Mⁿ)→Ω^p+1(Mⁿ) as dθα=dα+θ∧αfor α∈Ω^p(Mⁿ).Let d^∗ be the formalL²adjointofdwithrespecttog.Wecanalsodefineanoperatord^∗_θastheformal L²adjointofd_θwithrespecttog.Further,Δ_θ=d_θd^∗_θ+d^∗_θd_θisthecorrespondingLapla- cian. These operators are lower-order perturbations of the corresponding operators in theusualHodge-deRhamtheoryandthereforehavemuchthesameanalyticproperties.

For example, the usual proof of the Hodge decomposition theorem goes through, and oneobtainsanorthogonaldecomposition

Ω^p(Mⁿ) =H^p(Mⁿ)⊕dθ(Ω^p−1(Mⁿ))⊕d^∗_θ(Ω^p+1(Mⁿ)),

whereH^p(Mⁿ) isthespaceofΔθharmonicforms,whichisisomorphictoH^p(Mⁿ,θ).

LetdV bethevolumeformofgandX thedualvectorfieldofθdefinedbyg(X,Y)= θ(Y).ChooseaΔ_θ harmonicformαinH^p(Mⁿ,θ).Then

d_θα=dα+θ∧α= 0 d^∗_θα=d^∗α+iXα= 0.

The following integral formula and its Corollary 4.3 will be crucial in the proof of Theorem1.4.

Theorem4.1.

Mⁿ

|X|²|α|²dV =1 2

Mⁿ

α∧[LX,∗]α,

where[L_X,∗]α=L_X∗α− ∗L_XαandL_XαistheLiederivative ofαinthedirectionX.

Remark 4.2.When θ is exact and X = ∇f for some smooth function f on Mⁿ, we believethat the integral formula inTheorem 4.1 is the sameas [7]. It is also possible toadaptthemethodin[7] toproveTheorem4.1.However,wepresentadifferent proof here.

Corollary4.3.

Mⁿ

|X|²|α|²dV ≤C_n

Mⁿ

(|∇X|+|div(X)|)|α|²dV

forsomeconstant Cn depending only onn.

Proof. TheRiemannianmetricgonMⁿinducesalinearmapbetweenT MⁿandT^∗Mⁿ definedby

g:T Mⁿ→T^∗Mⁿ

< g(X), Y >=g(X, Y),∀X, Y ∈T Mⁿ.

Let g⁻¹ be the inverse of the above map g and h the endomorphism of the bundle T^∗Mⁿ→Mⁿ by

h=LXg◦g⁻¹.

ThederivationoftheGrassmannalgebraΛT^∗Mⁿinducedbyhisdenotedbyi(h).This is alinearmapsuchthat,ifγ∈T^∗Mⁿ, theni(h)(γ)=h(γ),and

i(h)(ω1∧ω2) = (i(h)ω1)∧ω2+ω1∧(i(h)ω2) (4.1) forany ω1,ω2∈ΛT^∗Mⁿ. Thefollowing formulaisprovedin[21].

[LX,∗]ω= (i(h)−1

2T rh)∗ω (4.2)

forany ω∈ΛT^∗Mⁿ.

Letdiv(X) bethedivergenceofX withrespectto g.As (L_Xg)(Y, Z) = g(∇YX, Z) + g(Y,∇ZX)

forallY,Z∈T Mⁿ,we seethatT rh= 2div(X).ThenbyTheorem4.1,we get

Mⁿ

|X|²|α|²dV ≤Cn

Mⁿ

(|∇X|+|divX|)|α|²dV

forsomeconstantC_n depending onlyonn.

Now weproveTheorem 4.1.Wefirstlyneedthefollowinglemmas.

Lemma 4.4.Forany pformω,wehave

∗iXω= (−1)^p−1θ∧ ∗ω, (4.3) where ∗istheHodgestaroperatorwith respecttog.

Proof. Foranyp−1 formξ,wehave

Mⁿ

ξ∧ ∗iXω=

Mⁿ

g(ξ, iXω)dV

=

Mⁿ

g(θ∧ξ, ω)dV =

Mⁿ

θ∧ξ∧ ∗ω

= (−1)^p−1

Mⁿ

ξ∧θ∧ ∗ω.

Hence

∗iXω= (−1)^p−1θ∧ ∗ω.

Lemma4.5. Letβ=∗α,then

dβ−θ∧β= 0.

Proof. Asd^∗α= (−1)^n(p+1)+1∗d∗αand d^∗α+i_Xα= 0,weget (−1)^n(p+1)+1∗d∗α+i_Xα= 0.

Hence

(−1)^n(p+1)+1∗ ∗d∗α+∗iXα= 0.

ByLemma4.4,wehave

∗iXα= (−1)^p−1θ∧ ∗α.

Itfollowsthat

(−1)^pd∗α+ (−1)^p−1θ∧ ∗α= 0 So

dβ−θ∧β= 0.

Nowweproceed toproveTheorem 4.1.Asdα+θ∧α= 0,weget iXdα+iX(θ∧α) = 0.

So

iXdα∧β+|X|²α∧β−θ∧iXα∧β = 0. (4.4) Ontheotherhand,asdβ−θ∧β= 0,we get

i_Xdβ−i_X(θ∧β) = 0.

So

iXdβ∧α− |X|²β∧α+θ∧iXβ∧α= 0.

Then

α∧iXdβ− |X|²α∧β+ (−1)^pθ∧α∧iXβ = 0. (4.5)

By(4.4),(4.5),weget

−iXdα∧β+α∧iXdβ−2|X|²α∧β+θ∧iXα∧β+ (−1)^pθ∧α∧iXβ = 0. (4.6) Combined with

θ∧i_Xα∧β+ (−1)^pθ∧α∧i_Xβ=θ∧i_X(α∧β)

=|X|²α∧β−iX(θ∧α∧β) =|X|²α∧β, we get

−i_Xdα∧β+α∧i_Xdβ=|X|²α∧β. (4.7) Since

d(iXα∧β) =diXα∧β+ (−1)^p−1iXα∧dβ, we get

Mⁿ

i_Xα∧dβ= (−1)^p

Mⁿ

di_Xα∧β. (4.8)

Ontheother hand,wehave

0 =i_X(α∧dβ) =i_Xα∧dβ+ (−1)^pα∧i_Xdβ. (4.9) Combining (4.8),(4.9),weget

Mⁿ

α∧i_Xdβ=−

Mⁿ

di_Xα∧β. (4.10)

From (4.7),(4.10),weget

Mⁿ

|X|²α∧β=−

Mⁿ

iXdα∧β−

Mⁿ

diXα∧β =−

Mⁿ

LXα∧β

=−

Mⁿ

LX(α∧β) +

Mⁿ

α∧LXβ=

Mⁿ

α∧LXβ. (4.11)

As β=∗α,weget

Mⁿ

α∧LXβ =

Mⁿ

α∧LX∗α=

Mⁿ

α∧ ∗LXα+

Mⁿ

α∧[LX,∗]α. (4.12)

Moreover,

Mⁿ

α∧ ∗L_Xα=

Mⁿ

L_Xα∧ ∗α

=

Mⁿ

LX(α∧ ∗α)−

Mⁿ

α∧LX∗α=−

Mⁿ

α∧LX∗α

=−

Mⁿ

α∧ ∗LXα−

Mⁿ

α∧[LX,∗]α.

Hence

Mⁿ

α∧ ∗L_Xα=−1 2

Mⁿ

α∧[L_X,∗]α. (4.13)

By(4.11),(4.12),(4.13),weget

Mⁿ

|X|²|α|²dV =1 2

Mⁿ

α∧[L_X,∗]α.

5. ProofofTheorem1.4

Inthis sectionwe giveaproof ofTheorem 1.4. Theproof isbasedon Corollary4.3.

Anothercrucialtoolis thefollowing Poincaré-Sobolevinequality([2],page397).

Theorem5.1. Let(Mⁿ,g)be a closed smoothRiemannian manifoldsuchthat forsome constant b>0,

r_min(g)D²(g)≥ −(n−1)b²,

whereD(g)isthediameterof g,Ric(g) istheRiccicurvature ofg and

rmin(g) =inf{Ric(g)(u, u) :u∈T M, g(u, u) = 1}. LetR= _bC(b)^D(g),whereC(b)istheuniquepositiveroot oftheequation

x b

0

(cht+xsht)ⁿ⁻¹dt= π

0

sinⁿ⁻¹tdt.

Thenforeach 1≤p≤_n−q^nq ,p<∞and f ∈W^1,q(Mⁿ),we have f− 1

V(g)

Mⁿ

f dVp≤Sp,qdfq

fp≤Sp,qdfq+V(g)^1/p−1/qfq,

where V(g)isthevolumeof (Mⁿ,g),S(p,q)= (V(g)/vol(Sⁿ(1))^1/p−1/qRΣ(n,p,q) and Σ(n,p,q)is theSobolevconstant of thecanonicalunitsphereSⁿ defined by

Σ(n, p, q) =sup{fp/dfq:f ∈W^1,q(Sⁿ), f = 0,

Sⁿ

f = 0}.

Let p = _n−2²ⁿ ,q = 2 in Theorem 5.1 and apply Theorem 3and Proposition6 in [2]

pages395-396, thenwegetthefollowingmeanvalueinequality.

Theorem 5.2.Let n ≥ 3 and (Mⁿ,g) be a closed n-dimensional smooth Riemannian manifold suchthatforsome constant b>0,

rmin(g)D²(g)≥ −(n−1)b².

If f ∈W^1,2(Mⁿ)isa nonnegative continuousfunction suchthat fΔf ≥ −cf² (here Δ is anegativeoperator)inthesense of distribution forsomepositivenumber c,then

max_x∈Mⁿ|f|²(x)≤B_n(σ_nRc^1/2)

Mⁿf²dV V(g) ,

where σ_n =vol(Sⁿ)^1/nΣ(n,_n−2²ⁿ ,2) andB_n :R₊→R₊ isa functiondefinedby

Bn(x) = ∞ i=0

(xνⁱ(2νⁱ−1)^−1/2+ 1)^2ν−i, ν = n n−2. The functionBn satisfiestheinequalities

Bn(x)≤exp(2x√ ν/(√

ν−1)),0≤x≤1 Bn(x)≤Bn(1)x^2ν/(ν−1), x≥1.

In particular,lim_x→0+Bn(x)= 1 andBn(x)≤Bn(1)xⁿ forx≥1.

Let Mⁿ be aclosed Riemannian manifold with nonzero first de Rham cohomology groupandadmits asequenceofRiemannianmetricsg_i suchthat

Ric(gi)≥ −n−1 i D(g_i)≤1.

Moreover,thecurvatureoperatorofg_iisuniformlyboundedfrombelowby−Id.Forany [θ]∈H_dR¹ (Mⁿ),[θ]= 0, we aregoing to provethatthereexists somet ∈R,t= 0 such

thatH^p(Mⁿ,tθ)= 0 forany p. Ifn= 2,sincethefirstBetti numberofM²isbounded by2 (seee.g.[2]),thegenusofM²isatmost1 andH^p(M²,tθ)= 0 byExample1.Now weassumethatn≥3.Letd^∗ betheformalL² adjointofdwithrespecttogi.ByHodge theory,wecanchooseaharmoniconeformθ_iinthecohomologyclass[θ].Then

dθi= 0 d^∗θ_i = 0

θ_i= 0.

Let ti = ( ^V(gi)

M n|Xi|²dVi)^1/2 > 0, where V(gi) is the volume of (Mⁿ,gi), dVi is the volume form of gi, |Xi|² = gi(Xi,Xi) and Xi is thedual vector field of θi defined by g_i(X_i,Y) = θ(Y). We claim that for sufficiently large i, H^p(Mⁿ,t_iθ_i) = 0 for any p.

ChooseaΔ_tiθi harmonicform α_i inH^p(Mⁿ,t_iθ_i).Then dαi+tiθi∧αi= 0 d^∗αi+itiXiαi= 0.

Thegoalis to provethatα_i = 0. ByTheorem 2.1, wecanassume that1≤deg(α_i)≤ n−1.AsRic(g_i)≥ −ⁿ⁻¹_i ,applyingBochnerformula toX_i [20], weget

1

2Δ|Xi|²=|∇Xi|²+Ric(gi)(Xi, Xi)≥ |∇Xi|²−n−1

i |Xi|², (5.1) whereΔ istheLaplacianactingonfunctionswhichisanegativeoperator.Then

Mⁿ

|∇Xi|²dVi ≤n−1 i

Mⁿ

|Xi|²dVi. (5.2)

Letdiv(Xi) bethedivergenceofXi withrespect togi.Asθiisaharmoniconeform, weseediv(X_i)= 0 (seee.g.Proposition31in[20] page206).ByCorollary4.3,wehave

Mⁿ

t²_i|X_i|²|α_i|²dV_i≤C_n

Mⁿ

t_i|∇X_i||α_i|²dV_i, (5.3)

for someconstantC_n depending only onn. Applying Hölder’s inequalityon (5.3) and using(5.2),weget

Mⁿ

t²_i|X_i|²|α_i|²dV_i≤C_n

Mⁿ

t_i|∇X_i||α_i|²dV_i

≤Cn(

Mⁿ

t²_i|∇Xi|²dVi)¹²(

Mⁿ

|αi|⁴dVi)¹²