Conformal metrics on the unit ball with the prescribed mean curvature

CONFORMAL METRICS ON THE UNIT BALL WITH THE PRESCRIBED MEAN CURVATURE ZHANG HONG (B.Sc., M.Sc., ECNU, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2014 To my wife: Jiang Zewei DECLARATION I hereby declare that the thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Zhang Hong March 2014 Acknowledgements I would like to thank my thesis advisor Professor Xu Xingwang for bringing this particularly interesting topic to me and sincerely appreciate his constant help, support and encouragement. Moreover, I would like to thank Professor Pan Shengliang at Shanghai Tongji University who provided me a lot of helps and make me gain much confidence with mathematical research during my master study. I also would like to thank my friends: Ngo Quoc Anh, Ruilun Cai and Jiuru Zhou for their valuable comments on my thesis. During the preparation of the thesis project, I have had a lot of helpful discussion with them, which make the thesis more rigorous. A special appreciation goes to my parents and wife for their continued love, encouragement and support. v Contents Acknowledgements v Summary ix Introduction Conclusions The flow and elementary estimates 3.1 Flow equation and its energy . . . . . . . . . . . . . . . . . . . . . . . 3.2 Uniform lower bound of the mean curvature . . . . . . . . . . . . . . 12 3.3 Long time existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 Lp convergence 21 Blow-up analysis 33 5.1 Normalized flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2 Concentration-compactness . . . . . . . . . . . . . . . . . . . . . . . 35 Finite-dimensional dynamics 59 6.1 Estimate of ||ξ||L∞ and ||divS n ξ||L∞ . . . . . . . . . . . . . . . . . . . 59 6.2 Estimate of the change rate of F2 (t) . . . . . . . . . . . . . . . . . . . 62 vii viii Contents 6.3 The shadow flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 Existence of conformal metrics 75 Bibliography 107 Appendix A 111 Appendix B 121 Appendix C 125 Summary This thesis focuses on the prescribed mean curvature problem on the unit ball in the Euclidean space with dimension three or higher. Such problem is well known and attracts a lot of attention. If the candidate f for the prescribed mean curvature is sufficiently close to the mean curvature of the standard metric in the sup norm, then the existence of solution has been known for more than fifteen years. It is interesting to investigate how large that difference can be. This thesis partially achieves this goal using the mean curvature flow method. More precisely, we assume that the given candidate f is a smooth positive Morse function which is non-degenerate in the sense that |∇f |2S n + (∆S n f )2 = and maxS n f /minS n f < δn , where δn = 21/n , when n = and δn = 21/(n−1) , when n ≥ 3. We then show that f can be realized as the mean curvature of some conformal metric provided the Morse index counting condition holds for f . This shows that the possible best difference in the sup norm may be the number (δn − 1)/(δn + 1). ix Chapter Introduction The problem of finding a conformal metric on a manifold with certain prescribed curvature has been extensively studied during the last few decades, see for instance [9, 20, 31, 42] and references therein. Among them, a typical one is the prescribing scalar curvature problem on n (n ≥ 3) dimensional compact Riemannian manifolds without boundary, which can be described as follows. Let (M, g0 ) be an n (n ≥ 3) dimensional compact manifold without boundary with Riemannian metric g0 . Let f (x) be a smooth function on M . The problem is to find a conformal metric g = u4/(n−2) g0 such that the scalar curvature of the new metric g is equal to f (x). It is well known that this problem is equivalent to seeking the positive solutions to the following nonlinear partial differential equation: − n+2 4(n − 1) ∆g0 u + R0 u = f (x)u n−2 , n−2 (1.1) where R0 is the scalar curvature under the metric g0 . When the prescribed function f (x) is a constant function, the problem above is the well known Yamabe problem. In 1960, Yamabe [43], by using variational techniques, claimed to have solved this problem. Unfortunately, a serious gap was found in his proof later. Following Yamabe’s argument, Trudinger [41] was able to fill the gap in the case of small Yamabe invariant which is defined by Y (M, g0 ) = u∈C inf ∞ (M ),u>0 M 4(n−1) |∇u|2g0 n−2 2n + Rg0 u2 dvg0 u n−2 dvg0 M n−2 n 118 Chapter A. integrating by parts, we can obtain the estimate + B2R (y0 ) η |∇ log u|2 dVgE ≤ + B2R (y0 ) |∇η|2 dVgE + C(n, P ) ∂ ≤ C(n, P, gE )R n−1 + B2R (y0 ) η dµgE . Using H¨older’s inequality, we have |∇ log u|dVgE ≤ C(n, P, gE )Rn−1 . + BR (y0 ) + Since r is sufficiently small, B3r (x0 ) is homeomorphic to small half ball around the n+1 origin in R+ . Hence, from the John-Nirenberg estimate (see [24, Theorem 7.21]), it follows that there exists some p∗ > such that ep ∗ | log u−log u| + B3r (y0 ) dVgE ≤ C(n, P, gE , r), where log u = −B + (x0 ) log u dVgE . 3r Since m is chosen sufficient large, p0 will be sufficiently small. Hence p0 < p∗ . Therefore, we can estimate + B3r (x0 ) u−p0 dVgE = + B3r (x0 ) + B3r (x0 ) up0 dVgE ep0 | log u−log u| dVgE + B3r (x0 ) e−p0 | log u−log u| dVgE ≤ e p0 | log u−log u| e p∗ | log u−log u| + B3r (x0 ) dVgE ≤ + B3r (x0 ) dVgE ≤ C(n, P, gE , r). That is + + ||u||Lp0 (B3r (x0 )) ≤ C(n, P, gE , r)||u||L−p0 (B3r (x0 )) . Combining (A.14), (A.16) and (A.17) yields + ||u||Lp (B2r inf u. (x0 )) ≤ C(n, P, r, p0 , gE ) + Br (x0 ) (A.17) 119 Since p0 now is just a fixed number, we have C(n, P, r, p0 , gE ) = C(n, P, r, gE ). The proof of Claim is completed. Claim 2. For sufficiently small r > 0, there exists a constant C(n, P, gE , r) > such that u dVgE ≤ Cu(y), Br+ (x) for all x, y ∈ S n . From Claim 1, it follows that + (x) B2r u, u dVgE ≤ L0 inf + Br (x) for some constant L0 . In particular, we have + B2r (x) u dVgE ≤ L1 Br+ (x) u dVgE , where the constant L1 = L0 /Vol(Br+ (x)). We claim that Br+ (x) u dVgE ≤ L0 Lm−1 u(y), whenever d(x, y) ≤ mr. In fact, if d(x, y) ≤ mr, then we can find a sequence of points xk : ≤ k ≤ m such that x1 = x, d(xm , y) ≤ r, and d(xk , xk+1 ) ≤ r for all ≤ k ≤ m − 1. From this, it follows that + B2r (xm ) u dVgE ≤ L0 +inf u ≤ L0 u(y). Br (xm ) Moreover, we have Br+ (xk ) u dVgE ≤ + B2r (xk+1 ) u dVgE ≤ L1 Br+ (xk+1 ) for all ≤ k ≤ m − 1. Therefore, we obtain Br+ (x) u dVgE ≤ L0 Lm−1 u(y). u dVgE , 120 Chapter A. Claim 3. There exists a constant C > 0, depending only on P and gE , such that n+1 n−1 2n Sn u n−1 dµgE ≤ Cinf u sup u n S . Sn Let be the original point of the unit ball B n+1 . Denote by Bs (0) the ball with radius s. Consider the annulus A := B1− r4 (0)\B1− r2 (0). Then we can find finitely many points x1 , x2 , . . . , xl ∈ S n such that A ⊂ ∪li=1 Br+ (xi ). For any y ∈ S n , it follows from Claim that u dVgE ≤ Cu(y). A Hence, A u dVgE ≤ Cinf u. n (A.18) S Since u is harmonic in B n+1 , we have, for < s ≤ 1, that s−n u dVgE = Integrating the above equality from − r to − (1 − r/4)n+1 − (1 − r/2)n+1 n+1 n+1 (1 − r/4) − (1 − r/2)n+1 = n+1 Cinf u ≥ n S u dµgE . Sn ∂Bs (0) r and using (A.18), we can obtain u dµgE Sn n+1 Sn (1 − r/4)n+1 − (1 − r/2)n+1 ≥ sup u n+1 Sn That is u Sn 2n n−1 2n u− n−1 u n−1 dµgE n+1 − n−1 Sn n+1 n−1 dµgE ≤ Cinf u sup u n S Sn 2n u n−1 dµgE . . Appendix B In this Appendix B, we generalize Proposition A in [10] to the mean curvature case. The following result will be used to control the upper bound of the normalized flow in the blow-up argument. Theorem B.1. On (B n+1 , gE ), if g = u n−1 gE with u > is a metric satisfying 2n (i) −S n u n−1 dµS n = 1; 2n (ii) −S n |Hg |p u n−1 dµS n ≤ ap for some p > n; (iii) < Λ ≤ λ1 (g), where λ1 (g) is the first non-zero Steklov eigenvalue associated with g, where ap , Λ are positive constants and (iv) assuming the condition: dµS n ≥ l0 ωn , (B.1) {x∈S n ;u(x)≥χ0 } for some constants χ0 , l0 . Then there exist n, p, ap , Λ, χ0 , l0 such that and a constant C0 depending only on 2n − u n−1 + ≤ C0 . Sn 2n Proof: Let dµg = u n−1 dµS n , H = Hg . Since   ∆gE u = 0,   ∂u n−1 ∂ν0 Multiplying the first equation by 2β (n − 1)ωn in B n+1 , n+1 + u = Hu n−1 , on S n , uβ n−1 for any β > and integrating by parts yields n+1 uβ−1 |∇u|2 dVgE + − uβ+1 dµS n = − Hu n−1 +β dµS n . B n+1 Sn Sn 121 122 Chapter B. Set w = u β+1 . Then |∇w|2 dVgE (n − 1)ωn B n+1 (β + 1)2 (β + 1)2 − w2 dµS n + − Hu n−1 w2 dµS n . =− 4β 4β Sn Sn (B.2) By Sobolev’s trace inequality, we have − w 2n n−1 n−1 n dµS n Sn |∇w|2 dVgE + − w2 dµS n (n − 1)ωn B n+1 Sn 2 (β + 1) ≤ − Hu n−1 w2 dµS n + − w2 dµS n . 4β Sn Sn ≤ (B.3) For a sufficiently large number b > 0, we obtain 2n bp 2n |H|p u n−1 dµS n ≤ ap ωn , u n−1 dµS n ≤ Sn {|H|>b} which implies that 2n u n−1 dµS n ≤ {|H|>b} ap ωn . bp From H¨older’s inequality, it follows that |H|u n−1 w2 dµS n {|H|>b} p ≤ |H| u 2n n−1 p u dµS n Sn p ≤ ap 2n n−1 − p1 n w dµS n − p1 n w 2n n−1 n−1 n dµS n n−1 n dµS n Sn {|H|>b} ap bp 2n n−1 ωnn . (B.4) Sn On the other hand, by the Raleigh-Ritz characterization for λ1 , we have − ϕ2 dµg ≤ Sn − ϕ dµg Sn + λ−1 ωn B n+1 |∇ϕ|2g dVg , for any ϕ ∈ H (B n+1 ). Now let ϕ = u with < < 1, then 123 − u 2n +2 n−1 dµS n ≤ − u Sn 2n + n−1 + Λ−1 dµS n Sn ωn u2 |∇u |2 dVgE . B n+1 (B.5) By the condition (B.1), we have dµS n ≥ l0 ωn , i.e., |Eχ0 | ≥ l0 ωn , Eχ0 where Eχ0 = {x ∈ S n ; u(x) ≥ χ0 } and |Eχ0 | = Vol(Eχ0 , gS n ). Hence 2n u n−1 + dµS n Sn 2n 2n u n−1 + dµS n + = u n−1 + dµS n c Eχ Eχ0 2n 2n 2n (u n−1 − χ0n−1 )u dµS n + χ0n−1 = Eχ0 ≤ c Eχ Eχ0 (u 2n n−1 2n n−1 − χ0 )u2 dµS n (u Eχ0 2n u n−1 + dµS n u dµS n + 2n n−1 2n n−1 − χ0 ) dµS n + C(χ0 ). Eχ0 For any < η < 1, we have u 2n + n−1 dµ Sn Sn 2n 2n (u n−1 − χ0n−1 )u2 dµS n ≤ (1 + η) Eχ0 2n 2n (u n−1 − χ0n−1 ) dµS n × + (1 + η −1 )C(χ0 )2 Eχ0 2n ≤ (1 + η)(ωn − χ0n−1 |Eχ0 |) −1 + (1 + η )C(χ0 ) . 2n u n−1 +2 dµS n Sn (B.6) 2n Without loss of generality, assume that χ0n−1 |Eχ0 | ≤ ωn , otherwise choose χ0 small 2n 2n enough. Since χ0n−1 |Eχ0 | ≥ χ0n−1 l0 ωn > 0, we may choose sufficiently small η > 2n such that (1 + η)(1 − χ0n−1 l0 ) ≤ − δ, for some positive constant δ = δ(χ0 , l0 ). From (B.5) and (B.6), it follows that 124 Chapter B. 2n δ− u n−1 +2 dµS n ≤ C(χ0 , l0 ) + Λ−1 − u2 |∇u |2 dµS n . Sn Sn 1+β Choose β = + , then w = u = u1+ . By (B.2), we have 2n − u n−1 +2 dµS n ≤ Cδ −1 + Sn (n − 1) − Hu n−1 w2 dµS n . 2(1 + )δΛ S n (B.7) Set I = −S n Hu n−1 w2 dµS n . Then it follows from (B.4) that p I ≤ ap ≤ app ap bp ap bp − p1 n w n−1 n 2n n−1 dµS n ωnn + b− u n−1 w2 dµS n Sn Sn − p1 n n−1 n 2n w n−1 dµS n ωnn + bCδ −1 + Sn b (n − 1) I. 2(1 + )δΛ (B.8) Now choose b sufficiently large and p ap ap bp − p1 n ωnn < sufficiently small such that b (n − 1) and − ≥ . 16 2(1 + )δΛ Then it follows from (B.8) that 2n I≤ − w n−1 dµS n Sn n−1 n + C. By the choice of β, we have (1 + β)2 /4β ≤ 2. Hence, (B.3) implies that − w 2n n−1 n−1 n dµS n ≤ C + C− w2 dµS n . Sn Sn Since is sufficiently small, < min{1, n−1 }. Therefore − w dµS n = − u Sn 2+2 dµS n ≤ Sn − u 2n n−1 (n−1)(2+2 ) 2n dµS n Sn It follows from (B.7) that 2n − u n−1 +2 dµS n ≤ C0 , Sn with =2 . = 1. Appendix C For the sake of having a self-contained thesis, we will provide the details of the proof of Proposition 6.7 in the Appendix C which will follow from the following several lemmas. Lemma C.1. With error o(1) → and O(1) ≤ C as t → ∞ there holds b − b, θ θ = b, θ = ∂f ωn α (θ) + o(1) , i = 1, ., n; n ∂xi   −4πα | log | ∆S n f (θ) + O(1)|∇f (θ)|2S n + o(1)  − ω α n(n−2) n ∆S n f (θ) + O(1)|∇f (θ)|2S n + o(1) n=2 n ≥ 3. Proof: Using the equation −∆S n x = nx and the Kazdan-Warner condition ∇x, ∇Hh S n dµh = 0, Sn we can obtain nx(αfΦ − Hh ) dµh = − nb = Sn ∇x, α∇fΦ − ∇Hh = ∆S n x(αfΦ − Hh ) dµh Sn Sn dµh + R1 Sn ∇x, ∇fΦ = α Sn dµS n + R1 + R2 , (C.1) Sn with error R1 = 2n n−1 ∇x, ∇v S n (αfΦ n+1 − ∇Hh )v n−1 dµS n Sn 125 126 Chapter C. bounded by |R1 | ≤ C||∇v||L2 ||αfΦ − Hh ||L2 (S n ,h) ≤ C||v − 1||H F22 . (C.2) and ∇x, ∇fΦ R2 = α 2n S n (v n−1 − 1) dµS n Sn 2n x(fΦ − f (θ))(v n−1 ) dµS n = nα Sn 2n − α n−1 ∇x, ∇v n+1 Sn fΦ − f (θ) v n−1 dµS n , Sn bounded by |R2 | ≤ C ||v − 1||L2 + ||∇v||L2 ||fΦ − f (θ)||L2 ≤ C||v − 1||H ||fΦ − f (θ)||L2 . (C.3) We henceforth focus on the term ∇x, ∇fΦ Sn x(fΦ − f (θ)) dµS n . dµS n = n Sn (C.4) Sn In the following, we choose the coordinate as in the Section 6.3. With this coordinate at hand, we then have x(fΦ − f (θ)) dµ Sn = Sn B1/ 2n dz + R3 , x(f (Ψ (z)) − f (θ)) (1 + |z|2 )n (0) where x = Ψ(z), with error |R3 | ≤ C z: |z|>1 dz ≤ C n. (1 + |z|2 )n Inserting the expansion (6.11) into the first term of right-hand side of the above equation yields B1/ 2n dz x(f (Ψ (z)) − f (p)) (1 + |z|2 )n (0) = xdf (θ)z B1/ (0) 2n+1 dz (1 + |z|2 )n 127 + x ∇df (θ)(z, z) B1/ (0) 2n+1 dz + R4 (1 + |z|2 )n := I + II + R4 , (C.5) with error R4 bounded by |R4 | ≤ C B (0) |z|3 dz (1 + |z|2 )n 1/    C n=2   ≤ C | log | n =    C n ≥ 4. To estimate b − b, θ θ, we have, by symmetry, that I − I, θ θ = B1/ (0) 2z + |z|2 |z |2 = B1/ (0) n i=1 n+1 ∂f dz i (θ)z i ∂z (1 + |z|2 )n 2n+2 dz ∂f n ∂f (θ), . . . , |z | (θ) ∂z ∂z n (1 + |z|2 )n+1 = C1 df (θ) + R5 , where C1 = Rn 2n+2 |z|2 dz = ωn , n+1 n(1 + |z| ) n and the error R5 is bounded by |R5 | ≤ C Rn \B and II − II, θ θ = B1/ (0) 1/ (0) |z|2 dz ≤C (1 + |z|2 )n+1 n+1 . 2z 2n+1 dz = 0. ∇df (θ)(z, z) + |z|2 (1 + |z|2 )n Now summing all the estimates above up, we find b − b, θ θ = α x(fΦ − f (θ))dµS n + Sn = ωn α df (θ) + R6 , n (R1 + R2 ) n (C.6) 128 Chapter C. with error |R6 | ≤ C |Ri | ≤ C + C(||v − 1||2H + ||αfΦ − Hh ||2L2 + ||fΦ − f (θ)||2L2 ) i=1 ≤ C + CF2 + C||fΦ − f (θ)||2L2   C | log |2 + C|b|2 n = ≤  C + C|b|2 n≥3 in view of (C.2), (C.3), and Lemmas 6.3, 6.6 and 6.4. Now, we deal with b, θ . Notice that by symmetry we have − |z|2 + |z|2 I, θ = B1/ (0) n i=1 n+1 ∂f dz i (θ)z = 0. i ∂z (1 + |z|2 )n and by using the bound for n = 4|z|4 dz − 4π| log | ≤ C, (1 + |z|2 )3 B1/ (0) and the following fact for n ≥ Rn 2n+1 (1 − |z|2 )|z|2 dz =− ωn , n+1 n(1 + |z| ) n(n − 2) we have II, θ = xn+1 ∆S n f (p) B1/ (0) = ∆S n f (p) B = (0) 2n+1 |z|2 dz n(1 + |z|2 )n 2n+1 (1 − |z|2 )|z|2 dz n(1 + |z|2 )n+1  1/  −4π | log |∆S f (p) + R73  − ω ∆S n f (p) n(n−2) n + R7n+1 n=2 n ≥ 3, with error term |R7n+1 | ≤ C n . Hence, we obtain b, θ xn+1 (fΦ − f (θ))dµS n + = α Sn (R1 + R2 ) n 129 =   −4πα | log |∆S n f (θ) + R83  − ω α ∆S n f (θ) n(n−2) n + R8n+1 n=2 (C.7) n ≥ 3, with error |R8n+1 | ≤C |Ri | + R7n+1 ≤   C + C|b|2 + C|∇f (θ)|2S n  C | log | + C|b|2 + C|∇f (θ)|2 n S i=1 n=2 n ≥ 3. in view of Lemma 6.3. From (C.6) and (C.7), it follows that |b|2 = |b − b, θ θ|2 + | b, θ |2 ≤ C|∇f (θ)|2S n + O( )(1 + |∆S n f (θ)|2 ) + C|b|4 . Since |b|2 → as t → ∞, by choosing t sufficiently large such that C|b|2 ≤ then absorbing the last term to the left hand side, we complete the proof. and Observe that Lemmas 6.6, C.1, 6.4 and 6.3 yield respectively the bound F2 ≤ C(|∇f (p)|2S n ) and ||v − 1||2H ≤ + O( ) (C.8)   C(|∇f (p)|2S ) | log | + O( ) n =  C(|∇f (p)|2 n ) S + O( ) n ≥ 3. Lemma C.2. As t → ∞ there holds    8π ( dqdt , dqdt , − dr ) + O(|∇f (p)|2S ) | log | + O( ) n = dt b=   2ωn ( dq1 , ., dqn , − dr ) + O(|∇f (p)|2 n ) + O( ) n ≥ 3. S n+1 dt dt dt Proof: From (5.6) it follows that n n b= 2 (αfΦ − Hh ) dµh = Sn ξ dµh = X + I, Sn where 2n ξ(v n−1 − 1) dµS n . I= Sn (C.9) 130 Chapter C. The claim then follows from (6.8)-(6.9) and the estimate 2n |I| ≤ C||ξ||L∞ ||v n−1 − 1||H ≤ CF22 ||v − 1||H   C(|∇f (p)|2S ) | log | + O( ) n = 2 ≤ C(F2 + ||v − 1||H ) ≤  C(|∇f (p)|2 n ) + O( ) n ≥ 3, S implied by Lemmas 6.1, 6.4, 6.3 and (C.8) and (C.9). The proof of the next lemma is identical to the proof of Lemma 6.10 in [16] and we omit it. Lemma C.3. With error o(1) → as t → ∞ there holds dΘi dq i = (2 + o(1)) , dt dt and Finally, i = 1, ., n, d dr (1 − |Θ(t)|2 ) = (2 + o(1))(1 − |Θ(t)|2 ) . dt dt and |Θ(t)| can be related as follows. Lemma C.4. With error o(1) → as t → ∞ there holds − |Θ(t)|2 =   (4 + o(1)) | log | n =  ( 4n n−2 + o(1)) n ≥ 3. Proof: With our choice of coordinates, we find − |Θ(t)|2 = − |Θn+1 |2 = (1 − Θn+1 )(1 + Θn+1 ) (1 − Φn+1 ) dµS n = (2 + o(1)) Sn n+1 = = + o(1) ωn 2n+2 + o(1) ωn dz (1 + |z|2 )n | z|2 dz . (1 + | z|2 )(1 + |z|2 )n (1 − Ψn+1 (z)) Rn Rn With the estimate Rn \B1/ (0) | z|2 dz ≤ (1 + | z|2 )(1 + |z|2 )n Rn \B1/ (0) dz ≤ C n, |z|2n 131 the claim now follows from for n = lim →0 B1/ (0) and for n ≥ lim →0 B1/ (0) |z|2 dz /| log | = π, (1 + | z|2 )(1 + |z|2 )2 |z|2 dz nωn . = n 2 n (1 + | z| )(1 + |z| ) (n − 2) Proof of Proposition 6.7: (i) The first claim is a direct consequence of Lemmas C.1-C.4. (ii) From Lemma C.4 and (i) we obtain − |Θ|2 ≥ n=2 C0 t d (1 dt − |Θ|2 ) ≤ C(1 − |Θ|2 )2 , that is, with some constant C0 > 0. From Lemma C.4 it follows that for | log | ≥ C1 , 1+t and for n ≥ C1 t for sufficiently large t. Hence for both cases n = and n ≥ ≥ ≥ C1 t| log t| (C.10) for some suitable t0 ≥ and all t ≥ t0 , with a uniform constant C1 > 0. From (i) we then deduce df (θ) dΘ C2 = ∇S n f (θ) + o(1) ≥ (|∇f (θ)|2S n + o(1)). dt 2|Θ| dt t| log t| Since the integral of (t| log t|)−1 is divergent, the flow must accumulate at a critical point Q of f . Now if ∆S n f (Q) > 0, by the first claim in the Proposition 6.7 we then have dtd (1 − |Θ|2 ) > for sufficiently large t. However it will contradict with the fact that − |Θ|2 → as t → ∞. Hence, at the only concentration point Q we have ∆S n f (Q) ≤ 0. CONFORMAL METRICS ON THE UNIT BALL WITH THE PRESCRIBED MEAN CURVATURE ZHANG HONG NATIONAL UNIVERSITY OF SINGAPORE 2014 Conformal metrics on the unit ball with the prescribed mean curvature Zhang Hong 2014 [...]... theory Chapter 2 Conclusions In this thesis, we studied the problem of the existence of conformal metrics on n + 1 dimensional unit ball in Euclidean space with the prescribed mean curvature The major finding of the thesis is the following Theorem 2.1 Let n ≥ 2 and f : S n → R be a positive smooth Morse function satisfying the non-degeneracy condition (1.5) and the simple bubble condition max f Sn min... that the existence of a positive solution if the prescribed function f satisfies the nondegeneracy condition (1.5) and the index counting condition (1.6) Notice that when n = 2, the index counting condition and the Morse index condition are equivalent from the previous argument Hence, it is reasonable that Theorem 2.1 still holds true if the simple bubble condition is removed The question is that whether... n−1 , on ∂M, ∂ where ∂ν0 is the normal derivative operator with respect to outward normal ν and to the metric g0 and R0 and H0 are respectively scalar curvature and mean curvature of the metric g0 In this thesis, we consider the case that the underlying manifold is the unit ball in the Euclidean space Let (B n+1 , gE ) be the n + 1 (n ≥ 2) dimensional unit ball with Euclidean metric gE Then the boundary... we consider the following evolution equation ∂g = (αf − H)g (3.1) ∂t on ∂B n+1 , and R=0 (3.2) in B n+1 , where H and R are respectively the mean curvature of S n and scalar curvature of B n+1 with respect to the metric g(t) From the equation (3.1), our metric flow preserves the conformal class If we 4 write the metric in the form g(t) = u n−1 gE , together with the equation (3.2), we can rewrite the. .. proved the existence of a positive solution if the candidate f satisfies the non-degeneracy condition (∆S n f )2 + | f |2 n = 0 on S n , S (1.5) and the index counting condition (−1)ind(f,y) = (−1)n , {y∈S n , (1.6) f (y)=0,∆f (y) . theory. Chapter 2 Conclusions In this thesis, we studied the problem of the existence of conformal metrics on n + 1 dimensional unit ball in Euclidean space with the prescribed mean curvature. The major. CONFORMAL METRICS ON THE UNIT BALL WITH THE PRESCRIBED MEAN CURVATURE ZHANG HONG (B.Sc., M.Sc., ECNU, China) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL. the prescribed function f satisfies the non- degeneracy condition (1.5) and the index counting condition (1.6). Notice that when n = 2, the index counting condition and the Morse index condition