SYMMETRIC EXISTENCE RESULTS, AND AGGREGATED, TOWERING BLOW-UP SEQUENCES FOR THE PRESCRIBED SCALAR CURVATURE EQUATION ON S N 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 DEDICATION This thesis is dedicated to my late father Zhou Baoping His words of inspiration and encouragement in pursuit of excellence, still linger on Acknowledgements The completion of this PhD thesis has been a long journey I would like to take this opportunity to extend my utmost appreciation to the people who have encouraged and supported me throughout the process of writing this thesis My heartfelt and profoundest gratitude goes first and foremost to Professor Leung Man Chun, my supervisor, for his constant encouragement and enlightening guidance that have significantly helped me in uniting this thesis, and his enthusiasm and immense knowledge that have inspired me throughout my graduate studies I have been extremely lucky to have a supervisor who cared so much about my research I deeply appreciate his insightful guidance, valuable advice and periodic constructive suggestions Without his consistent and illuminating instruction, this thesis could not have reached its present form Studying under his guidance is an enjoyable, exhilarating and rewarding experience I would like to thank the Department of Mathematics at the National University of Singapore, especially those members of the Thesis Defense Committee, Professor Chua Seng Kee and Professor Xu Xingwang, for their input, valuable discussions and 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 Mathematics for the simulating discussions They always help me in exchanging ideas and give the enjoyable environment They made my life at NUS a truly memorable experience and their friendships are invaluable to me Last but not the least, my very profound gratitude would go to my beloved parents for their unfailing support and endless love throughout my years of studies I could not have finished my thesis without their understanding and encouragement This thesis is dedicated to the memory of my father and to my mother Some material presented in this thesis was adapted from an article published in Proceedings of the American Mathematical Society Zhou Feng September 2015 Contents Acknowledgements Summary vii xiii List of Symbols and Conventions Introduction I xv Prescribed scalar curvature equation on S n in the pres- ence of reflection or rotation symmetry Description of the main results 11 Symmetry and blow-up point for the flow equation 13 3.1 Long time existence, uniqueness and symmetry of flow equation 13 Proof of the Main Theorem 2.1 19 4.1 Projection from the north pole N 19 4.2 Choosing the initial data uo 20 ix x Contents 4.3 Proof of Main Theorem 2.1 22 Examples concerning the bound (2.4) II 25 Construction of blow-up sequences for the prescribing scalar curvature on S n : aggregated blow-up and towering blow-up 27 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 The Lyapunov - Schmidt reduction scheme on the perturbed functional: 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 7.4 First order property – weak interaction between the two bubbles 38 7.5 Second order property – solving the equation in the perpendicular 36 directions 41 7.6 Finite dimensional reduction 45 7.7 Coupled quasi - hyperbolic gradient 45 7.8 Estimates of wzσ 7.9 Extracting the key term in the reduced functional 49 ∇ and λk · Dk wzσ Juxtaposed annular domains ∇ in concrete setting 47 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 F.2 Estimate of Rn |(z + w)τ − zτ − τ zτ −1 w| · |h| with τ = n+2 n−2 155 Here C is a positive constant depending on the dimension n Set a := c with b > b Then we have 1+ =⇒ b =⇒ if |c| b n+2 n−2 n+2 n−2 c b n+2 n−2 c 1+ b · n+2 c · n−2 b −1− n+2 −1− n+2 (b + c) n−2 − b n−2 − n+2 n−2 −b n+2 c · n−2 b (b + c) =⇒ (b + c) n−2 − b n−2 − n+2 c b c ≤ b 2 c b n+2 ≤ C · b n−2 · n + n−2 ·b ·c n−2 c2 ≤ C· n−6 b n−2 n + n−2 − ·b ·c n−2 n+2 n−2 =⇒ n+2 ≤ C· ≤ C· n + n−2 ·b ·c n−2 n−6 n−2 |c| b n+2 · |c| n−2 n+2 ≤ C · γ · |c| n−2 , n−6 n−2 < γ Now we let b := z∫ and c := wz∫ It follows that z∫ + wz∫ n+2 n−2 n + n−2 · z ∫ · w z∫ n−2 n+2 − z∫n−2 − Let Ωγ := y∈R w z∫ z∫ n ≤ C · γ · w z∫ n+2 n−2 n−6 n−2 < γ Hence z∫ + wz∫ n+2 n−2 n+2 − z∫n−2 − Ωγ ≤ C ·γ· w z∫ n+2 n−2 n + n−2 · z∫ · wz∫ · |h| n−2 · |h| Ωγ ≤ C ·γ· wz∫ n+2 2n · n−2 n+2 n+2 2n · h Ωγ ≤ C · γ · w z∫ n+2 n−2 · h On the other hand, z∫ + wz∫ Rn \Ωγ n+2 n−2 n+2 − z∫n−2 − n + n−2 · z∫ · wz∫ · |h| n−2 156 Chapter F Estimates on the derivatives of the solution to the auxiliary equation ≤ n+2 n−2 z∫ + w z ∫ Rn \Ωγ n+2 n−2  = Rn \Ωγ z∫ +1 w z∫  n+2 n−2 n−2 n + n−2 · z ∫ · w z∫ n−2 n+2 n−2 z∫ w z∫ + + γ − n−6 ≤ n+2 + z∫n−2 + + n+2 n−2 n−2 + γ − n−6 + Rn \Ωγ n−6 n−2 w z∫ z∫ n+2 ≤ C γ − n−6 wz ∫ n+2 · n−2 n−2 z∫ w z∫  n+2 n−2  · w z∫ n−2 n−2 n+2 · γ − n−6 n−2 · wz ∫ n+2 n−2 · |h| · |h| n−2 z∫ ≤ γ − n−6 wz∫ ≥ γ =⇒ n+2 n−2 · |h| · |h| Rn \Ωγ n+2 n−2 n+2 ≤ C γ − n−6 · wz∫ · h Thus we obtain z ∫ + w z∫ n+2 n−2 Rn F.3 n+2 − z∫n−2 − n + n−2 · z∫ · wz∫ · |h| ≤ C · wz∫ n−2 Estimates of wz∫ n+2 n−2 and λk · Dk wz∫ · h in con- crete setting ➀ Estimate of wz∫ : Let wz∫ ∈ ⊥ σ be the unique solution to the auxiliary equation [cf (7.31)] Under the conditions in the weak interaction lemma 7.4 , for n ≥ , and |ε| ≤ ε¯o , and h := λ2 ¯1 , ≥ h λ1 we have w z∫ ≤ C· |ε · H| Rn Starting with the auxiliary equation = Pσ ◦ Iε (z∫ + wz∫ )[h] 2n n+2 2n n−2 · z∫ n+2 2n + h n+2 − o (1) and λk · Dk wz∫ F.3 Estimates of wz∫ in concrete setting ∇(z∫ + wz∫ ), ∇h − n(n − 2) z∫ + wz∫ = Rn − c˜n [ ε · H ] · z∫ + wz∫ Rn n+2 n−2 + n+2 n−2 + 157 ·h · h for h ∈ ⊥ σ , we obtain n+2 ∇z∫ , ∇h − n(n − 2)z∫n−2 · h = Rn ∇wz∫ , ∇h − n(n + 2)z∫n−2 · wz∫ · h + Rn − n(n − 2) z∫ + w z ∫ n+2 n−2 Rn − c˜n [ ε · H ] · z∫ + wz∫ Rn ➁ Estimate of λk · Dk wz∫ n+2 n−2 + n+2 − z∫n−2 − n + n−2 · z∫ · wz∫ n−2 ·h · h : Starting with = Pσ ◦ Iε z∫ + wz∫ [h] = Pσ ◦ Io z∫ + wz∫ [h] + Pσ ◦ [ε · G ] z∫ + wz∫ [h] , we have = Pσ ◦ Iε z∫ + wz∫ λk · Dk z∫ + λk · Dk wz∫ = Pσ ◦ Io z∫ + wz∫ λk · Dk z∫ + λk · Dk wz∫ + Pσ ◦ [ε · G ] z∫ + wz∫ λk · Dk z∫ + λk · Dk wz∫ Note that for h ∈ D1 , , λk · Dk z∫ + λk · Dk wz∫ h Io z∫ + wz∫ = Io (z∫ ) λk · Dk wz∫ h − n(n + 2) z∫ + wz∫ Rn Io (z∫ ) λk · Dk z∫ h + n−2 + − z∫n−2 · λk · Dk z∫ + λk · Dk wz∫ · h 158 Chapter F Estimates on the derivatives of the solution to the auxiliary equation When n ≥ 6, the function f (x) = x n−2 is concave on [0, ∞) as f (x) < for all x > Then we have f (a) + f (b) = f (a + b) · a a+b +f (a + b) · b a+b a b · f (a + b) + · f (a + b) = f (a + b) a+b a+b ≥ This implies that n−2 z ∫ + w z∫ + − z∫n−2 ≤ wz ∫ n−2 We encounter estimates of the following type For f, g, h ∈ D1 , , f n−2 · g · h Rn ≤ n−2 f n+2 2n 2n n+2 · g · h Rn 2n n−2 n−2 2n Rn applying H¨older’s inequality with p = ≤ C1 f · 2n n−2 n+2 · g 2n 2n and q = n+2 n−2 n+2 2n 2n n+2 · h (applying Sobolev inequality) Rn ≤ f C1 n+2 · 2n · n+2 n−2 n+2 · 2n n+2 · n+2 n−2 g Rn C1 f 2n n−2 · n+2 n+2 2n · g Rn ≤ C2 n+2 2n · h Rn applying H¨older’s inequality with p = = n−2 n+2 2n n−2 n+2 n+2 and q = n−2 n−2 n+2 · n+2 2n · h Rn f 2n n−2 n · g · h (applying Sobolev inequality) Rn = C2 f 2n n−2 n−2 2n n−2 · g · h Rn ≤ C3 f n−2 · g · h (applying Sobolev inequality) and λk · Dk wz∫ F.3 Estimates of wz∫ in concrete setting 159 Then n−2 z ∫ + w z∫ + Rn ≤ wz∫ n−2 · − z∫n−2 · λk · Dk z∫ + λk · Dk wz∫ · h · |h| λk · Dk z∫ + λk · Dk wz∫ Rn ≤ n−2 w z∫ · λk · Dk z∫ · |h| + w z∫ Rn n−2 · λk · Dk wz∫ · |h| Rn ≤ w z∫ · 2n n−2 n+2 · λk · Dk z∫ 2n n+2 n+2 2n · h Rn w z∫ + · 2n n−2 n+2 · λk · Dk wz∫ 2n n+2 n+2 2n · h Rn λk · Dk z∫ ≤ 2n n+2 · n+2 n−2 n−2 n+2 · n+2 2n · Rn w z∫ · n+2 n+2 2n · h Rn n−2 n+2 · n+2 2n 2n n+2 · n+2 n−2 λk · Dk wz∫ + · Rn · n+2 n+2 2n · 2n · n+2 n−2 n+2 w z∫ · h Rn λk · Dk z∫ = · 2n · n+2 n−2 n+2 · + λk · Dk wz∫ 2n n−2 w z∫ n−2 · 2n n−2 · h Rn λk · Dk z∫ ≤ + λk · Dk wz∫ ≤ C · C + λk · Dk wz∫ n−2 · w z∫ · |ε · H| 2n n+2 Rn Next we consider the term ε·G = − · c˜n · [ε · H] · z∫ + wz∫ Rn ≤ C Rn z∫ + w z ∫ 2n n−2 · z∫ Rn + n−2 h n+2 · h − o (1) λk · Dk z∫ + λk · Dk wz∫ h |ε · H| · z∫n−2 + wz∫ n−2 · n−2 + : · λk · Dk z∫ + λk · Dk wz∫ · h λk · Dk z∫ + λk · Dk wz∫ ≤ C n+2 2n λk · Dk z∫ + λk · Dk wz∫ h z ∫ + w z∫ n+2 n−2 ε·G · h |ε · H| · z∫n−2 · λk · Dk z∫ · |h| + C · |h| Rn |ε · H| · z∫n−2 · λk · Dk wz∫ · |h| 160 Chapter F Estimates on the derivatives of the solution to the auxiliary equation n−2 |ε · H| · wz∫ +C · λk · Dk z∫ · |h| Rn n−2 |ε · H| · wz∫ +C · λk · Dk wz∫ · |h| Rn ≤ C Rn |ε · H| · z∫n−2 · z∫ · |h| + C n−2 |ε · H| · wz∫ +C Rn |ε · H| · z∫n−2 · λk · Dk wz∫ · |h| · z∫ · |h| + C |ε · H| · wz∫ Rn ≤ C· n−2 · λk · Dk wz∫ · |h| Rn |ε · H| 2n n+2 Rn |ε · H| +C· n+2 2n · n−2 n+2 |ε · H| 2n n−2 2n n−2 Rn + C · |ε| · wz∫ · h · z∫ Rn +C· n+2 2n n−2 n−2 2n · 2n n−2 n−2 · z∫ 2n n−2 · λk · Dk wz∫ n−2 2n · z∫ · w z∫ · λk · Dk wz∫ n−2 · h · h · h Since Pσ ◦ Io (z∫ ) λk · Dk wz∫ h = − n(n + 2) z ∫ + w z∫ Rn + ε · [Pσ ◦ G ] z∫ + wz∫ + n−2 + Pσ ◦ Io (z∫ ) λk · Dk z∫ h − z∫n−2 · λk · Dk z∫ + λk · Dk wz∫ · h λk · Dk z∫ + λk · Dk wz∫ h , it follows that Pσ ◦ Io (z∫ ) λk · Dk wz∫ h ≤ Pσ ◦ Io (z∫ ) λk · Dk z∫ h + z∫ + w z ∫ Rn n−2 + − z∫n−2 · λk · Dk z∫ + λk · Dk wz∫ · h + ε · [Pσ ◦ G ] z∫ + wz∫ ≤ C· h − o (1) · h λk · Dk z∫ + λk · Dk wz∫ F.4 Proof of Proposition F.2 161 + C · C + λk · Dk wz∫ · |ε · H| · z∫ Rn + C· |ε · H| 2n n+2 Rn + C· |ε · H| 2n n−2 Rn + C· |ε · H| 2n n−2 Rn + C · |ε| · wz∫ n−2 + n−2 h n+2 − o (1) · h n+2 2n n+2 2n · n−2 n+2 · z∫ · h · 2n n−2 n−2 n−2 2n · λk · Dk wz∫ · z∫ 2n n−2 n+2 2n 2n n−2 2n n+2 n−2 2n · w z∫ · z∫ · λk · Dk wz∫ n−2 · h · h · h That is, λk · Dk wz∫ ≤ C· h1 − o (1) + C· |ε · H| 2n n+2 Rn |ε · H| + C· 2n n+2 Rn + C· |ε · H| Rn 2n n−2 n+2 2n 2n n−2 · z∫ 2n n−2 + n−2 h n+2 −o(1) n+2 2n · z∫ 2n n−2 n−2 2n · z∫ · |ε · H| 2n n+2 Rn 2n n−2 · z∫ n+2 2n + h n+2 − o (1) Therefore, we obtain λk · Dk wz∫ F.4 ≤ C h − o (1) |ε · H| + Rn 2n n+2 2n n−2 · z∫ 2(n+2) n(n−2) Proof of Proposition F.2 Proposition F.2 For n ≥ , let z∫ = Vλ1 , ξ1 + Vλ2 , ξ2 We assume the followings (i) ε and (λ1 , ξ1 ; λ2 , ξ2 ) ∈ (R+ × Rn ) × (R+ × Rn ) satisfy the conditions n−2 162 Chapter F Estimates on the derivatives of the solution to the auxiliary equation |ξ1 − ξ2 | λ2 ¯1 ≤ C1 , and h := ≥ h λ1 λ1 |ε| ≤ ε¯1 , (ii) |H| ≤ C2 in Rn (iii) h1 − o (1) |ε · H| ≥ 2n n−2 2n n+2 2(n+2) n(n−2) · z∫ Rn at (λ1 , ξ1 ; λ2 , ξ2 ) Then we have ≤ C¯ · (λ · ∇) IRH (z∫ ) − ε · GH (z∫ ) h n+2 − o (1) Here C¯ is independent on H , ε and z∫ In addition, o(1) → 0+ when h → ∞ Proof We start with the definition of IRH (z∫ ) IRH (z∫ ) = Iε z∫ + wz∫ = ∇(z∫ + wz∫ ), ∇(z∫ + wz∫ ) − Rn [ ε · H ] · z∫ + wz∫ + c¯−1 Rn λk · Dk IRH (z∫ ) = 2n n−2 + (n − 2)2 z∫ + w z ∫ Rn 2n n−2 + ∇(z∫ + wz∫ ), ∇ λk · Dk z∫ + λk · Dk wz∫ Rn − n(n − 2) z∫ + wz∫ Rn − c˜n [ ε · H ] · z ∫ + w z∫ Rn Recall that c¯−1 = − n+2 n−2 + · λk · Dk z∫ + λk · Dk wz∫ n+2 n−2 + · λk · Dk z∫ + λk · Dk wz∫ n−2 · c˜n That is, 2n n+2 λk · Dk IRH (z∫ ) = ∇z∫ , ∇ λk · Dk z∫ Rn − n(n − 2) λk · Dk z∫ · z∫n−2 ↑ = Io (z∫ ) λk · Dk z∫ ∇z∫ , ∇ λk · Dk wz∫ + Rn Rn ∇wz∫ , ∇ λk · Dk wz∫ + Rn ∇wz∫ , ∇ λk · Dk z∫ + F.4 Proof of Proposition F.2 163 − n(n − 2) λk · Dk z∫ · z∫ + wz∫ Rn λk · Dk wz∫ · z∫ + wz∫ − n(n − 2) Rn n+2 n−2 + n+2 − z∫n−2 n+2 n−2 + n+2 − c˜n Rn [ ε · H ] · λk · Dk z∫ · z∫n−2 ε · GH (z∫ ) ↑ = λk · Dk − c˜n [ ε · H ] · λk · Dk z∫ · (key term) z∫ + w z ∫ Rn − c˜n [ ε · H ] · λk · Dk wz∫ · z∫ + wz∫ Rn n+2 n−2 + n+2 − z∫n−2 n+2 n−2 + It follows that λk · Dk IRH (z∫ ) = λk · Dk ε · GH (z∫ ) + Io (z∫ ) λk · Dk z∫ ∇wz∫ , ∇ λk · Dk z∫ = Rn − n(n + 2) z∫n−2 · λk · Dk z∫ · wz∫ ↑ = − n(n − 2) Io (z∫ ) λk · Dk z∫ wz∫ λk · Dk z∫ · z∫ + wz∫ Rn − n(n − 2) λk · Dk wz∫ · z∫ + wz∫ Rn n+2 n−2 + n+2 − z∫n−2 − n+2 n−2 z∫n−2 · wz∫ n+2 n−2 + − ∆z∫ · λk · Dk wz∫ + n(n + 2) Rn Rn z∫n−2 · λk · Dk wz∫ · wz∫ n+2 n+2 + Vλn−2 ↑ ∆z∫ = −n(n − 2) Vλn−2 , ξ1 , ξ2 ∇wz∫ , ∇ λk · Dk wz∫ + Rn ↑ = − n(n + 2) z∫n−2 · λk · Dk wz∫ · wz∫ Io (z∫ ) λk · Dk wz∫ wz∫ 164 Chapter F Estimates on the derivatives of the solution to the auxiliary equation − c˜n [ ε · H ] · λk · Dk z∫ · n+2 n−2 z ∫ + w z∫ + Rn − c˜n n+2 − z∫n−2 n+2 n−2 [ ε · H ] · λk · Dk wz∫ · z∫ + wz∫ + Rn So we have λk · Dk IRH (z∫ ) = λk · Dk ε · GH (z∫ ) + Io (z∫ ) λk · Dk z∫ + Io (z∫ ) λk · Dk z∫ wz∫ Io (z∫ ) λk · Dk wz∫ wz∫ + + I + II + III + IV + V , where I = −n(n − 2) λk · Dk z∫ · z∫ + wz∫ Rn II = −n(n − 2) n+2 n−2 + z∫ + wz∫ [ ε · H ] · λk · Dk z∫ · z ∫ + w z∫ + [ ε · H ] · λk · Dk wz∫ · z∫ + wz∫ + Rn III = −˜ cn IV = −˜ cn Rn n+2 V = −n(n − 2) Rn − z∫n−2 − + Rn n+2 n−2 n+2 n+2 n−2 λk · Dk wz∫ · n+2 n−2 n+2 − z∫n−2 − n+2 n−2 n+2 n−2 z∫n−2 · wz∫ , z∫n−2 · wz∫ , n+2 − z∫n−2 , , n+2 n+2 λk · Dk wz∫ · z∫n−2 − Vλn−2 + Vλn−2 , ξ1 , ξ2 From Lemma F.1 , we have Io (z∫ ) λk · Dk z∫ ≤ C· h n+2 − o (1) , (i) and Io (z∫ ) λk · Dk z∫ wz∫ ≤ C· h − o (1) · wz∫ (ii) Using the uniform bound of the operator Io (z∫ ), one has Io (z∫ ) λk · Dk wz∫ wz∫ ≤ C · λk · Dk wz∫ · w z∫ (iii) In the following, we specify n ≥ , |I| ≤ C · wz∫ n+2 n−2 · λk · Dk z∫ ≤ C · w z∫ n+2 n−2 ; (iv) F.4 Proof of Proposition F.2 165 n+2 n−2 |II| ≤ C · wz∫ · λk · Dk wz∫ (v) By (A.7) , we have for a > b > and τ > , ≤ Cτ · aτ −1 · b (a + b)τ − (aτ + bτ ) Here Cτ is a positive constant depending on τ , but not on a and b It follows that |III| = c˜n | ε · H | · λk · Dk z∫ · z ∫ + w z∫ Rn n+2 n−2 + n+2 − z∫n−2 | ε · H | · λk · Dk z∫ · z∫n−2 · wz∫ + wz∫ ≤ C Rn n+2 n−2 n+2 | ε · H | · z∫n−2 · wz∫ ≤ C Rn |ε · H | ≤ C 2n n+2 Rn |ε · H | ≤ C 2n n+2 Rn 2n n−2 | ε · H | · z ∫ · w z∫ + C n+2 n−2 Rn n+2 2n · z∫ · w z∫ + C · z∫ · wz∫ n+2 2n · n−2 n+2 n+2 2n Rn 2n n−2 n+2 2n · z∫ · w z∫ + C · w z∫ n+2 n−2 (vi) Let us continue |IV| = c˜n | ε · H | · λk · Dk wz∫ · n+2 n−2 z ∫ + w z∫ + Rn n+2 | ε · H | · z∫n−2 · λk · Dk wz∫ + C ≤ C Rn |ε · H | ≤ C 2n n+2 Rn + C · w z∫ n+2 |V| ≤ C Rn 2n n−2 n+2 · Rn ≤ C Rn ; · λk · Dk wz∫ n+2 · n−2 Vλ21 , n−2 ξ · V λ2 , ξ n n−2 (vii) n+2 z∫n−2 − Vλn−2 + Vλn−2 , ξ1 , ξ2 ≤ C n n−2 Vλ1 , ξ1 · Vλ2 , ξ2 · λk · Dk wz∫ Rn · λk · Dk wz∫ · λk · Dk wz∫ n+2 n+2 n−2 n+2 2n · z∫ n+2 n−2 | ε · H | · w z∫ · λk · Dk wz∫ n+2 2n · λk · Dk wz∫ Rn 2n n−2 n−2 2n 166 Chapter F Estimates on the derivatives of the solution to the auxiliary equation ≤ C· ≤ C· n+2 2n · λk · Dk wz∫ n h − o (1) h n+2 − o (1) · λk · Dk wz∫ (viii) The condition h − o (1) ≥ |ε · H| 2n n+2 Rn n+2 · 2n n−2 2n n−2 · z∫ implies that n−2 h − o (1) ≥ |ε · H| 2n n+2 Rn n+2 2n 2n n−2 · z∫ We have w z∫ ≤ C· h n+2 − o (1) ≤ C· h n−2 , and − o (1) λk · Dk wz∫ ≤ C· h − o (1) This gives us Io (z∫ ) λk · Dk z∫ ≤ C· Io (z∫ ) λk · Dk z∫ wz∫ ≤ C· Io (z∫ ) λk · Dk wz∫ wz∫ ≤ C· |I| ≤ C · |II| ≤ C · |III| ≤ C · |IV| ≤ C · |V| ≤ C · h n+2 h n+2 h n+2 h n+2 h n+6 h n+2 h n+2 h n+6 − o (1) − o (1) − o (1) − o (1) − o (1) − o (1) − o (1) − o (1) ; (i) ; (ii) ; (iii) ; (iv) ; (v) ; (vi) ; (vii) (viii) Hence λk · Dk IRH (z∫ ) − ε · GH (z∫ ) ≤ C¯ · h n+2 − o (1) , F.4 Proof of Proposition F.2 for k = , , 167 = , , , n Therefore, we conclude that (λ · ∇) IRH (z∫ ) − ε · GH (z∫ ) ≤ C¯ · h n+2 − o (1) Here C¯ is independent on H , ε and z∫ , and o (1) → 0+ when h → ∞ We remark that the situation for juxtaposed bubbles is similar SYMMETRIC EXISTENCE RESULTS, AND AGGREGATED, TOWERING BLOW-UP SEQUENCES FOR THE PRESCRIBED SCALAR CURVATURE EQUATION ON S N ZHOU FENG NATIONAL UNIVERSITY OF SINGAPORE 2015 Symmetric existence results, and aggregated, towering blow-up sequences ZHOU FENG 2015 [...]... results on blow - up sequences of infinite number of solutions for the prescribed (and fixed) scalar curvature equation on S n (n ≥ 6) , including aggregated and towering blow ups The constructions make use of the Lyapunov - Schmidt reduction method, count on the hyperbolic structure on the collection of standard bubbles, and apply a degree theory for the quasi - hyperbolic gradient xiii List of Symbols and. .. 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 of Main Theorem 2.1 is that it requires information only on specific critical points in the fixed point... point set Chapter 3 Symmetry and blow- up point for the flow equation 3.1 Long time existence, uniqueness and symmetry of flow equation In the first part of this thesis, we consider only smooth and positive initial data for equation (1.7) Thus let uo ∈ C+∞ (S n ) Via Lemmas 2.7 and 2.11 in [20] and the proceeding statements, equation (1.7) has a unique smooth solution u(t, x) on [0, ∞) × S n such that...  Condition (3.11) not only confines the blow - up to one point but also makes the blow - up ‘simple’ (that is, developing only one bubble) ; cf the last estimate above in curly brackets and [34] The case of exactly one ‘bubble’ is analyzed in [20] Specifically, Theorem 4.1, Proposition 6.1 and Lemma 6.11 in [20] lead to the following Lemma 3.5 Under the conditions and notation in Lemmas 3.3 and. .. 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. .. function The question asks if one can find a conformal metric g such that the scalar curvature becomes the prescribed function K This kind of problem of finding metrics, on closed manifolds, with prescribed scalar curvature, has been studied ongoing for several decades When the underlying manifold is the standard sphere S n , the problem becomes much harder and more interesting A great deal of mathematical... 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 solutions {vi }∞ i=1 , which... enough there exists a finite dimensional manifold Z ⊂ H made of almost critical points of Iε ( in the sense that the differential Iε is small enough on Z ) If one also knows that the second differential Iε 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 Iε = 0 onto the orthogonal... flow equation for the prescribed scalar curvature equation to obtain existence theorems in cases where the prescribed function K exhibits reflection or rotation symmetry (with fixed point set denoted by F ) We also demonstrate that the “one bubble” condition, namely, 2· max K F τ cannot be totally taken away Here τ = n−2 2 max K n S τ < In the second part, by using annular domains, we obtain constructive... J Moser [41] in the early 1970s He shows that the problem has a solution for a function K on S 2 which is invariant under the antipodal map K(x) = K(−x) for x ∈ S 2 ⊂ R3 ( K > 0 on S 2 ) Then Escobar and Schoen [23] generalize this result to dimension 3 They obtain an existence result for prescribed functions K > 0 satisfying the symmetry condition K(γ(x)) = K(x) (1.3) for γ ∈ Γ and x ∈ S 3 ⊂ R4 ... prescribed scalar curvature equation on the standard sphere S n In the first part, we use the negative gradient flow equation for the prescribed scalar curvature equation to obtain existence theorems... Symbols and Conventions Introduction I xv Prescribed scalar curvature equation on S n in the pres- ence of reflection or rotation symmetry Description of the main results 11 Symmetry and blow-up. .. point for the flow equation 3.1 Long time existence, uniqueness and symmetry of flow equation In the first part of this thesis, we consider only smooth and positive initial data for equation (1.7)