SYMMETRIC EXISTENCE RESULTS, AND AGGREGATED, TOWERING BLOW UP SEQUENCES FOR THE PRESCRIBED SCALAR CURVATURE EQUATION ON SN

22 5 Examples concerning the bound 2.4 25 II Construction of blow-up sequences for the prescribing scalar curvature on Sn: aggregated blow-up and towering blow-up 27 6 Description of the

SYMMETRIC EXISTENCE RESULTS, AND AGGREGATED, TOWERING BLOW-UP SEQUENCES FOR THE PRESCRIBED

ZHOU FENG

(B.Sc., Nankai University, P R China)

A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF MATHEMATICS

NATIONAL UNIVERSITY OF SINGAPORE

2015

This thesis is dedicated to my late father Zhou Baoping.

His words of inspiration and encouragement in pursuit of excellence,

still linger on.

The completion of this PhD thesis has been a long journey I would like to takethis opportunity to extend my utmost appreciation to the people who have encour-aged and supported me throughout the process of writing this thesis

My heartfelt and profoundest gratitude goes first and foremost to Professor ung Man Chun, my supervisor, for his constant encouragement and enlighteningguidance that have significantly helped me in uniting this thesis, and his enthusi-asm and immense knowledge that have inspired me throughout my graduate studies

Le-I have been extremely lucky to have a supervisor who cared so much about my search I deeply appreciate his insightful guidance, valuable advice and periodicconstructive suggestions Without his consistent and illuminating instruction, thisthesis could not have reached its present form Studying under his guidance is anenjoyable, exhilarating and rewarding experience

re-I would like to thank the Department of Mathematics at the National University

of Singapore, especially those members of the Thesis Defense Committee, ProfessorChua Seng Kee and Professor Xu Xingwang, for their input, valuable discussionsand accessibility Deepest gratitude is also due to committee member, Professor

vii

viii Acknowledgements

Daniel Burns, for his expertise, patience and generous help

I owe my sincere gratitude to my brilliant friends from Department of ematics for the simulating discussions They always help me in exchanging ideasand give the enjoyable environment They made my life at NUS a truly memorableexperience and their friendships are invaluable to me

Math-Last but not the least, my very profound gratitude would go to my beloved ents for their unfailing support and endless love throughout my years of studies Icould not have finished my thesis without their understanding and encouragement.This thesis is dedicated to the memory of my father and to my mother

par-Some material presented in this thesis was adapted from an article published inProceedings of the American Mathematical Society

Zhou Feng September 2015

3.1 Long time existence, uniqueness and symmetry of flow equation 13

4.1 Projection from the north pole N 194.2 Choosing the initial data uo 20

ix

x Contents

4.3 Proof of Main Theorem 2.1 22

5 Examples concerning the bound (2.4) 25 II Construction of blow-up sequences for the prescribing scalar curvature on Sn: aggregated blow-up and towering blow-up 27 6 Description of the main results 29 6.1 Description of the main result of the case on aggregated blow - up 29

6.2 Description of the main result of the case on towering blow - up 31

7 The Lyapunov - Schmidt reduction scheme on the perturbed func-tional: the case of two bubbles 33 7.1 The flow chart 34

7.2 The perturbed functional 36

7.3 Separation inequalities and weak interaction lemmas 36

7.4 First order property – weak interaction between the two bubbles 38

7.5 Second order property – solving the equation in the perpendicular directions 41

7.6 Finite dimensional reduction 45

7.7 Coupled quasi - hyperbolic gradient 45

7.8 Estimates of k wzσk∇ and k λk· Dk`wzσk∇ in concrete setting 47

7.9 Extracting the key term in the reduced functional 49

8 Juxtaposed annular domains 53 8.1 The key term ( ε · Gk0/0) 54

8.2 Estimate of (i) in (8.8) 57

8.3 Spacing of the annular domains and estimation of (ii) in (8.8) 58

Contents xi

8.4 Estimate of (iii) in (8.8) 61

8.5 Estimate on the integral in (A)ii in Theorem 7.15 63

8.6 Blow - up sequence of solutions with juxtaposed bubbles 65

9 Superimposed annular domains 69 9.1 The key term ( ε · GHs j //) 70

9.2 Interference of the other annular domain – estimate of (i) in (9.6) 73

9.3 Inner and outer interferences – estimate of (ii) in (9.6) 76

9.4 Weak interaction – estimate of (iii) in (9.6) 78

9.5 Estimate on the integral in (T)ii in Theorem 7.15 79

9.6 Blow - up sequence of solutions with superimposed bubbles 81

Bibliography 84 Appendices 91 A Preliminaries 93 A.1 Non - compactness of critical Sobolev embedding 93

A.2 Transformation rule for conformal Laplacian 94

A.3 Separation inequalities 99

B Proofs of Weak Interaction Lemmas 103 B.1 Proof of Weak Interaction Lemma for juxtaposed bubbles 103

B.2 Proof of Weak Interaction Lemma for superimposed bubbles 110

C Second order information Io00 for juxtaposed bubbles 115 C.1 Uniform invertibility of Io00(Vλ, ξ) revisited 116

C.2 Arguments toward Lemma C.1 119

xii Contents

C.3 Restrictions to the balls and ‘almost’ perpendicularity 121C.4 Completion of the proof of Lemma C.1 128C.5 Estimate ofR

RnV14/(n−2)· |f2− f2

1| 131C.6 Estimate of |f | = |h1| on ∂Bo(R) 135C.7 Improved non - degeneracy 136

F Estimates on the derivatives of the solution to the auxiliary

F.1 Estimates of Io0(z∫)λk· Dk`z∫

 and Io00(z∫)λk· Dk`z∫ wz∫
151F.2 Estimate ofR

Rn|(z + w)τ − zτ − τ zτ −1w| · |h| with τ = n+2n−2 154F.3 Estimates of wz∫ 5 and λk· Dk`wz∫ 5 in concrete setting 156F.4 Proof of Proposition F.2 161

F ) We also demonstrate that the “one bubble” condition, namely,  max

on the hyperbolic structure on the collection of standard bubbles, and apply a degreetheory for the quasi - hyperbolic gradient

xiii

List of Symbols and Conventions

We list some symbols and conventions which will be used throughout this thesis

Sn n - sphere of radius 1, i.e (x1, · · · , xn+1) ∈ Rn+1 | x2

1+ · · · + x2n+1 = 1

g1 Induced metric on Sn

∆1 Laplace - Beltrami operator associated with g1 on Sn

go Euclidean metric on Rn

∆ Laplace - Beltrami operator associated with go on Rn

h · , · i Inner product defined via Euclidean metric go on Rn

k · k Norm defined via Euclidean metric go on Rn

Chapter 1

Introduction

In this thesis we consider the prescribed scalar curvature problem on the standardsphere (Sn, g1) , n ≥ 3 Let K be a fixed smooth function The question asks if onecan find a conformal metric g such that the scalar curvature becomes the prescribedfunction K

This kind of problem of finding metrics, on closed manifolds, with prescribed scalarcurvature, has been studied ongoing for several decades When the underlying man-ifold is the standard sphere Sn, the problem becomes much harder and more in-teresting A great deal of mathematical effort has been devoted to the study ofthe problem on Sn Earlier studies may be found in [16, 38, 46, 48] and referencestherein Letting g = un−24 · g1 on Sn with n ≥ 3 , it is well - known that the prob-lem is equivalent to finding a positive solution to the following prescribed scalarcurvature equation

2 Chapter 1 Introduction

also Lemma A.1

When K equals to a positive constant, say K = 4 n (n − 1) , i.e ˜cnK = n (n − 2) ,equation (1.1) has a family of positive solutions

y = ˙P(x) for x ∈ Sn\ {N} In particular, any sequence { Uλi, ξi}∞i=1 with λi → 0+

and ξi → ~0 forms a blow - up sequence

One of the earliest existence results for the prescribing scalar curvature problem isobtained by J Moser [41] in the early 1970s He shows that the problem has asolution for a function K on S2 which is invariant under the antipodal map

K(x) = K(−x)for x ∈ S2 ⊂ R3 ( K > 0 on S2) Then Escobar and Schoen [23] generalizethis result to dimension 3 They obtain an existence result for prescribed functions

K > 0 satisfying the symmetry condition

for γ ∈ Γ and x ∈ S3 ⊂ R4, where Γ is a finite group of isometries acting withoutfixed point except the identity on S3 In higher dimensions (n ≥ 4) , they requirethe prescribed function K to satisfy also the (n − 2) - flatness condition: there is

a point xM ∈ Sn such that

K(xM) = max

S n K and ∇jK(xM) = 0 for j = 1 , , n − 2 (1.4)See [23] In this thesis, we uniformly assume that K > 0 on Sn and is smooth,although some results can hold with weaker assumption on the sign and regularity

In [19] , W - X Chen deals with the situation when Γ consists non - empty fixedpoints Relevant to the discussion in this thesis, we highlight one of the consequences

of Chen’s work [19] : in the case ΓM = { R , Id } , where R is a ‘mirror’ reflection

upon a hyperplane H ⊂ Rn+1 passing through the origin [ see (2.1) ] , and there is a

point ¯xm ∈ H ∩ Sn so that

K(¯xm) = max

H∩S nK and ∆1K(¯xm) > 0 , (1.5)then together with the help of the (n − 2) - flatness condition (1.4) , one can find a

smooth positive solution U to the prescribing scalar curvature equation

∆1U − ˜cnn (n − 1) U + (˜cnK) Un+2n−2 = 0 (1.6)

on Sn, where K > 0 is smooth, n ≥ 3 , and ˜cn = 4(n−1)n−2 Refer to condition (5) of

the main theorem in [19] (p 355)

In recent years, there has been tremendous interest in developing the parabolic

method which has been used as an important tool to study equation (1.1) , yielding

new perspectives and insight into the equation See [11, 47, 48, 20] and the references

S n

K · [u(t , ·)]n−22n dVg1

(1.9)

is included to keep the volume of (Sn, gu) constant along the flow ( see [20] ) The

right hand side of (1.7) is equal to a positive constant times the negative gradient

4 Chapter 1 Introduction

of the energy quotient functional:

QK[u] =

1

kSnkZ

S n

Rgu· un−22n dVg1



1

kSnkZ

S n

K · un−22n dVg1

for u ∈ C+∞(Sn) That is, for u(t , ·) satisfying (1.7) ,

QK[ u(t + s , ·) ] ≤ QK[ u(t , ·) ] for s > 0 (t ≥ 0) (1.11)

See [20] We will summarize in Chapter 3 , the main features known so far for theflow equation

A significant development comes about when Ambrosetti and Malchiodi apply theelegant Lyapunov - Schmidt reduction method to find infinite number of so-lutions to the Yamabe equation (see [10, 12, 28] for further development) Thetechnique of Lyapunov - Schmidt reduction method is inspired by Sacks and Uh-lenbeck’s pioneering work [43] By considering the perturbed energy functionals,Sacks and Uhlenbeck have introduced the blow - up analysis of harmonic maps indimension 2 and established many existence results of minimizing harmonic maps

in homotopy classes Another inspiration to Lyapunov - Schmidt reduction methodcomes from the famous technique of “gluing an infinite number of instantons” in thegauge theory to study the Yang - Mills equations Refer to Taubes’ gauge - theoreticwork on the boundary of the moduli space of solutions to the Yang - Mills equations[49]

Ambrosetti and Malchiodi show the (n − 2) - flatness condition is not necessary if

K is sufficiently close to a constant [ (1.5) is still required; see Theorem 7.7 andRemark 7.10 in [3] for the full statement Cf [2] , also [35] ] Due to the nature ofthe method, the argument does not reveal an estimate on the ‘closeness’ This is insome sense provided by Main Theorem 2.1 in this thesis The proof of Theorem 2.1

is represented in Chapter 4

As illustrated by the non - existence example in [8] and [9], the (n − 2) - flatness

con-dition cannot be totally taken away if we allow K > 0 to vary far from a constant

We will discuss this aspect in chapter 5

Lyapunov - Schmidt reduction method also allows us to construct simple blow - up

sequence of positive solutions of equation (1.1) [35] Using similar finite dimension

reduction method, Wei and Yan consider in their ingenious work [51] the existence

of infinite number of solutions, amounting to the case of cluster blow - up

In the second part of thesis, we seek to apply the wonderful results of Lyapunov

-Schmidt reduction method [3] to construct non - constant function K such that the

prescribed scalar curvature equation (1.1) has an infinite number of positive

so-lutions {vi}∞i=1, which comprise an aggregated blow - up sequence of solutions

(Theorem 6.1), or a towering blow - up sequence of solutions (Theorem 6.2)

Roughly, the technique of Lyapunov - Schmidt reduction method can be summarized

as follows ( see the monograph of Ambrosetti and Malchiodi [3] ): one is interested in

looking for critical points of a real valued smooth functional Iε from infinite

dimen-sional Hilbert space H , knowing that for ε > 0 small enough there exists a finite

dimensional manifold Z ⊂ H made of almost critical points of Iε ( in the sense

that the differential Iε0 is small enough on Z ) If one also knows that the second

differential Iε00 restricted to the orthogonal complement of the tangent space to Z is

non - degenerate, then one can solve an auxiliary equation ( given by the projection

of the equation Iε0 = 0 onto the orthogonal complement of T Z ) and reduce the

prob-lem of finding critical points of the functional Iε : H → R to finding critical points

of a suitable functional Φε : Z → R , i.e a functional of finitely many variables

More precisely, via the stereographic projection ˙P, equation (1.1) is transformed

into

6 Chapter 1 Introduction

in Rn See (7.3) for the relations between u and v , and between K and K Wewrite

K = 4 n (n − 1) + ε H Then the Euler functional corresponding to question (1.1) can be expressed as

Iε = Io + ε Gwhere Io plays the role of the unperturbed functional and G is the perturbation.Refer to S 7.2 for more details

The key objects is the collection of functions

We consider the interaction of function H with Z :

For ε ∈ R small, the question on finding a critical point of Iε is reducing to finding

a stable critical point of the perturbation GH(z) constrained on Z Refer to [35] Next we introduce annular domains (1.14) , and determine precisely the critical

points of the reduced functional, and show that the critical points are non

-degenerate, hence a stable critical point By juxtaposing or superimposing the

annular domains, and estimating the gradient interference, we are able to find

infi-nite number of stable critical points via degree theories for maps

Let

Ooη = Bη( t + ∆ ) \ Bη( t − ∆ ) , (1.14)where η ∈ Rn, t > ∆ > 0 are given real numbers In this thesis, very often, either

η = 0 or η → ~0 Given the standard bubbles

the situation of a single annular domain [ say, η = 0 ] is analyzed in [35] , in which

we show that the restricted ‘functional’

is the geometric mean of the outer and inner radii

The gradient changes for the function defined in (1.16) is estimated in [35] When

there are two annulus domains, the gradient contributions can be symmetric (in

the juxtaposition case) or asymmetric (in the superposition case) To handle the

asymmetric situation, we introduce the quasi - hyperbolic gradient (see S 7.7) ,

which can be observed naturally once we put the hyperbolic structure into the

picture Unexpectedly, we emphasize that this notation enables us to fuse together

the two cases ( ‘horizontal’ and ‘vertical’ ) for this problem

To obtain the results, we introduce three stages of separations [ cf (8.8) and (9.6) ],

and apply a degree theory for the quasi - hyperbolic gradient given in S 7.7 Finally,

the proofs of Theorem 6.1 and Theorem 6.2 are presented in Chapter 8 and Chapter

9 , respectively

Part I

Prescribed scalar curvature

equation on S n in the presence of reflection or rotation symmetry

9

Chapter 2

Description of the main results

In the first part of this thesis, we consider two types of symmetries

(2.1) K is symmetric for a mirror reflection upon a hyperplane H ⊂ Rn+1 ( H passesthrough the origin ) As the situation is invariant under a rotation, withoutloss of generality, we assume that H is the hyperplane perpendicular to the

x1- axis In this way, the symmetry is expressed as

K(γm(x)) = K(x),

where γm : Sn → Snis given by γm(x1, x2, , xn+1) = (−x1, x2, , xn+1)for x = (x1, x2, , xn+1) ∈ Sn As such F = { (0 , x2, , xn+1) ∈ Sn} =

H ∩ Sn is the fixed point set

(2.2) K is invariant under a rotation γθ of angle θ = π

k (with the axis being astraight line in Rn+1 passing through the origin) Here k > 1 is an integer.Likewise, without loss of generality, we take it that indeed the straight line isthe xn+1- axis In this case the fixed point set is F = {N, S}, where N is thenorth pole and S is the south pole

Theorem 2.1 Suppose that K > 0 is a smooth function on Sn which is invariant

11

12 Chapter 2 Description of the main results

under the symmetry described in either (2.1) or (2.2) Assume that

xm ∈ F with K(xm) = max

F K & ∆1K(xm) > 0 , (2.3)and

max

S n Kτ < 2 ·max

where τ = n−22 Then equation (1.6) has a smooth positive solution V for K

Condition (2.3) can be seen as a type of “saddle point behavior” on the restrictedmaximum on F The inequality in (2.4) is not altogether technical In Chapter 5 ,

we describe examples for K which fulfill symmetry condition (2.1) and Laplaciancondition (2.3) , but with

max

S n Kmax

F K >

1

ε  1 ,

so that equation (1.6) does not have a positive solution However, the construction

in Chapter 5 reveals only a rough estimate (non - sharp) on ε

In the literature, there are major works done on the prescribed scalar equation(1.6) via blow - up analysis and Morse theory We refer the interested readers to[7, 16, 18, 34, 38, 46], and the references therein In comparison, one advantage ofMain Theorem 2.1 is that it requires information only on specific critical points inthe fixed point set

Chapter 3

Symmetry and blow-up point for the flow equation

symme-try of flow equation

In the first part of this thesis, we consider only smooth and positive initial datafor equation (1.7) Thus let uo ∈ C∞

+(Sn) Via Lemmas 2.7 and 2.11 in [20] andthe proceeding statements, equation (1.7) has a unique smooth solution u(t, x) on[0, ∞) × Sn such that

14 Chapter 3 Symmetry and blow-up point for the flow equation

for all x ∈ Sn and t ∈ [ 0, ∞)

Proof Let Φ : Ω → Sn be a local chart, with (y1, , yn) the local coordinates.Here Ω is an open domain in Rn For f ∈ C∞(Sn) , by the diagram

∂yj

,

we obtain

for x ∈ Sn Note that γ is an isometry Consequently [ cf (1.8) ] ,

∆1u(t , x) − ˜cnn (n − 1) u(t , x) + ˜cnRgu(t , ·)(x) [ u(t , x) ]n+2n−2 = 0

=⇒ Rgu(t , γ(·))

x

= Rgu(t , ·)

... degenerate, then one can solve an auxiliary equation ( given by the projection

of the equation Iε0 = onto the orthogonal complement of T Z ) and reduce the

prob-lem...

In the literature, there are major works done on the prescribed scalar equation( 1.6) via blow - up analysis and Morse theory We refer the interested readers to[7, 16, 18, 34, 38, 46], and the. ..

1

= L1−2n · QK[ui]

1

≤ L1−2n · QK[uo]