The Q-curvature flow in a closed CR 3-manifold

The Q-curvature flow in a closed CR 3-manifold

Takanari Saotome

Department of Mathematics, Academia Sinica, Taipei 10617, Taiwan

e-mail : tsaotome@math.tsukuba.ac.jp Shu-Cheng Chang

Department of Mathematics, National Taiwan University, Taipei 10617, Taiwan

Taida Institute for Mathematical Sciences (TIMS), National Taiwan University,

Taipei 10617, Taiwan

e-mail : scchang@math.ntu.edu.tw

(2010 Mathematics Subject Classification : 52V20, 53C44.)

Abstract. Let (M, J,[θ0]) be a closed CR 3-manifold. We deform the contact form ac- cording to a CR analogue ofQ-curvature flow. This is a fourth-order evolution equation.

We show that the solution of CR Q-curvature flow exists for all time and has smoothly asymptotic convergence onM×[0,∞) under certain conditions. In particular, we obtain existence theorems for zero CRQ-curvature in a closed CR 3-manifold.

1 Introduction

Let (M³, J, θ) be a 3-dimensional closed pseudohermitian manifold (see next section for more details). In the paper of [6], [9], [11], and [10], the CR Paneitz operator P with respect to (J, θ) is defined by

P λ= ∆²_bλ+T²λ+ 4Im(A₁₁λ11+A_11,1λ1) and the CRQ-curvature is defined by

(1.1) Q=−2

3(∆bW + 2ImA_11,11),

Note that for (M³, J) being the boundary of a bounded strictly pseudoconvex do- main inC²,the CR Paneitz operatorPappears as the compatibility operator for the Key words and phrases: CR Manifold, Subelliptic, Embedable, Tanaka-Webster Cur- vature, Torsion, Q-curvature, Paneitz operator, Sub-Laplacian, Kohn Laplacian.

* This work was supported in part by the NSC of Taiwan.

57

degenerate Laplacian in the paper of [10] and a constant multiple of CRQ-curvature is the coefficient of logarithmic singularity of Fefferman’s asymptotic expansion of the Szeg¨o kernel ([11], [6]).

For a real valued function λon (M³, J,[θ₀]), we consider a conformal transfor- mationθ=e^2λθ₀. Under this conformal change, we have :

(1.2) P=e^−4λP0 and Q=e^−4λ(Q0+ 2P0λ),

where P0 and Q0 denote the CR Paneitz operator and the CR Q-curvature with respect to (J, θ0), respectively. For 3-dimensional pseudohermitian manifolds, it follows from the definition of theQ-curvature (1.1) that

(1.3)

∫

M

Q θ∧dθ= 0.

Then it is natural to ask if (M³, J,[θ₀]) admits a contact form θ =e^2λθ₀ so that theQ-curvature vanishes pointwise.

Conjecture 1.1. ([6]) There exists a contact form θ in [θ0] with zero CR Q- curvature in a closed CR3-manifold(M, J,[θ0])with(Q0)ker= 0, where(Q0)kerde- notes the component ofQ0 inkerP0 with respect to the decompositionC^∞(M;R) = kerP0⊕(kerP0)^⊥.

In this paper, we obtain a partial answer to this conjecture by using method of the CRQ-curvature flow. We define the energy functional Eon (M³, J,[θ0]) by

(1.4) E(θ) =

∫

M

P0λ·λdµ0+

∫

M

Q0λdµ0

withθ=e^2λθ₀ and dµ₀=θ₀∧dθ₀. Then, for minimizingE(θ) in [θ₀],we consider the fourth-order CRQ-curvature flow in a closed CR 3-manifold (M³, J,[θ₀]) :

(1.5)















∂λ

∂t =−(Q^⊥₀ + 2P0λ) +r, θ=e^2λθ₀; λ(p,0) =λ₀(p),

∫

M

e^4λ0dµ0=

∫

M

dµ0.

Here r =V⁻¹

∫

M

(Q^⊥₀ + 2P0λ)dµ, V =

∫

M

dµ. Note that dµ=e^4λdµ0 and V =

∫

M

dµ is invariant under the flow (1.5). Moreover, (1.5) is the negative gradient flow ofE(θ).(see Lemma 4.2)

Concerning global existence and a priori estimates for the solution of (1.5), we introduced the following notions.

Definition 1.1. (i) On a closed pseudohermitian (2n+ 1)-manifold (M, J, θ), we call the CR Paneitz operatorP nonnegative if

∫

M

P φ·φ dµ≥0, for all realC^∞-functions φ.

(ii) On a closed pseudohermitian manifold (M, J, θ), we call the CR Paneitz operator P essentially positive if there exists a constant Υ>0 such that

(1.6)

∫

M

P φ·φ dµ≥Υ

∫

M

φ² dµ,

for all realC^∞-functionsφ∈(kerP)^⊥ (i.e. perpendicular to the kernel ofP in the L²-norm with respect to the volume formdµ).

Definition 1.2. We say that condition (∗) is satisfied if there exist constants 0≤ ϵ <1 andC(ϵ)≥0 such that

(∗)

∫

M

|∇²bφ|²dµ≤(2 +ϵ)

∫

M

(∆_bφ)²dµ+C(ϵ)

∫

M

φ²dµ, for all realC^∞-functionφ∈(kerP)^⊥.

In this paper, we will show the following two theorems:

Theorem 1.2. Let (M, J,[θ₀])be a closed CR 3-manifold. Suppose that condition (∗) and (∆_b)₀(kerP₀) ⊂ kerP₀ are satisfied and the CR Paneitz operator P₀ is essentially positive. Then the solution of (1.5) exists on M ×[0,∞)and converges smoothly to λ_∞ ≡λ(·,∞) as t → ∞. Moreover, the contact form θ_∞ =e^2λ∞θ0

has the CR Q-curvature with

Q_∞=e^−4λ∞(Q0)ker. In additional if(Q₀)_ker= 0, thenQ_∞= 0.

Theorem 1.3. Let (M, J,[θ0]) be a closed CR 3-manifold. Suppose that M is embeddable in C² and the CR Paneitz operatorP0 is nonnegative. Then the same result as Theorem 1.2 holds.

Remark 1.4. (i)Note that if the pseudohermitian torsion is vanishing, thenM is embeddable and the CR Paneitz operator P is nonnegative ([16], [3]).

(ii)In the paper [4], it is shown that under the condition of nonnegativity ofP, the embeddability of a closed CR 3-manifold M follows from the positivity of the CR Yamabe constant.

2 Preliminary

We introduce some basic materials in a pseudohermitian 3-manifold (see [14], [15] for more details). LetM be a closed 3-manifold with an oriented contact form θ withP= kerθ. The Reeb vector field ofθ is the unique vector fieldT such that θ(T) = 1 andLTθ= 0. A CR structure is a smooth endomorphismJ :P−→Psuch thatJ²=−Id. The CR structureJdecomposesC⊗P=T1,0⊕T0,1. A CR structure is called integrable, if the condition [T1,0, T1,0]⊂T1,0is satisfied. A pseudohermitian (CR) structure is a integrable CR structure J together with a global contact form θ. Let {T, Z₁, Z¯1} be a frame of C⊗T M, where Z₁ ∈ T_1,0, Z¯1 = Z₁ ∈ T_0,1. Then {θ, θ¹, θ^¯1}, the coframe dual to {T, Z1, Z¯1}, satisfies dθ = ih_1¯1θ¹∧θ^¯1 for some nonzero real function h_1¯1. If h_1¯1 is positive, we call such a pseudohermitian structure (J, θ) positive. Throughout this paper, we assume pseudohermitianCR structure is positive andh_1¯1= 1.

The pseudohermitian connection∇of (M, J, θ) is given in terms of a local frame Z1∈T1,0 by

dθ¹=θ¹∧θ₁¹+θ∧τ¹ τ¹≡0 modθ^¯1 (2.1)

θ01=θ0

¯1= 0, θ11+θ¯1

¯1= 0,

We callτ¹the pseudohermitian torsion, and putτ¹=A¹¯1θ^¯1. The scalar curvature of∇is called Tanaka-Webster curvature and is denoted by W.

We define a canonical metric g_θ on (M, J, θ) by

g_θ(X, Y) =−dθ(X, J Y) +θ(X)θ(Y).

To consider smoothness for functions, we recall below what the Folland-Stein space S^k,p(M) is. Let D denote a differential operator acting on functions. We say D has weightm, denotedw(D) =m,ifmis the smallest integer such thatD can be locally expressed as a polynomial of degreemin vector fields tangent to the contact bundleP.We define the Folland-Stein spaceS^k,p(M) of functions onM by

S^k,p(M) ={φ∈L^p(M)|Dφ∈L^p(M) wheneverw(D)≤k}.

So it is natural to define the S^k,p-norm∥φ∥S^k,p ofφas follows:

∥φ∥S^k,p≡



 ∑

0≤j≤k

∥∇^jbφ∥^pL^p





1/p

.

The function space S^k,p(M) with the above norm is a Banach space fork≥0,1<

p <∞. There are also embedding theorems of Sobolev type. For instance,S^2,2 ⊂ S^1,4 (for dimM = 3). We refer the reader to, for instance, [8] and [7] for more discussions on these spaces.

3 Subelliptic Estimate for the CR Paneitz Operator

Suppose that (M³, θ, J) is a closed pseudohermitian 3-manifold.

Definition 3.1. ([14]) We define an operatorP1that characterizes CR-pluriharmonic functions by P1φ=φ¯1

¯1

1+iA11φ¹.

The CR Paneitz operatorP is defined byP φ= 2(

δb((P1φ)θ¹) +δb((P1φ)θ^¯1)) , where δb is the divergence operatorδb(σ1θ¹) =σ1,1.

We first recall an intergral CR Bochner formula as following.

Lemma 3.1. ([3]) For any λ∈C^∞(M),we have

(3.1) 0 =

∫

M

(∆_bλ)²dµ−

∫

M

|∇²bλ|²dµ+ 2

∫

M

(λ₀)²dµ−ρ+ 2α.

Here, α= Im

∫

M

A¹¹λ1λ1dµandρ=

∫

M

W|∇bλ|²dµ.

Proposition 3.2. We assume the CR Paneitz operator P satisfies the condition (∗)and ∆b(kerP)⊂kerP. Then P is subelliptic on the orthogonal complement of kerP. That is

∥λ∥²S^4,2 ≤C(∥P λ∥²L²+∥λ∥²L²), for some constant C independent of λ∈(kerP)^⊥.

Proof. By the definition of CR Paneitz operator, we have

∫

M

P λ·λ dµ=

∫

M

|∆_bλ|²dµ−

∫

M

|λ₀|²dµ−8α.

Using the CR integral Bochner formula (Lemma 3.1), and the condition (∗),

(3.2) (1−ϵ)

∫

M

|∆bλ|²dµ−2

∫

M

P λ·λ dµ−ρ−6α≤0.

On the other hand, by Young inequality, we can estimate the term ofαandρby C₁α+C₂ρ≤ϵ

∫

M

|∆_bλ|²dµ+C(ϵ)

∫

M

|λ|²dµ.

Hence from (3.2) we have , (1−ϵ−ϵ^′)

∫

M

|∆bλ|²dµ−2

∫

M

P λ·λ dµ−C(ϵ, ϵ^′)

∫

M

|λ|²dµ≤ 0. Therefore,

(3.3)

∫

M

|∆_bλ|²dµ≤ 2 ϵ₀

∫

M

P λ·λ dµ+C(ϵ₀)

∫

M

|λ|²dµ.

Since the sub-Laplacian ∆_b is subelliptic onC^∞(M), there exists a constantC >0 such that

(∗∗) ∥λ∥²S^2,2 ≤C[(P λ, λ)L²+∥λ∥²L²].

forλ∈(kerP)^⊥.

Next it follows from [10] thatP = ∆²_b+T²−2S^′, S^′λ=i((A₁₁λ¹)^,1−(A11λ¹)^,1).

Therefore,

[∆b, P]λ= [∆b,∆²_b+T²−2S^′]λ= [∆b, T²]λ−2[∆b, S^′]λ.

Since the order of the operatorS^′ is 2, we can ignore the second part of the above.

Moreover, we have [∆b, T²]λ= [∆b, T]T λ+T[∆b, T]λ= [(ImS)T+T(ImS)]λ,for Sλ= 2i(A₁₁λ¹)^,1. Hence we have∥[∆b, P]λ∥L² ≤C∥λ∥S^4,2.

It follow from (∗∗) that

(3.4)

∥∆_bλ∥S^2,2 ≤ C[(P∆_bλ,∆_bλ)_L2+∥∆_bλ∥²_L2]

≤ C((P λ,∆²_bλ)_L2+ϵ∥λ∥²_S4,2+C(ϵ)∥λ∥²_S2,2)

≤ C[(ϵ+ϵ^′)∥λ∥²_S4,2+C(ϵ^′)∥P λ∥²_L2+C(ϵ)∥λ∥²_S2,2].

These estimates show that

(3.5) ∥λ∥²S^4,2≤C(∥P λ∥²L²+∥λ∥²L²),

for some constantCindependent ofλ∈(kerP)^⊥.

Next, we recall that if the range of∂b is closed inL²(M), we have a subelliptic estimate for∂_b on the orthogonal complement of kerb ([12], [13]).

Proposition 3.3. If the range of ∂_b is closed in L²(M), then the CR Paneitz operator P is subelliptic on the orthogonal complement of kerP.

Proof. Since the Kohn Laplacian (for functions) isb=−∆_b+iT, we have P λ=bbλ−2Sλ=bbλ−2Sλ.

We observe that the kernel of P₁ is just the space of all CR-pluriharmonic functions. Letw=u+iv∈kerb, u, v∈C^∞(M;R). By Lemma 3.1. of [14],

P1u=P1v= 0 and u, v∈kerP.

Therefore if λ∈(kerP)^⊥∩C^∞(M;R),then it holds that (λ, w)L² = (λ, u)L²+i(λ, v)L² = 0.

This concludes that

(kerP)^⊥⊂(kerb)^⊥.

Letλ∈(kerP)^⊥∩C^∞(M;R). Thenλ∈(kerb)^⊥∩(kerb)^⊥, so that by the closedness of the range of b, there exists λ^′ ∈(kerb)^⊥ ⊂ C^∞(M;C) such that λ=bλ^′.Since kerb⊂kerP,ifw∈kerb,then we have

(bλ, w)_L2= (λ^′,bbw)_L2 = (λ^′,2Sw)_L2.

In particularbλ−2Sλ^′∈(kerb)^⊥.Moreover, sinceλ^′∈(kerb)^⊥, we have,

∥Sλ^′∥L²≤C∥λ^′∥S^2,2 ≤C^′∥λ∥L². Therefore

∥bλ−2Sλ^′∥²S^2,2 ≤ 2(∥bλ∥²S^2,2+ 4∥Sλ^′∥²S^2,2)

≤ C^′(∥bλ∥²S^2,2+∥λ∥²S^2,2) (3.6)

≤ C^′′∥bλ∥²S^2,2.

Similarly, ∥bλ∥²S^2,2 ≤ C∥bλ−2Sλ^′∥²S^2,2. It follows from the subellipticity of b,b on their orthogonal complement of the kernel, forλ∈(kerP)^⊥

∥λ∥S^4,2 ≤ C₁∥bλ∥S^2,2 ≤C₂∥bλ−2Sλ^′∥S^2,2

≤ C3∥b(bλ−2Sλ^′)∥L²

≤ C₄(∥bbλ∥L²+∥λ∥L²).

Hence bb has the subellipticity on (kerP)^⊥. Since P = bb−2S and S is a lower order operator, we obtain the subelliptic estimate forP on (kerP)^⊥.

As a consequence of Proposition 3.3 and J.J.Kohn’s results of embeddability ([12], [13]), one obtains

Corollary 3.4. Let(M, J, θ)be a closed pseudohermitian3-manifold. Suppose that M is embeddable, then the CR Paneitz operator P is subelliptic on(kerP)^⊥.

4 A Priori Uniformly Estimates

LetT be the maximal time for a solutionλof the flow (1.5) onM×[0,T). We will derive the uniformlyS^4k,2-norm estimate for the solutionλ.

First we recall a CR pseudohermitian Moser inequality ([3]). Let∥ · ∥p denote theL^p-norm with respect to the volume formdµ0.

Lemma 4.1. ([3]) Let (M, J, θ0) be a closed pseudohermitian 3-manifold. Then there exist constants C, κ,andν such that for all φ∈C^∞(M), there holds

(4.1)

∫

M

e^φdµ₀≤Cexp (

κ∇⁰bφ⁴

4+ν∥φ∥⁴4

) .

Lemma 4.2. Let λbe a solution of the flow (1.5) onM×[0,T). Then there exists a constant β=β(Q₀, θ₀)>0 such that

(4.2) E(θ) =

∫

M

P₀λ·λdµ₀+

∫

M

Q₀λdµ₀≤β², for allt≥0.

Proof. It follows from (1.3) and the second formula of (1.2) that

(4.3)

d

dtE(θ) = 2

∫

M

P0λ·∂λ

∂t dµ0+

∫

M

Q0

∂λ

∂t dµ0

= −

∫

M

(Q^⊥₀ + 2P0λ)²dµ0<0.

In particular, the functionalEis non-increasing along the flow (1.5).

We also observe a following elementaly fact.

Lemma 4.3. Let f : [0,T)−→R be a C¹-function satisfyingf^′ ≤ −L1f+L2 for some constantsL1, L2>0. Then, fort∈[0,T),

f(t)≤f(0)e^−L1t+L2

L1

.

Now we writeλ=λker+λ^⊥, Q0= (Q0)ker+Q^⊥₀ according to the decomposition C^∞(M) = kerP0⊕(kerP0)^⊥.Comparing both sides of the flow (1.5), we obtain

(4.4) ∂λ^⊥

∂t =−(Q^⊥₀ + 2P0λ^⊥) and ∂λker

∂t =r(t).

Proposition 4.4. Let (M, J, θ0) be a closed pseudohermitian 3-manifold and λ be a solution of the flow (1.5). Suppose that the CR Paneitz operator P0 of θ0 is subelliptic on(kerP0)^⊥ and essentially positive. Then there exists a positive constant C(k,Υ, θ0, β)>0 such that

∥λ∥S^4k,2≤C(k,Υ, θ0, β) for allt≥0.

Proof. We devide the proof in three parts.

Part 1: From the second formula of (4.4), we haveλ_ker(t,·)−λ_ker(0,·) =

∫ t 0

rdt.

Since

∫ t 0

rdt

∫

M

Q₀dµ₀= 0,

∫

M

Q0λkerdµ0=

∫

M

Q0[λker(0, p) +

∫ t 0

rdt]dµ0=

∫

M

Q0λker(0, p)dµ0≥ −C.

Furthermore it follows from the self-adjointness of P₀ that

∫

M

P₀λ^⊥·λ_kerdµ₀ = 0, and

E(θ) =

∫

M

P₀λ·λdµ₀+

∫

M

Q₀λdµ₀≥

∫

M

P₀λ^⊥·λ^⊥dµ₀+

∫

M

Q₀λ^⊥dµ₀−C.

It then follows from Lemma 4.2 and the essensial positivity of P0 that there exists a constantC(Υ, Q₀, θ₀)>0 such that

(β²+C)≥

∫

M

P0λ^⊥·λ^⊥dµ0+

∫

M

Q0λ^⊥dµ0≥Υ 2

∫

M

(λ^⊥)²dµ0−C(Υ, Q^⊥₀, θ0).

So we have theL²-estimate,

∫

M

(λ^⊥)²dµ₀≤C for allt≥0.

Part 2: Next, we show theS^4k,2-estimate forλ^⊥−λ^⊥, whereλ^⊥=V₀⁻¹

∫

M

λ^⊥dµ0

is the mean value with respect todµ0. We compute, for all positive integersk, d

dt

∫

M

|P₀^kλ^⊥|²dµ0=−2

∫

M

(P₀^kλ^⊥)(P₀^kQ^⊥₀)dµ0−4

∫

(P₀^kλ^⊥)(P₀^k+1λ^⊥)dµ0.

By essential positivity ofP0,we have

∫

M

(P₀^kλ^⊥)(P₀^k+1λ^⊥)dµ0≥Υ

∫

M

(P₀^kλ^⊥)²dµ0. Therefore there exists a constantC such that

(4.5) d

dt

∫

M

|P₀^kλ^⊥|²dµ0≤ −3Υ

∫

M

|P₀^kλ^⊥|²dµ0+C(k, Q^⊥₀,Υ).

By Lemma 4.3, we obtain

∫

M

|P₀^kλ^⊥|²dµ0≤C.

By the subellipticity ofP₀and theL²-estimate for (λ^⊥)² of Part 1, we obtain

∥λ^⊥−λ^⊥∥S^4k,2 ≤C(k, Q^⊥₀,Υ) for allt≥0.

Part 3: We note that λ−λ= [λ^⊥−λ^⊥] + [λ_ker(0,·)−λ_ker(0,·)],and hence (4.6) ∥λ−λ∥S^4k,2 ≤C(λker(0,·), k) +∥λ^⊥−λ^⊥∥S^4k,2≤C(k, Q^⊥₀,Υ).

In particular, there holds∥λ−λ∥S^4,2 ≤C.Therefore by the Sobolev embedding theorem, we haveS^4,2⊂S^1,8 and∥λ−λ∥S^1,8 ≤C.

Now using Lemma 4.1 (CR Moser inequality), we get e^−4λV0=

∫

M

e^4(λ−λ)dµ0≤C1exp(C2∥λ−λ∥S^1,4)≤C(Q^⊥₀,Υ).

Here we usedV₀=V =

∫

M

e^4λdµ₀. Hence we concludeC ≥λ≥ −C(Q^⊥₀,Υ),and we finally obtain

(4.7) ∥λ∥S^4k,2≤C(Q^⊥₀,Υ) for allt≥0.

5 Long-Time Existence and Asymptotic Convergence

Due to the a prioriS^4k,2-estimate for a solution of (1.5) of the previous section, we can prove the long time existence and asymptotic convergence of solutions of (1.5).

Proof of Theorem 1.2. SinceP0is nonnegative and subelliptic on (kerP0)^⊥, there exists a unique C^∞-smooth solutionλ^⊥ for a short time. On the other hand, we can apply the contraction mapping principle to show the short time existence of a uniqueC^∞ smooth solution λker to (4.4). Extending the solution at the maximal time T, the long time exitence then follows from Proposition 4.4 and the Sobolev embedding theorem.

Starting from (4.3), we compute d²

dt²E(θ) =−4

∫

M

(Q^⊥₀ + 2P0λ)P0

∂λ

∂tdµ0≥4Υ

∫

M

(Q^⊥₀ + 2P0λ)²dµ0

Therefore−d dtE(θ) =

∫

M

e^4λ[Q−e^−4λ(Q0)ker]²dµ(≥0) is nonincreasing.

By the a priori estimate (4.7), we can find a sequence of times t_j such that λ_j ≡ λ(·, t_j) converges to λ_∞ in C^∞-topology as t_j → ∞. On the other hand, integrating (4.3) gives

(5.1) E(λ_∞)−E(λ₀) =−

∫ _∞

0

∫

M

e^4λ[Q−e^−4λ(Q₀)_ker]²dµ dt <∞. In view of (5.1), we obtain

(5.2) 0 = lim

t→∞

∫

M

e^4λ[Q−e^−4λ(Q0)ker]²dµ=

∫

M

e^4λ∞[Q_∞−e^−4λ∞(Q0)ker]²dµ_∞ It follows thatQ_∞=e^−4λ∞(Q0)ker,and 2P0λ^⊥_∞+Q^⊥₀ = 0.

In the following, we are going to prove the smooth convergence for all time.

Write λ_∞ = λ^⊥_∞+ (λ_∞)_ker. Observe that ∥λ^⊥_∞−λ^⊥∥2 ≤ C∥2P₀(λ^⊥_∞−λ^⊥)∥2 = C∥2P0λ^⊥+Q^⊥₀∥2. We compute

|E(λ^⊥_∞)−E(λ^⊥)| = ∫ 1

0

d

dsE(λ^⊥+s(λ^⊥_∞−λ^⊥))ds

≤

∫ 1 0

∥2P₀(λ^⊥+s(λ^⊥_∞−λ^⊥)) +Q^⊥₀∥2∥λ^⊥_∞−λ^⊥∥2ds

≤ C1∥2P0λ^⊥+Q^⊥₀∥²2.

By the trick used in [18] (see also, [3]), we estimate∥E(λ^⊥)−E(λ^⊥_∞)∥^1−ϑ, 0≤ϑ <¹₂, and we can obtain

(5.3)

∫ _∞

0

∂λ

∂t

⊥

2

dt <+∞.

Observing that d

dt(λ^⊥−λ^⊥_∞) = ˙λ^⊥= dλ^⊥

dt and−d

dt∥λ^⊥−λ^⊥_∞∥²2≤C∥λ˙^⊥∥²2,we can then deduce theL²-convergence of the flow:

(5.4) lim

t→∞∥λ^⊥−λ^⊥_∞∥²2= 0.

On the other hand, we have estimate|r(t)| ≤C₂∥2P₀λ^⊥+Q^⊥₀∥2=C₂∥λ˙^⊥∥2. It also follows from (5.3) that

∫ _∞

0

|r(t)|dt <+∞.So in view of (4.4),λ_ker→(λ_∞)_ker= (λker)_∞ast→ ∞(not just a sequence of times). Sinceλker−(λ_∞)ker=

∫ _∞

t

r(t)dt is a function of time only, we also have

(5.5) lim

t→∞∥λker−(λ_∞)ker∥S^k,2 = 0.

Withλ^⊥ replaced byλ^⊥−λ^⊥_∞ in the argument to deduce (4.5), we obtain (5.6) d

dt

∫

M

(P₀^k(λ^⊥−λ^⊥_∞))²dµ₀≤ −4Υ

∫

M

(P₀^k(λ^⊥−λ^⊥_∞))²dµ₀+C(k)∥λ^⊥−λ^⊥_∞∥2. By Lemma 4.3 and (5.4), we get lim

t→∞

∫

M

(P₀^k(λ^⊥−λ^⊥_∞))²dµ₀ = 0. Hence by the subellipticity ofP₀^k,we conclude from (5.4) that lim

t→∞∥λ^⊥−λ^⊥_∞∥S^4k,2 = 0.Together with (5.5), this shows thatλconverges toλ_∞ smoothly ast→ ∞.

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