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The Yamabe Problem The Yamabe Problem Selim Tawfik Abstract The Yamabe problem asks if any Riemannian metric g on a compact smooth man- ifold M of dimension n ≥ 3 is conformal to a metric with constant scalar curvature. The problem can be seen as that of generalizing the uniformization theorem to higher dimensions, since in dimension 2 scalar and Gaussian curvature are, up to a factor of 2, equal. 1 Introduction In 1960, Yamabe claimed to have found a solution to the problem that would later come to bear his name. Eight years following, however, Neil Trudinger discovered a serious error in Yamabe’s proof. Trudinger was able to salvage some of Yamabe’s work but only by introducing further assumptions on the manifold M. In fact, Trudinger showed that there is a positive constant α(M) such that the result is true when λ(M) < α(M) (λ(M) is the Yamabe invariant, to be defined later). In particular, if λ(M) ≤ 0, the question is resolved. In 1976, Aubin improved on Trudinger’s work by showing that α(M) = λ(Sn), where the n-sphere is equipped with its standard metric. Moreover, Aubin showed that if M has dimension n ≥ 6 and is not locally conformally flat, then λ(M) < λ(Sn). The remaining cases proved to be more difficult and it was not until 1984 that they had been resolved by Richard Schoen, thereby completing the solution to the Yamabe problem. In order to gain an appreciation for these developments and the final solution, it is necessary to look at Yamabe’s approach. First we reiterate the statement of the problem. The Yamabe problem: Let (M,g) bea C∞ compact Riemannian manifold of dimension n ≥ 3 and S its scalar curvature. Is there a metricg ˜ conformal to g such that the scalar curvature S˜ of this new metric is constant? 2 Yamabe’s approach In this section we will detail Yamabe’s attack on the problem. As we will see, he reduced the question of finding a conformal metric with constant scalar curvature to that of finding a smooth non-negative solution to a certain differential equation, the Yamabe equation. Because Krondrakov’s theorem does not apply directly to the situation presented by that equation, Yamabe turned his attention to a family of approximating differential equations, which turn out to be the Euler-Lagrange equations of certain functionals. He proves smooth and non-negative solutions exist, and, claiming that these solutions are uniformly bounded 1 and hence uniformly continuous, concludes that a solution Yamabe equation exists. An error had slipped into his argument however; the uniform boundedness is not true in general, and an explicit counter-example is found on the sphere. 2.1 The differential equation Let M be a smooth, connected, and compact Riemannian manifold with metric g. We put 2f ∞ g˜ = e g with f ∈ C (M). In a local chart, let Γij and Γ˜ij denote the Christoffel symbols of g andg ˜ respectively (throughout this section, given a symbol associated with (M,g), the same symbol with a tilde above denotes its counterpart in (M, g˜)). Let R and S denote the Ricci and scalar curvatures of (M,g) respectively. Their difference is given by: kl Γij − Γ˜ij = (gkl∂if + gki∂j f − gij ∂kf) g The Ricci curvatures are then related in the following way: 2 R˜jk = Rjk − (n − 2)∇jkf + (n − 2)∇j f∇kf + (∇f − (n − 2)|∇f| )gjk from which we derive: S˜ = e−2f (S + 2(n − 1)∇f − (n − 1)(n − 2)|∇f|2) where ∆ is the Laplace-Beltrami operator. The above formula simplifies if we put e2f = ϕp−2 with p =2n/(n − 2) andg ˜ = ϕp−2g: n − 1 S˜ = ϕ1−p 4 ∆ϕ + Sϕ (1) n − 2 n−1 p−2 We use the notation a = 4 n−2 and = a∆+ S. The above implies thatg ˜ = ϕ g has constant scalar curvature iff it satisfies the Yamabe equation: ϕ = λϕp−1 (2) This can be seen as a nonlinear eigenvalue problem. In fact, the equation ϕ = λϕq depends heavily on q. When q is close to 1, its behavior is quite similar to the that of the eigenvalue problem for . If q is very large however, linear theory is not useful. It turns out that the exponent in the Yamabe equation is the critical value below which the equation is easy to solve and above which it may be unsolvable. In fact Yamabe had studied the more general equation ∆ϕ + h(x)ϕ = λf(x)ϕN−1 (3) ∞ where N = p, h,f ∈ C (M), and ϕ ≡ 0 is a non-negative function in H1 (the first Sobolev space). He wished to find λ ∈ R such that the above problem admits a solution. He considered the functional −2/q i 2 q Iq(ϕ)= ∇ ϕ∇iϕ dV + h(x)ϕ dV f(x)ϕ (4) ˆM ˆM ˆM 2 where 2 < q ≤ N. The denominator is defined since H1 ⊆ LN ⊆ Lq (2.21). Yamabe next shows the above functional with q = N has (3) for its Euler-Lagrange equation. Define µq = inf{Iq(ϕ): ϕ ∈ H1, ϕ ≥ 0, ϕ ≡ 0} As we will soon see, showing µN (as N = p, this is the case we are most interested in) is attained directly is difficult since we cannot invoke Krondrakov’s theorem where we would like. Thus Yamabe turned his attention to the approximate equations with q<N : ∆ϕ + h(x)ϕ = λf(x)ϕq−1 (5) ∞ Theorem (Yamabe): For 2 <q<N, there exists a C strictly positive ϕq satisfying (5) with λ = µq and Iq(ϕq)= µq. Proof: The proof is split into several parts. a) For 2 < q ≤ N, µq is finite. Indeed −2/q 2 −2 Iq(ϕ) ≥ inf (0,h(x)) sup f(x) ||ϕ||2||ϕ||q x∈M x∈M and 2 −2 1−2/q 2/n ||ϕ||2||ϕ||q ≤ V ≤ sup(1, V ) with V = M dV . On the other hand, ´ −2/q µq ≤ Iq(1) = h(x)dV f(x)dV ˆM ˆM q b) Let {ϕi} be a minimizing sequence such that M f(x)ϕi dV = 1: ´ ϕi ∈ H1, ϕi ≥ 0, lim Iq(ϕi)= µq i→∞ First we prove that the set of the ϕi is bounded in H1, 2 2 2 2 2 ||ϕi||H1 = ||∇ϕi||2 + ||ϕi||2 = Iq (ϕi) − h(x)ϕi dV + ||ϕi||2 ˆM Since we can suppose that Iq (ϕi) <µq + 1, then 2 2 ||ϕi||H1 ≤ µq +1+ 1 + sup |h(x)| ||ϕi||2 x∈M and 3 −2/q 2 1−2/q 2 1−2/q ||ϕi||2 ≤ [V ] ||ϕi||q ≤ [V ] inf f(x) x∈M c) If 2 <q<N, there exists a non-negative function ϕq ∈ H1 satisfying Iq(ϕq)= µq and q f(x)ϕq dV =1 ˆM Indeed for 2 <q<N the imbedding H1 ⊆ Lq is compact by Kondrakov’s theorem (A2) and since the bounded closed sets in H1 are weakly compact (A7), there exists a subsequence of {ϕi}, {ϕj}, and a function ϕq ∈ H1 such that 1. ϕj → ϕq in Lq. 2. ϕj → ϕq weakly in H1. 3. ϕj → ϕq a.e.. The last assertion is true by A8. The first assertion implies q f(x)ϕq dV =1 ˆM and the third implies ϕq ≥ 0. Finally the second implies ||ϕq||H1 ≤ lim inf ||ϕj ||H1 i→∞ Hence Iq(ϕq) ≤ lim Iq(ϕj )= µq j→∞ because ϕj → ϕq in L2 according to the first assertion above, since q ≥ 2. Therefore, by definition of µq, Iq(ϕq )= µq. d) ϕq satisfies (5) weakly in H1. We compute the Euler-Lagrange equation. Set ϕ = ϕq +νψ with ψ ∈ H1 and ν ∈ R small. An asymptotic expansion gives: −2/q q−1 Iq (ϕ) = Iq(ϕ) 1+ νq f(x)ϕq ψdV ˆM i +2ν ∇ ϕq∇iψdV + h(x)ϕq ψdV + O(ν) ˆM ˆM thus ϕq satisfies for all ψ ∈ H1: 4 i q−1 ∇ ϕq∇iψdV + h(x)ϕq ψdV = µq f(x)ϕq ψdV ˆM ˆM ˆM which is the weak form of (5) with λ = µq. To check that the preceding computation is correct, we note that since D(M) is dense in H1 and ϕ ≡ 0, then inf Iq(ϕ) = inf∞ Iq (ϕ) = inf∞ Iq(|ϕ|) ≥ inf Iq(ϕ) ≥ inf Iq(ϕ) ϕ∈H1 ϕ∈C ϕ∈C ϕ∈H1,ϕ>0 ϕ∈H1 ∞ Iq(ϕ)= Iq(|ϕ|) when ϕ ∈ C because the set of points P where ϕ(P ) = 0 and |∇ϕ(P )| =0 is null. ∞ e) ϕq ∈ C for 2 ≤ q<N and the functions ϕq are uniformly bounded for 2 ≤ q ≤ q0 <N. Let G(P,Q) be the Green’s function. ϕq satisfies the integral equation −1 ϕq(P ) = V ϕq(Q)dV (Q) (6) ˆM + G(P,Q) µ f(Q)ϕq−1 − h(Q)ϕ dV (Q) ˆ q q q M r0 We know that ϕq ∈ L with r0 = N. Since by A6 part 3 there exists a constant B such that |G(P,Q)| ≤ B[d(P,Q)]2−n, then according to Sobolev’s lemma A1 and its corollary, r1 ϕq ∈ L for 2 < q ≤ q0 with 1 n − 2 q − 1 q − 1 2 = + 0 − 1= 0 − r1 n r +0 r0 n q−1 and there exists a constant A1 such that ||ϕq ||r1 ≤ A1||ϕq||r0 . By induction we see that rk ϕq ∈ L with 1 q − 1 2 (q − 1)k 2 (q − 1)k − 1 = 0 − = 0 − 0 rk rk−1 n r0 n q0 − 2 (q−1)k and there exists a constant Ak such that ||ϕq||rk ≤ Ak||ϕq||r0 .
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