thesis

PRESCRIBING SCALAR CURVATURE

LIU HONG(B.Sc., Peking University)

A thesis submitted for theDegree of Doctor of Philosophy

Supervisor

Professor Xingwang Xu

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2015

Acknowledgements

I would like to thank all people who have helped and inspired me through my thesis.

First and foremost I would like to express the deepest appreciation and sincerest gratitude

to my supervisor, Professor Xingwang Xu, who has supported me during my doctoral study

with his patience and knowledge. Without his guidance, encouragement and persistent help,

this dissertation would not have been possible. I am grateful to be his student, and his

perpetual energy and enthusiasm in research enabled me to develop a good understanding of

research.

I would like to thank Professor G. Tian, Professor P. Yang, Professor A. Chang, Professor

S. T. Yau, Professor R. Schoen and Professor J. Case for their kind help and encouragement.

I thank Professor Fei Han, Professor Lei Zhang, Dr. Hengfei Lu, Dr. Feng Zhou, Dr. Hong

Zhang, Dr. Ruilun Cai, Dr. Jize Yu, Dr. Caihua Luo, Dr. Yuke Li, Dr. Ran Wei, Dr. Liuqin

Yang, Dr. Ying Cui, Dr. Yufei Zhao, Dr. Ruixiang Zhang, Dr. Xiaowei Jia and his girlfriend

Master Ya Sun, Dr. Chen Yang and his wife Master Dan Zheng, Dr. Yan Wang and his

boyfriend Dr. Xiaofei Zhao, Dr. Jinjiong Yu and his girlfriend Master Yile Li, Dr. Hawk,

Master Yuchen Wang, Miss Andrea, Dr. Yuke Li, Master Qiong Liu, Miss Jiayu Zhu and Miss

Xueliang Hu for their assistance, stories and jokes. Thanks, Dr. Han.

I would like to thank my grandmother for her spiritual support. I would like to thank

Mr. Bo Yang, Ms. Zhao Han, Miss Mengqi Hong and their families for their aid in my most

difficult time.

Finally, I would like to thank my father Heng Long Hong and mother Liu Ying Xie for

giving my life in the first place for unconditional support and love.

i

Contents

Acknowledgements i

Summary iv

List of Symbols v

1 Introduction 1

2 Elementary estimates and long time existence 3

3 Reduction 18

4 The case u∞ = 0 20

5 Convergence 26

iii

Summary

The prescribing scalar curvature problem originated from the classical Yamabe problem. Since

Yamabe constant is a conformal invariant, we can divide conformal closed manifolds into

positive, negative and zero Yamabe cases. The negative Yamabe case is well understood. So

we focus on and solve the positive case here.

iv

Symbol

Symbol Definition

N a conformal closed manifold

n dimension of N

χ(N) Euler characteristic of N

Sg, S scalar curvature

[g0] conformal class of g0

Y (N, [g0]) Yamabe constant

Eg[u] Yamabe energy

Sn standard sphere

expp exponential map from p

Tp(N) tangent space at p

∆ Laplacian operator

∇ gradient operator

2∗ 2nn−2

V ol volume

∂ partial differential operator

o(1) tending to zero

v

Chapter 1

Introduction

Prescribing scalar curvature problem is a fundamental problem in modern geometry. This

problem in diffeomorphic class is well understood from the work of J. L. Kazdan and F. W.

Warner [15],[16]. For closed 2-manifolds, Kazdan and Warner proved in 1975 [15] that the

obvious sign condition demanded by the Gauss-Bonnet Theorem:

(a) K positive somewhere if χ(N) > 0,

(b) K changes sign unless K ≡ 0 if χ(N) = 0,

(c) K negative somewhere if χ(N) < 0,

is sufficient for a smooth function K on a given closed 2-manifold N to be the Gaussian

curvature of some metric on N. In case of dimension n ≥ 3, they proved if f is negative

somewhere, then f could be the scalar curvature of some smooth metric in [16]. Furthermore,

if N admits a metric with positive scalar curvature, any smooth function could be a scalar

curvature of some Riemannian metric (see [16]). They also proved if N is non-compact of

dimension n ≥ 3 diffeomorphic to an open submanifold of some closed manifold, then every

smooth function could be a scalar curvature (see [16]).

So we consider the prescribing scalar curvature problem in conformal class which is more

subtle and plays a crucial role in modern conformal geometry. The Yamabe constant of a

conformal manifold (N, [g0]) is defined as

Y (N, [g0]) = infg∈[g0]

∫NSgdµg

(∫Ndµg)

n−2n

,

where Sg is the scalar curvature of g and µg is the volume density of the metric g. The Yamabe

1

CHAPTER 1. INTRODUCTION 2

energy of a function u on (N, g) of class H1 is defined as

Eg[u] =

∫N

(cn|∇gu|2 + Sgu2)dµg

V ol(N, g), cn =

4(n− 1)

n− 2.

Since Yamabe constant is a conformal invariant, we can divide conformal manifolds into

positive, negative and zero Yamabe cases. The negative Yamabe case is easier and well

understood. So we focus on positive Yamabe case in this paper. In 1976, T. Aubin proved

a perturbation theorem with non-sphere condition. Namely, if (N, g) with positive Yamabe

constant is not conformal to standard sphere, then there is κ > 1 depending on (N, g) such that

any smooth positive function f satisfying sup f ≤ κ inf f is the scalar curvature of a conformal

metric(see [1]). In 1986, J. F. Escobar and R. M. Schoen proved in [12] that for given closed

3-manifold of positive scalar curvature which is not conformally equivalent to standard sphere,

any smooth function which is positive somewhere could be the scalar curvature of a conformal

metric. They treated the case of dimension n ≥ 4 with locally conformally flat restriction.

The most subtle case is standard sphere. In 1991, A. Bahri and J. M. Coron proved in [6]

a result on S3 with an index condition. In the same year, A. Chang and P. C. Yang proved

the perturbation theorem for general Sn in [10] with an analogous index condition. In 2010,

X. Chen and X. Xu gave a concrete bound for the perturbation theorem using the scalar

curvature flow in [11].

During 2005-2007, S. Brendle provided a new proof of Yamabe problem by constructing

a family of test functions in [3] and [4] which is reminiscent of Schoen’s proof of Yamabe

problem [19]. Both proofs study the role of Green’s functions on manifolds and rely on Yau’s

positive mass theorem(see [22]).

Since Q. A. Ngo and X. Xu [17] have solved the zero Yamabe case, we treat the non-sphere

positive Yamabe case here as the last piece of the jigsaw puzzle. Our main result is:

Theorem 1. Given a closed conformal manifold (N, [g0]) which is not conformally equivalent

to the round sphere and satisfies Y (N, [g0]) > 0. If n ≥ 3, then any positive smooth function

on N could be the scalar curvature of some g ∈ [g0].

It seems that G. Bianchi and E. Egnel [7] have constructed a counter example to Theorem

1. But we have clarified their case in [14].

Chapter 2

Elementary estimates and long time

existence

From now on, we always assume (N, [g0]) is a manifold satisfying the conditions in Theorem

1. Let f be a positive smooth function on N. We can choose g0 in the conformal class such

that the scalar curvature S0 of g0 is a positive constant. Since N is compact, 0 < m ≤ f ≤M.

We write one conformal metric g as g = u4

n−2 g0 . To find the desired metric we need to find a

positive solution of the following equation

−cn∆u+ S0u = fun+2n−2 .

Consider the scalar curvature flow

ut =n− 2

4[α(t)f − S(t)]u

where S(t) is the scalar curvature of g(t) which has standard formula

S = u−n+2n−2 (−cn∆u+ S0u)

where cn = 4(n−1)n−2

,∆ = ∆g0 and S0 is the scalar curvature of g0. By multiplying a constant,

we can assume ∫N

fdµg0 = 1 .

3

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 4

From now on, the integration is always on the whole manifold N. First let us define

E[u] :=

∫(cn|∇u|2 + S0u

2)dµ =

∫Sdµg

Ef [u] :=E[u]

(∫fdµg)

n−2n

where dµg is the volume density of g and dµ = dµg0 , | · | = | · |g0 .The factor α(t) is chosen such that

0 =d

dt

∫fdµg =

d

dt

∫fu2∗dµ0 = 2∗

∫fu

n+2n−2utdµ

=n

2α

∫f 2dµg −

n

2

∫fSdµg ,

where 2∗ = 2nn−2

is the critical exponent and ∇ = ∇g0 . Hence the natural choice for this factor

is

α(t) =

∫fSdµg∫f 2dµg

.

For p ≥ 1, we define

Fp(g(t)) =

∫|αf − S|pdµg .

Lemma 1. If a positive smooth function u satisfies

ut =n− 2

4(αf − S)u ,

S = u−n+2n−2 (−cn∆u+ S0u) ,

thend

dtE[u] =

d

dtEf [u] = −n− 2

2F2 .

Proof.

d

dtEf [u] =

dEdt

(∫fdµg)

n−2n

−2E[u]

∫fu

n+2n−2utdµ

(∫fdµg)

n−2n

+1.

Sinced

dt(|∇u|2) = 2〈∇u,∇ut〉

= 2div(ut∇u)− 2ut∆u ,

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 5

we havedE[u]

dt=

d

dt

∫(cn|∇u|2 + S0u

2)dµ

= 2

∫(−cn∆u+ S0u)utdµ ,

d

dtEf [u] = 2(

∫fdµg)

2−nn

∫(−cn∆u+ S0u− αfu

n+2n−2 )utdµ

= −2(

∫fdµg)

2−nn

∫(αf − S)u

n+2n−2utdµ

= −n− 2

2(

∫fdµg)

2−nn

∫(αf − S)2dµg

= −n− 2

2

∫(αf − S)2dµg .

The identity holds because Ef [u] = E[u] in our flow.

Hence Ef [u(t)] is decreasing. For any initial function u(0) = u0 ∈ H1(N, g0), we have

E[u(t)] = Ef [u(t)](

∫fdµg)

n−2n ≤ CEf [u0] <∞ , C = C(N, u0) .

Lemma 2. There exists a constant α1 depending on f and u0 such that

|α(t)| ≤ α1 .

Proof.

|α(t)| =|∫fSdµg|∫f 2dµg

≤ C|∫fSdµg|

(∫fdµg)2

≤ C|∫fSdµg| ,∫

fSdµg = −cn2

∫(∆f)u2dµ+ cn

∫f |∇u|2dµ+

∫fS0u

2dµ .

Thus,

|α(t)| ≤ CE[u] ≤ CE[u0] .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 6

Lemma 3.

(αf − S)t = (n− 1)∆g(αf − S) + S(αf − S) + α′f .

Proof. By the formula

S = u−n+2n−2 (−cn∆u+ S0u) ,

we have

St = u−n+2n−2 (−cn∆ut + S0ut)−

n+ 2

n− 2u−

2nn−2ut(−cn∆u+ S0u)

=n− 2

4u−

n+2n−2−cn[u∆(αf−S)+2〈∇(αf−S),∇u〉+(αf−S)∆u]+S0(αf−S)u−n+ 2

4(αf−S)S

= −n+ 2

4(αf−S)S+

n− 2

4u−

n+2n−2 (αf−S)(−cn∆u+S0u)−(n−1)u−

n+2n−2 [u∆(αf−S)+2〈∇(αf−S),∇u〉] .

Since

∆g = u−2nn−2 det(g0)−

12∂j(u

2 det(g0)12 gij0 ∂i)

= u−4

n−2 ∆ + 2u−n+2n−2 〈∇u,∇·〉 ,

we have

St = −(αf − S)S − (n− 1)∆g(αf − S) ,

(αf − S)t = (n− 1)∆g(αf − S) + (αf − S)S + α′f .

Since the case of dimension three has been proved by Escobar and Schoen, we always

assume n ≥ 4 from now on.

Lemma 4. F2 is bounded.

Proof.

d

dtF2 = −n− 4

2[cn

∫|∇(αf−S)|2gdµg+

∫|(αf−S)|2gdµg]−cn

∫|∇(αf−S)|2gdµg+

n

2

∫αf(αf−S)2dµg

≤ n

2

∫αf(αf − S)2dµg since Y (N) > 0

≤ CF2

for some constant C > 0. Thus,

F2(g(t)) ≤ F2(g(0)) + C

∫ t

0

F2(g(s))dµg(s)

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 7

≤ F2(g(0)) +2C

n− 2E[u0] .

In later estimates, we will constantly utilize positive Yamabe condition.

Lemma 5. There exists a constant C such that α′ ≤ CF12

2 .

Proof.

|α′| = (

∫f 2dµg)

−1| ddt

(

∫fSdµg)− α

d

dt(

∫f 2dµg)|

≤ C| ddt

(

∫fSdµg)− α

d

dt(

∫f 2dµg)|

≤ C| − (n− 1)

∫f∆g(αf − S)dµg +

n− 2

2

∫fS(αf − S)dµg −

nα

2

∫f 2(αf − S)dµg|

≤ C| − (n− 1)

∫(∆gf)(αf − S)dµg|+ C1F1 + C2F2 + C3F

12

2 .

By Cauchy inequality and boundness of F2, we have

|α′| ≤ C|∫

(∆gf)(αf − S)dµg|+ CF12

2 .

Moreover,∫(∆gf)(αf − S)dµg =

∫(αf − S)(∆f)u−

4n−2dµg + 2

∫(αf − S)〈∇u,∇f〉u−

n+2n−2dµg.

By Cauchy inequality,

|∫

(αf − S)(∆f)u−4

n−2dµg| ≤ (sup |∆f |)(∫u

2(n−4)n−2 dµ)

12F

12

2 .

Since 2(n−4)n−2

< 2nn−2

,∫u

2(n−4)n−2 dµ is bounded.

|∫

(αf − S)〈∇u,∇f〉u−n+2n−2dµg| ≤ sup(|∇f |)(

∫|∇u|2u−

4n−2dµ)

12F

12

2 .

Hence we only need to bound∫|∇u|2u−

4n−2dµ.

If n = 6, ∫|∇u|2u−

4n−2dµ =

∫|∇u|2u−1dµ

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 8

= −∫

∆u√u

(√u lnu)dµ

≤ (

∫(∆u)2u−1dµ)

12 (

∫u(lnu)2dµ)

12 .

Since 2×66−2

= 3, we have∫u(lnu)2dµ =

∫u≥1

u(lnu)2dµ+

∫u<1

u(lnu)2dµ

≤∫u3dµ+

∫4

e2dµ

≤ C.

Since S = u−2(S0u− 5∆u), we have∫(∆u)2u−1dµ =

1

25

∫(S2

0u− 2S0Su2 + S2u3)dµ.

Since∫fu3dµ = 1,

∫(αf − S)2u3dµ and α are bounded, we know the above term is bounded

too and the desired estimate follows.

If n 6= 6, ∫(S − S0u

− 4n−2 )u2dµ =

4(n− 1)(n− 6)

(n− 2)2

∫|∇u|2u−

4n−2dµ .

So the result follows from ∫S0u

2n−8n−2 dµ ≤ C ,∫

Su2dµ ≤ (

∫S2dµg)

12 (

∫u

2(n−4)n−2 dµ)

12 ≤ C .

Now we are able to give a lower bound of the scalar curvature S.

Lemma 6.

S − αf ≥ γ, t ≥ 0 ,

where γ := minS0 − α(0)f,− 2√3

√max0, α′f + α2f 2,−αf.

Proof. Define

L = ∂t − (n− 1)∆g + αf − γ ,

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 9

then

LS = St − (n− 1)∆gS + (αf − γ)S

= S(S − γ)− (n− 1)∆g(αf) .

By using comparison function w(t) = αf + γ, we have

Lw = αtf − (n− 1)∆g(αf) + (αf − γ)(αf + γ)

≤ −1

4γ2 − (n− 1)∆g(αf)

≤ S(S − γ)− (n− 1)∆g(αf)

≤ LS .

But

w(0) ≤ α(0)f + γ ≤ S0 .

Thus we get a lower bound of S(t), t ≥ 0, by the maximum principle for a linear parabolic

differential equation.

The following estimates will help us to achieve the long time existence of the flow.

Lemma 7. For any T > 0, there exists C = C(T ) such that

C−1 ≤ u(x, t) ≤ C, (x, t) ∈ N × [0, T ) .

Proof. From the equation of the flow and lemma 6, we know

u(t) ≤ e−n−24γtu(0), t ∈ [0, T ] .

Define a constant

P = S0 + supt∈[0,T ]

supN

[−(αf + γ)u4

n−2 ] .

From Lemma 6 again, we have

0 ≤ (S − αf − γ)un+2n−2

= −cn∆u+ S0u− (αf + γ)un+2n−2

≤ −cn∆u+ Pu .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 10

Hence by the Corollary A.3 in [3], we have

1 = V ol(N, g0) ≤ C infNu(sup

Nu)

n+2n−2 .

So the assertion follows.

Define

δ = supt∈[0,T ]

‖αf + γ‖∞ + 1 .

From Lemma 6, we have

S + δ ≥ αf + γ + |αf + γ|+ 1 ≥ 1 .

Lemma 8. For any p > 2, we have

d

dt

∫(S + δ)p−1dµg = −4(p− 2)(n− 1)

p− 1

∫|∇g(S + δ)

p−12 |2gdµg

+(p− 1)(p− 2)(n− 1)

∫(S + δ)p−3〈∇g(S + δ),∇g(αf + δ)〉gdµg

−(n

2+ 1− p)

∫(S + δ)p−1(S − αf)dµg − (p− 1)δ

∫[(S + δ)p−2 − (αf + δ)p−2](S − αf)dµg

−(p− 1)δ

∫(αf + δ)p−2(S − αf)dµg .

Proof.d

dt

∫(S + δ)p−1dµg =

d

dt

∫(S + δ)p−1u

2nn−2dµ

=2n

n− 2

∫(S + δ)p−1u

n+2n−2utdµ+ (p− 1)

∫(S + δ)p−2Stdµg

=n

2

∫(S+δ)p−1(αf−S)dµg−(1−p)

∫(S+δ)p−2(S−αf)Sdµg−(p−1)(n−1)

∫(S+δ)p−2∆g(αf−S)dµg

= −n2

∫(S + δ)p−1(S − αf)dµg − (1− p)

∫(S + δ)p−2(S − αf)(S + δ − δ)dµg

−(p− 1)(n− 1)

∫(S + δ)p−2divg(∇g(αf − S))dµg

= −(n

2+ 1− p)

∫(S + δ)p−1(S − αf)dµg − (p− 1)δ

∫(S + δ)p−2(S − αf)dµg

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 11

−(p−1)(n−1)

∫divg[(S+δ)p−2∇g(αf−S)]−(p−2)(S+δ)p−3〈∇g(S+δ),∇g(αf−S)〉gdµg

= −(n

2+ 1− p)

∫(S + δ)p−1(S − αf)dµg − (p− 1)δ

∫[(S + δ)p−2 − (αf + δ)p−2](S − αf)dµg

−(p−1)δ

∫(αf+δ)p−2(S−αf)dµg+(p−1)(p−2)(n−1)

∫(S+δ)p−3〈∇g(S+δ),∇g(αf+δ−S−δ)〉gdµg .

Since

(S + δ)p−3〈∇g(S + δ),∇g(αf + δ)〉g − (S + δ)p−3|∇g(S + δ)|2g

= (S + δ)p−3〈∇g(S + δ),∇g(αf + δ)〉g −4

(p− 1)2|∇g(S + δ)

p−12 |2g ,

the identity follows.

For any p ≥ 2, we have

d

dtFp =

d

dt

∫|αf−S|pdµg = p

∫|αf−S|p−2(αf−S)(αf−S)tdµg+

n

2

∫|αf−S|p(αf−S)dµg

= p

∫|αf−S|p−2(αf−S)[(n−1)∆g(αf−S)+(αf−S)S+α′f ]dµg+

n

2

∫|αf−S|p(αf−S)dµg .

Since

|αf − S|p−2(αf − S)∆g(αf − S) = divg(|αf − S|p−2(αf − S)∇g(αf − S))

−〈∇g(|αf − S|p−2(αf − S)),∇g(αf − S)〉g ,

∇g(|αf − S|p−2(αf − S)) = (p− 1)|αf − S|p−2∇g(αf − S) ,

hence

d

dt

∫|αf − S|pdµg = −p(p− 1)(n− 1)

∫|αf − S|p−2|∇g(αf − S)|2gdµg + p

∫|αf − S|pSdµg

+pα′∫f |αf − S|p−2(αf − S)dµg +

n

2

∫|αf − S|p(αf − S)dµg

= −p(p− 1)(n− 1)

∫|αf − S|p−2|∇g(αf − S)|2gdµg + p

∫αf |αf − S|pdµg

pαt

∫f |αf − S|p−2(αf − S)dµg + (

n

2− p)

∫|αf − S|p(αf − S)dµg .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 12

Lemma 9. For any p > maxn2, 2, we have

d

dtFp + C1F

22∗pnn−2≤ C2(Fp + F

2p−n+22p−n

p ) ,

where the positive constants C1, C2 are independent of t.

Proof. Since

cn|∇g|αf − S|p2 |2g =

p2(n− 1)

n− 2|αf − S|p−2|∇g(αf − S)|2g ,

so from the derivative formula we have

d

dtFp = −(p− 1)(n− 2)

p

∫(cn|∇g|αf − S|

p2 |2g + S|αf − S|p)dµg

+[p+(p− 1)(n− 2)

p]

∫S|αf−S|pdµg+

n

2

∫|αf−S|p(αf−S)dµg+pα

′∫f |αf−S|p−2(αf−S)dµg .

By lemma 5 and Holder inequality, we have |α′| ≤ CF12

2 ≤ CF1pp , and thus

|pα′∫f |αf − S|p−2(αf − S)dµg| ≤ C|α′|Fp−1

≤ CFp .

With the help of positive Yamabe condition, we have

d

dtFp ≤ −C(

∫|αf − S|

npn−2dµg)

n−2n + CFp + CFp+1 .

By Holder inequality and Young inequality we deduce that

Fp+1 =

∫|αf − S|p+1dµg

≤ (

∫|αf − S|

npn−2dµg)

n−22p (

∫|αf − S|pdµg)

2p−n+22p

≤ ε(

∫|αf − S|

npn−2dµg)

n−2n + C(ε)(

∫|αf − S|pdµg)

2p−n+22p−n .

By choosing suitable ε, the assertion follows.

Lemma 10. For any T > 0, there exists C = C(T ) such that

F n2

2(n−2)

≤ C(T ), t ∈ [0, T ) .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 13

Proof. Since S + δ, αf + δ > 0 and p > 2, we deduce

[(S + δ)p−2 − (αf + δ)p−2](S − αf) = [(S + δ)p−2 − (αf + δ)p−2][(S + δ)− (αf + δ)] ≥ 0 .

So we can choose p = n+22> 2 in Lemma 8 to achieve

d

dt

∫(S + δ)

n2 dµg ≤ −

4(n− 1)(n− 2)

n

∫|∇g(S + δ)

n4 |2gdµg

+n(n− 1)(n− 2)

4

∫(S + δ)

n−42 〈∇g(S + δ),∇g(αf + δ)〉gdµg −

n

2δ

∫(αf + δ)

n−22 (S−αf)dµg .

From Lemma 6 we have

−n2δ

∫(αf + δ)

n−22 (S − αf)dµg ≤ −

n

2δγ

∫(αf + δ)

n−22 dµg

≤ C(T )

∫dµg

≤ C(T ) .

By Cauchy inequality and Young inequality we have

|∫

(S + δ)n−42 〈∇g(S + δ),∇g(αf + δ)〉gdµg| =

4

n|∫

(S + δ)n−44 〈∇g(S + δ)

n4 ,∇g(αf + δ)〉gdµg|

≤ 4

n

∫(S + δ)

n−44 |∇g(S + δ)

n4 |g|∇g(αf + δ)|gdµg

≤ ε

∫|∇g(S + δ)

n4 |2gdµg + C(ε)

∫|∇g(αf + δ)|2g(S + δ)

n−42 dµg .

We know from Lemma 7 that u is bounded on [0, T ). Hence by using Holder inequality we

can achieve

|∫

(S+δ)n−42 〈∇g(S+δ),∇g(αf+δ)〉gdµg| ≤ ε

∫|∇g(S+δ)

n4 |2gdµg+C(T, ε)[

∫(S+δ)

n2 dµg]

n−4n .

Set y(t) =∫

(S + δ)n2 dµg. By choosing small ε we can get

dy

dt+

∫|∇g(S + δ)

n4 |2gdµg ≤ C2(T )y

n−4n + C2(T ) .

We claim that y is bounded by a constant depending on T.

Since n ≥ 4, (y4n )′ ≤ C2 + C2y

4−nn . So y

4n ≥ 1⇒ (y

4n )′ ≤ 2C2 and the claim follows.

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 14

From the definition of y we know that y has the volume of N as a lower bound. Integrating

the inequalitydy

dt+

∫|∇g(S + δ)

n4 |2gdµg ≤ C2(T )y

n−4n + C2(T )

we get ∫ T

0

∫|∇g(S + δ)

n4 |2gdµgdt ≤ C3(T ) .

Due to our positive Yamabe condition, we have

(

∫|S + δ|

n2

2(n−2)dµ)n−2n ≤ C

∫(cn|∇|S + δ|

n4 |2 + S0|S + δ|

n2 )dµ ≤ C4(T ) .

So ∫ T

0

(

∫|S + δ|

n2

2(n−2)dµg)n−2n dt ≤ C5(T )

∫ T

0

(

∫|S + δ|

n2

2(n−2)dµ)n−2n dt ≤ C6(T ) .

Since S + δ, αf + δ > 0, we have

|S − αf | = |(S + δ)− (αf + δ)| ≤ max|S + δ|, C7(T )

and

|S − αf |p ≤ |S + δ|p + |αf + δ|p .

Since n−2n< 1, we have

(

∫|S − αf |

n2

2(n−2)dµg)n−2n ≤ (

∫|S + δ|

n2

2(n−2)dµg)n−2n + (

∫|αf + δ|

n2

2(n−2)dµg)n−2n .

Setting p = n2

2(n−2)in Lemma 9 we get

(logF n2

2(n−2)

)′ ≤ C + CFn−2n

n2

2(n−2)

.

So the assertion follows by integrating above inequality.

By the argument of Proposition 2.6 in [3] or Lemma 2.11 in [11], we achieve the following

inequality:

Lemma 11. For any λ ∈ (0,min 4n, 1), T > 0, there exists C = C(λ, T ) > 0 such that

|u(x, t)− u(y, s)| ≤ C[dN(x, y)λ + |t− s|λ2 ] ,

for any x, y ∈ N, t, s ∈ [0, T ), 0 < |t− s| < 1 .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 15

So the long time existence of the scalar curvature flow follows from the standard result of

parabolic equations. For example, one can read Theorem 8.3 and Theorem 8.4 in [24].

Lemma 12.∫∞

0F2dt <∞.

Proof. Since the flow has long time existence, from lemma 1 we know∫ ∞0

F2dt =2

n− 2(E[u0]− lim

t→∞E[u(t)]) <∞ .

Thus the assertion follows.

Lemma 13.

F2 → 0 .

Proof. In the proof of lemma 4, we know ddtF2 ≤ CF2. Thus, for any t > tν > 0, we have

F2(t) ≤ F2(tν) +

∫ ∞tν

F2 .

By lemma 12, we can pick a time sequence tν → ∞ such that F2(tν) → 0 and the result

follows.

Now we are going to achieve the convergence of α(t).

Lemma 14.

α = E[u] + o(1), t→∞.

Proof. Sinced

dtV ol(N, g(t)) =

n

2

∫(αf − S)dµg ,

by Lemma 13 and Cauchy inequality, we have ddtV ol(N, g(t))→ 0. We also have the identity

d

dtV ol(N, g(t)) =

n

2α

∫fdµg −

n

2E[u] =

n

2(α− E[u]) .

So the result follows.

Lemma 15. For any p ∈ [2, n2],

d

dtFp ≤ CFp .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 16

Proof. By the derivative formula in the proof of lemma 9 and the inequalities

|pα′∫f |αf − S|p−2(αf − S)dµg| ≤ CF

1pp Fp−1 ≤ CFp

αf − S ≤ −γ ,

we deduce the desired inequality.

Lemma 16. If 2 ≤ p < n2

and Fp is integrable over (0,∞), then Fp+1 is also integrable over

(0,∞).

Proof. By the formula

(n

2− p)

∫|αf − S|p(S − αf)dµg = p

∫αf |αf − S|pdµg

+pα′∫f |αf − S|p−2(αf − S)dµg −

d

dtFp −

4(p− 1)(n− 1)

p

∫|∇|αf − S|

p2 |2gdµg ,

we have ∫ T

0

∫|αf − S|p(S − αf)dµgdt ≤ C

∫ T

0

[Fp +2

n− 2pFp(0)]dt ≤ C .

Since αf − S ≤ −γ, we have∫ T

0

Fp+1dt ≤∫ T

0

∫|αf − S|p(S − αf + 2|γ|)dµgdt ≤ C

for any T > 0. As C is independent of T, the result follows.

Now we have the following fundamental result.

Lemma 17. For any p ∈ [1, n2], Fp → 0.

Proof. By lemma 12, lemma 16 and Holder inequality, Fp is integrable for any p ∈ [2, n2], and

the result follows from the same argument in the proof of lemma 13.

Define

α∞ := limt→∞

α(t) ,

E∞ := limt→∞

α(t) ,

Ef := infEf [u] : u ∈ H1, u is not constantly 0 .

CHAPTER 2. ELEMENTARY ESTIMATES AND LONG TIME EXISTENCE 17

Since E[u] is monotone, E∞ clearly exists and α∞ = E∞ by lemma 14.

Corollary 1. For any p ∈ [1, n2],∫|S − α∞f |pdµg → 0.

Proof. It is direct from lemma 14, lemma 17 and the inequality

|S − α∞f |p ≤ 2p−1(|S − αf |p + |α− α∞|p|f |p) .

Chapter 3

Reduction

The classical result of T. Aubin in Nonlinear Analysis on Manifolds. Monge-Ampere equations

131, shows that:

Lemma 18. If N is not a standard sphere, then Ef ≤ Y (sup f)−n−2n := n(n−1)ω

2nn (sup f)−

n−2n .

Moreover, if Ef < n(n − 1)ω2nn (sup f)−

n−2n , the prescribing scalar curvature problem of f has

a smooth positive solution.

Therefore, from now on, we only need to consider the case whereEf = n(n−1)ω2nn (sup f)−

n−2n .

For abbreviation, let uν = u(tν), λν = λu(tν), α(gν) = αν and gν = g(tν). We have the following

compactness result from Brendle [3]:

Lemma 19 (Bubble Decomposition).

After passing to a subsequence if necessary, we can find an l ∈ N, a smooth function u∞ and

a sequence of m-tuples (x∗k,ν , ε∗ν)1≤k≤l such that:

(1)4(n− 1)

n− 2∆u∞ − S0u∞ + α∞fu

n+2n−2∞ = 0 ,

where α∞ is the limit of the subsequence.

(2) For any i 6= j, we haved(x∗i,ν , x

∗j,ν)

ε∗ν

ν→∞−−−→∞ .

(3)

‖uν − u∞ −l∑

k=1

u(x∗k,ν ,ε∗ν)‖H1(N) −→ 0 ,

18

CHAPTER 3. REDUCTION 19

where u(p,ε) are a family of test functions satisfying

limε→0

εn−22 u(p,ε)(expp(εξ)) = (

1

1 + |ξ|2)n−22

for all p ∈ N and ξ ∈ TpN.

Proof. Since Ef [u(t)] is decreasing, uν is bounded in H1(N). By passing to a subsequence,

αν tends to a limit α∞ and u∞ is defined as the weak limit of uν. Then the argument from

Struwe’s proof of Proposition 2.1 in [20] works. The first and third results are from Struwe

[20], and the second statement is due to Bahri and Coron [5].

The recent work of Q. A. Ngo and X. Xu, [17] also gives a proof of the above lemma in the

special case of scalar curvature flow. A useful corollary is that, when E[u0] < 22nY (sup f)−

n−2n ,

the bubble number l ≤ 1. The reason is that, if l ≥ 2, then for sufficiently small ε > 0,

we have contradictory inequalities l2(sup f)

n−22 < l( Y

α∞)n2 ≤ lim supt→∞

∑l

∫B(x∗j ,ε)

fn2 dµg ≤

(sup f)n2−1 lim supt→∞

∫fdµg ≤ (sup f)

n−2n . We also refer the reader to [17] for the detailed

computation.

By the classical strong maximum principle (see Theorem 8.19 in [13]) we can deduce the

following proposition from

−4(n− 1)

n− 2∆u∞ + S0u∞ = α∞fu

n+2n−2∞ ≥ 0 .

Proposition 1. If u∞ vanishes at one point, then it vanishes everywhere.

From now on,as u∞ itself is a solution in case u∞ > 0, we only consider the case u∞ = 0.

Chapter 4

The case u∞ = 0

From now on, we restrict the initial data u(0) to satisfy Ef [u(0)] < 22nEf . Thus, the bubble

number l in the bubble decomposition could only be 1 noticing that Ef [u(t)] is decreasing

w.r.t. t, u∞ = 0 and the bubble decomposition is H1 convergence. For every time sequence

tν , let xν be one maximum point of u(tν). Since N is compact, by passing to a subsequence,

we can assume that xν → x∞ ∈ N. Set

kν = uν(xν)

2n−2

εν =1

kν

where uν = u(tν). We choose the normal coordinate system ξ w.r.t. g0 which means

(g0)ij = δij +O(|ξ|2) ,

dVg0 = (1 +O(|ξ|2))dξ .

Thus,

∆g0 = ∆ + bi∂i + dij∂2ij

where

bi = O(|ξ|2) ,

dij = O(|ξ|2) .

Now we follow the method from Q. A. Ngo and X. Xu [17] to construct the bubble decom-

position explicitly. First of all, we pick a smooth cut-off function on Rn for δ > 0 such

20

CHAPTER 4. THE CASE U∞ = 0 21

that

0 ≤ ηδ ≤ 1 ,

χB(0,δ) ≤ ηδ ≤ χB(0,2δ),

|∇ηδ| ≤C

δ.

ην(ξ) := ηδ(ξ

kν) .

We could write down the standard bubble as

Vν(ξ) = ην(ξ)(n(n− 1)

α∞f(x∞))n−24 (

2εν

(εν)2 + |ξ − ξν |2)n−22 .

Set τn = n(n−2)n(n−2)+2

and δν = (εν)τn . Given a pair (x, ε) ∈ M × [0,∞), we define the test

functions as the approximation of the bubble by

V(x,ε)(ξ) = χ(x∞,δν)(n(n− 1)

α∞f(x∞))n−24 (

2ε

ε2 + |ξ − ξx|2)n−22

:= χ(x∞,δν)(n(n− 1)

α∞f(x∞))n−24 v(x,ε)(ξ)

where ξx is the coordinate of x. Lemma 5.8, 5.9, 5.10 and 5.11 from [22] imply that uν−vν → 0

in H1 sense. By the Sobolev imbedding H1(N) → L2nn−2 (N), we have

1 = limν

∫fu

2nn−2ν dµ

= limν

∫fV

2nn−2ν dµ

= limν

∫fV

2nn−2ν dµ

= limν

∫B(O,2δν)

f(n(n− 1)

α∞f(x∞))n2 (

2εν

(εν)2 + |ξ − ξν |2)ndξ

= limν

∫B(O,2δν)

f(n(n− 1)

α∞f(x∞))n2 (

2εν

(εν)2 + |ξ|2)ndξ

= f(x∞)(n(n− 1)

α∞f(x∞))n2ωn .

Thus,

α∞ = f(x∞)2n−1n(n− 1)ω

2nn = f(x∞)

2n−1Y .

CHAPTER 4. THE CASE U∞ = 0 22

For every ν ∈ N, we defineAν = (x, ε, γ) ∈ (M×R+×R+) : d(x, xν) ≤ εν , 12≤ ε

εν≤ 2, 1

2≤

γ ≤ 2, . Letting the coordinate of xν be ξν , as in [3], we can find a 3-tuple (xν , εν , γν) ∈ Aνsuch that

E[uν − γνV(xν ,εν)] ≤ E[uν − γV(x,ε)]

for all (x, ε, γ) ∈ Aν . Since α is bounded, by passing to a subsequence we assume that αν → α∞

,and thus we have Eν → E∞.

Lemma 20.

‖uν − γνV(xν ,εν)‖H1ν→∞−−−→ 0 .

Proof. The assertion follows from the definition of (xν , εν , γν).

Lemma 21. We have

d(xν , xν) ≤ o(1)εν ,

ενεν

= 1 + o(1) ,

γν = 1 + o(1) .

In particular, (xν , εν , γν) is an interior point of Aν for sufficiently large ν.

Proof. We have

‖γνV(xν ,εν) − V(xν ,εν)‖H1 ≤ ‖uν − γνV(xν ,εν)‖H1 + ‖uν − V(xν ,εν)‖H1 = o(1) .

Thus the assertion follows from lemma 20.

Let us decompose uν into

uν = vν + wν

where

vν = γνV(xν ,εν) .

From Lemma 20, we know E[wν ] = o(1) .

Lemma 22. (1) We have

|∫V

n+2n−2

(xν ,εν)wνdµ| ≤ o(1)(

∫|wν |

2nn−2dµ)

n−22n .

CHAPTER 4. THE CASE U∞ = 0 23

(2) We have

|∫V

n+2n−2

(xν ,εν)

ε2ν − d(xν , x)2

ε2ν + d(xν , x)2

wνdµ| ≤ o(1)(

∫|wν |

2nn−2dµ)

n−22n .

(3) We have

|∫V

n+2n−2

(xν ,εν)

εν exp−1xν (x)

ε2ν + d(xν , x)2

wνdµ| ≤ o(1)(

∫|wν |

2nn−2dµ)

n−22n .

Proof. By the definition of (xν , εν , γν), we have∫(cn〈∇V(xν ,εν),∇wν〉+ S0V(xν ,εν)wν)dµ = 0 ,

hence ∫[cn∆V(xν ,εν) − S0V(xν ,εν)]wνdµ = 0 .

From the estimate

‖cn∆V(xν ,εν) − S0V(xν ,εν) + α∞fVn+2n−2

(xν ,εν)‖L 2nn+2

= o(1) ,

we can conclude that

|∫V

n+2n−2

(xν ,εν)wνdµ| ≤ o(1)‖wν‖L

2nn−2

.

This proves (1). The results in (2) and (3) follow from (1) and Lemma 18.

Lemma 23. If ν is sufficiently large, then we have

n+ 2

n− 2α∞

∫v

4n−2ν w2

νfdµ ≤ (1− c)∫

(cn|∇wν |2 + S0w2ν)dµ

for some positive constant c independent of ν.

Proof. By the definition of vν and lemma 21, we have∫|v

4n−2ν − V

4n−2

(xν ,εν)|n2 dµ = o(1) .

Therefore, we only need to prove that

n+ 2

n− 2α∞

∫V

4n−2

(xν ,εν)w2νfdµ ≤ (1− c)

∫(cn|∇wν |2 + S0w

2ν)dµ

for some positive constant c.

CHAPTER 4. THE CASE U∞ = 0 24

Suppose this is not true. Upon rescaling, we obtain a sequence of functions wν : ν ∈ Nsuch that ∫

(cn|∇wν |2 + S0w2ν)dµ = 1

and

limν→∞

n+ 2

n− 2α∞

∫V

4n−2

(xν ,εν)w2νfdµ ≥ 1 .

Note that∫|wν |

2nn−2dµ ≤ Y (N)−

nn−2 . Take a sequence Nν such that Nν → ∞, Nνεν → 0.

Let Ων = BNνεν (xν) \BNνεν (xν). We have

limν→∞

∫V

4n−2

(xν ,εν)w2νdµ > 0 ,

and

limν→∞

∫Ων

(cn|∇wν |2 + S0w2ν)dµ ≤ lim

ν→∞

n+ 2

n− 2α∞

∫V

4n−2

(xν ,εν)w2νfdµ

We now define a sequence of functions wν : TMxν → R by

wν(ξ) = εn−22

ν wν(expxν (ενξ)), ξ ∈ TMxν .

This sequence satisfies

limν→∞

∫ξ∈TMxν :|ξ|≤Nν

cn|∇wν(ξ)|2dξ ≤ 1

and

limν→∞

∫ξ∈TMxν :|ξ|≤Nν

|wν(ξ)|2nn−2dξ ≤ Y (N)−

nn−2 .

Hence, if we take the weak limit as ν →∞, then we obtain a function w : Rn → R such that∫(

1

1 + |ξ|2)2w(ξ)2dξ > 0 ,

and ∫|∇w(ξ)|2dξ ≤ n(n+ 2)

∫(

1

1 + |ξ|2)2w(ξ)2dξ .

Moreover, it follows from Lemma 18 that∫(

1

1 + |ξ|2)n+22 w(ξ)dξ = 0 ,

∫(

1

1 + |ξ|2)n+22

1− |ξ|2

1 + |ξ|2w(ξ)dξ = 0 ,

CHAPTER 4. THE CASE U∞ = 0 25∫(

1

1 + |ξ|2)n+22

ξ

1 + |ξ|2w(ξ)dξ = 0 .

Using a result of O. Rey, we conclude that w = 0 (see [18], Appendix D, pp. 49-51). This is

a contradiction.

Chapter 5

Convergence

Lemma 24. Let tν be a time sequence tending to infinity. For sufficiently large ν, there

exists a constant C such that

E[uν ]− α∞ ≤ C(

∫|S(gν)− α∞f |

2nn+2dµgν )

n+2n .

Proof. Using the identity

S(gν) = −u−n+2n−2

ν (cn∆uν − S0uν)

we obtain

E[uν ] =

∫(cn|∇uν |2 + S0u

2ν)dµ

=

∫(cn|∇vν |2 + S0v

2ν)dµ+ 2

∫un+2n−2ν S(gν)wvdµ−

∫(cn|∇wν |2 + S0w

2ν)dµ

= E[vν ] + 2

∫un+2n−2ν [S(gν)− α∞f ]wvdµ

−∫

(cn|∇wv|2 + S0w2v −

n+ 2

n− 2α∞fv

4n−2ν w2

v)dµ

+α∞

∫[−n+ 2

n− 2v

4n−2ν w2

ν + 2(vν + wν)n+2n−2wν ]fdµ .

Using Holder inequality, we obtain

|∫un+2n−2ν [S(gν)− α∞f ]wνdµ| ≤ (

∫u

2nn−2ν |S(gν)− α∞f |

2nn+2dµ)

n+22n (

∫|wν |

2nn−2dµ)

n−22n

≤ (

∫|S(gν)− α∞f |

2nn+2dµgν )

n+22n (

∫|wν |

2nn−2dµ)

n−22n

26

CHAPTER 5. CONVERGENCE 27

It follows from Lemma 23 that∫(cn|∇wv|2 + S0w

2v −

n+ 2

n− 2α∞fv

4n−2ν w2

v)dµ ≥ c

∫(cn|∇wv|2 + S0w

2v)dµ ,

hence ∫(cn|∇wv|2 + S0w

2v −

n+ 2

n− 2α∞fv

4n−2ν w2

v)dµ ≥ c(

∫|wν |

2nn−2dµ)

n−2n .

By the Cauchy inequality,

(

∫u

2nn−2ν |S(gν)− α∞f |

2nn+2dµ)

n+22n (

∫|wν |

2nn−2dµ)

n−22n

≤ C(

∫u

2nn−2ν |S(gν)− α∞f |

2nn+2dµ)

n+2n +

c

2(

∫|wν |

2nn−2dµ)

n−2n .

Moreover,

E[vν ]− α∞ = Ef [vν ](

∫fv

2nn−2ν dµ)

n−2n − α∞

= (Ef [vν ]− α∞)(

∫fv

2nn−2ν dµ)

n−2n + α∞[(

∫fv

2nn−2ν dµ)

n−2n − 1]

≤ (Ef [vν ]− α∞)(

∫fv

2nn−2ν dµ)

n−2n +

n− 2

nα∞[

∫fv

2nn−2ν dµ− 1] .

As calculated in [17] 7.3, we have the estimate

Ef [vν ] = Ef [V(xν ,εν)]

≤ α∞ + (ενδ−1ν )C[−1 +O((ενδ

−1ν )2)] +O(δ2

ν) .

By lemma 21, we have

Ef [vν ] ≤ α∞ + Cε2(n−2)

n(n−2)+2ν [−1 +O(ε

4n(n−2)+2ν )] +O(ε

2n(n−2)n(n−2)ν ) .

Thus, for sufficiently large ν,

Ef [vν ] ≤ α∞ − Cε2(n−2)

n(n−2)+2ν +O(ε

2n(n−2)n(n−2)ν ) ≤ 0 .

Now we only need to estimate the term

α∞

∫[−n+ 2

n− 2v

4n−2ν w2

ν+2(vν+wν)n+2n−2wν+

n− 2

nv

2nn−2ν −n− 2

n(vν+wν)

2nn−2 ]fdµ− c

2(

∫|wν |

2nn−2dµ)

n−2n .

CHAPTER 5. CONVERGENCE 28

We have a pointwise estimate due to Brendle [3] page 180,

| − n+ 2

n− 2v

4n−2ν w2

ν + 2(vν + wν)n+2n−2wν +

n− 2

nv

2nn−2ν − n− 2

n(vν + wν)

2nn−2 − n− 2

nw

2nn−2ν |

≤ C|vν |max(0, 6−nn−2

)|wν |min( 2nn−2

,3) .

Therefore,

| − n+ 2

n− 2v

4n−2ν w2

ν + 2(vν + wν)n+2n−2wν +

n− 2

nv

2nn−2ν − n− 2

n(vν + wν)

2nn−2 |

≤ C|vν |max(0, 6−nn−2

)|wν |min( 2nn−2

,3) +n− 2

n|w

2nn−2ν | ,

α∞

∫[−n+ 2

n− 2v

4n−2ν w2

ν + 2(vν + wν)n+2n−2wν +

n− 2

nv

2nn−2ν − n− 2

n(vν + wν)

2nn−2 ]fdµ

≤ C

∫|vν |max(0, 6−n

n−2)|wν |min( 2n

n−2,3)dµ+

n− 2

n

∫|w

2nn−2ν |dµ .

If n ≥ 6,

C

∫|vν |max(0, 6−n

n−2)|wν |min( 2n

n−2,3)dµ+

n− 2

n

∫|w

2nn−2ν |dµ

= C

∫|wν |

2nn−2dµ+

n− 2

n

∫|w

2nn−2ν |dµ

= o(

∫|wν |

2nn−2dµ)

n−2n ) .

If n < 6,

C

∫|vν |max(0, 6−n

n−2)|wν |min( 2n

n−2,3)dµ+

n− 2

n

∫|w

2nn−2ν |dµ

= C

∫|vν |

6−nn−2 |wν |3dµ+

n− 2

n

∫|w

2nn−2ν |dµ

≤ C(

∫|vν |

2nn−2dµ)

6−n2n (

∫|wν |

2nn−2dµ)

3(n−2)2n +

n− 2

n

∫|w

2nn−2ν |dµ

≤ C(

∫|wν |

2nn−2dµ)

3(n−2)2n +

n− 2

n

∫|w

2nn−2ν |dµ .

Since 3n−62n

> n−2n, the desired inequality holds for sufficiently large ν.

Corollary 2. For any c > 0 and 0 < γ < 1 there exists t0 > 0 such that, for any t > t0,

E[u]− α∞ ≤ c(

∫|S(g)− α∞f |

2nn+2dµg)

n+22n

(1+γ) .

CHAPTER 5. CONVERGENCE 29

Proof. If this is not true, there exists a γ and a time sequence tν such that

E[uν ]− α∞ > c(

∫|S(gν)− α∞f |

2nn+2dµgν )

n+22n

(1+γ) .

Meanwhile, for sufficiently large ν, we have

E[uν ]− α∞ ≤ C(

∫|S(gν)− α∞f |

2nn+2dµgν )

n+2n .

So we deduce

(

∫|S(gν)− α∞f |

2nn+2dµgν )

n+22n

(1−γ) >c

C

which is contradictory to corollary 1.

Lemma 25. For any 0 < γ < 1, there exists t0 > 0 and C > 0 such that, for any t > t0,

E[u]− α∞ ≤ C(

∫|S − αf |

2nn+2dµg)

n+22n

(1+γ) .

Proof. We have

(

∫|S(gν)− α∞f |

2nn+2dµgν )

n+22n

(1+γ) = ‖S − α∞f‖1+γ

L2nn+2

≤ C‖S − αf‖1+γ

L2nn+2

+ C‖α− α∞‖1+γ

L2nn+2

.

Since

α− α∞ = α− E[u] + E[u]− α∞

=

∫(αf − S)dµg + E[u]− α∞ ,

we have

‖α− α∞‖1+γ

L2nn+2≤ CF 1+γ

1 + C(E[u]− α∞)1+γ ,

(E[u]− α∞)1+γ ≤ c1+γ(

∫|S(gν)− α∞f |

2nn+2dµgν )

n+22n

(1+δ)(1+γ) ,

(E[u]− α∞)1+γ = o((

∫|S(gν)− α∞f |

2nn+2dµgν )

n+22n

(1+γ)) .

Thus the desired result follows.

By a similar argument in corollary 2, we can deduce

CHAPTER 5. CONVERGENCE 30

Corollary 3. For any c > 0 and 0 < γ < 1 there exists t0 > 0 such that, for any t > t0,

E[u]− α∞ ≤ c(

∫|S − αf |

2nn+2dµg)

n+22n

(1+γ) .

Proposition 2. ∫ ∞0

(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt ≤ C .

Proof. It follows from corollary 3 that there exists 0 < γ < 1, t0 > 0 and C > 0 such that, for

any t > t0,

E[u(t)]− α∞ ≤ C(

∫u(t)

2nn−2 |S − αf |

2nn+2dµ)

n+22n

(1+γ) .

It follows from Lemma 1 that

d

dt(E[u(t)]− α∞) ≤ −cF2 ≤ −cF

n+2n

2nn+2

≤ −c(E[u(t)]− α∞)2

1+γ , t ≥ t0.

Hence, for any t > t0,d

dt[(E[u(t)]− α∞)−

1−γ1+γ ] ≥ c ,

and

(E[u(t)]− α∞)−1−γ1+γ ≥ ct ,

E[u(t)]− α∞ ≤ Ct−1+γ1−γ .

It follows from Holder inequality that, for any T > t0,∫ 2T

T

(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt ≤ [T

∫ 2T

T

∫u(t)

2nn−2 (S − αf)2dµdt]

12

≤ C[T (E[u(2T )]− E[u(T )])]12

≤ C[T (E[u(2T )]− α∞)]12

≤ CT−γ

1−γ .

Finally, by picking L ∈ N such that 2L > t0, we have∫ ∞0

(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt =

∫ 2L

0

(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt

+∞∑k=L

∫ 2k+1

2k(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt

CHAPTER 5. CONVERGENCE 31

≤ C + C∞∑k=L

2−γ

1−γ k

≤ C .

From Proposition 2, we know that the volume does not concentrate:

Proposition 3. Given any η0 > 0, we can find some r > 0 such that∫Bx(r)

u(t)2nn−2dµ ≤ η0 , for all x ∈ N, t ≥ 0 .

Proof. Choose T > 0 such that∫ ∞T

(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt ≤ η0

n.

From the long time existence of the flow, we can choose r > 0 such that∫Bx(r)

u(t)2nn−2dµ ≤ η0 , for all x ∈ N, t ∈ [0, T ] .

Then, for t ≥ T we have∫Bx(r)

u(t)2nn−2dµ ≤

∫Bx(r)

u(T )2nn−2dµ+

n

2

∫ ∞T

(

∫u(t)

2nn−2 (S − αf)2dµ)

12dt

≤ η0 .

Proposition 4. There are positive constants c, C independent of t such that

c ≤ u(t) ≤ C, for all t ≥ 0 .

Proof. Fix n2< q < p < n+2

2. It follows from Lemma 2 and Lemma 14 that∫

|S|pdµg ≤ C

where C is a positive constant independent of t. By Proposition 3, we can find a constant

CHAPTER 5. CONVERGENCE 32

r > 0 independent of t such that∫Bx(r)

dµg ≤ η0 , for all x ∈ N, t ≥ 0 .

It follows from Holder inequality that∫|S|qdµg ≤ (

∫Bx(r)

dµg)p−qp (

∫|S|pdµg)

qp .

Hence, if we choose η0 small enough, then we have∫Bx(r)

u(t)qdµ ≤ η1 , for all x ∈ N, t ≥ 0 ,

where η1 is the constant in Proposition A.1 [3] and we can conclude that u(t) is uniformly

bounded from above. From Corollary A.3 [3] again, we know that u(t) is uniformly bounded

w.r.t. ‖(x, t)‖∞.

Thus, under the assumption that the weak limit u∞ = 0, we have proved u converges

strongly to a nonzero limit which is a contradiction. Thus the weakly limit is never zero

under small initial energy condition which proves Theorem 1.

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