The Space of Negative Scalar Curvature Metrics

Transcription

1 The Space o Negative Scalar Curvature Metrics Kyler Siegel November 12, Introduction Review: So ar we ve been considering the question o which maniolds admit positive scalar curvature (p.s.c.) metrics. On the one hand, we ve mentioned that there are obstructions to admitting p.s.c. metrics, or example coming rom the index theory o dirac operators or spin maniolds. On the other hand, we saw last time that the condition o admitting a p.s.c. metric is preserved under surgeries o codimension 3, and this allowed us to conclude that lots o maniolds admit p.s.c. metrics. We now want to consider the question at one meta level higher, namely what is the topology o the space o p.s.c. metrics?. Actually, this topology can be quite complicated in general, as the ollowing theorem illustrates. In what ollows, let R, R +, R denote the space o metrics with any scalar curvature, (strictly) positive scalar curvature, and (strictly) negative scalar curvature respectively. Theorem 1.1 [LM] Let X be a closed spin maniold o dimension n with R + (X). Then π 0 (R + (X)) i n 0, 1 (mod 8) π 1 (R + (X)) i n 0, 1 (mod 8) π 0 (R + (X)) is ininite i n = 4k 1 7 or some k N π 0 (R + (S 7 )/Di(S 7 )) is ininite In this talk we ll consider the related but much easier question: what is the topology o R (M) or a closed maniold M? Remark 1.2 By the Kazdan-Warner trichotomy, we know that R (M) is always nonempty. Here is our main result [Loh]: Theorem 1.3 (Lohkamp) For M a closed maniold o dimension n 3, π i (R (M)) = 0 or i 0. 1

2 Using general results on ininite dimensional maniolds, we get Corollary 1.4 (Palais-Whitehead) [Pal] R (M) is contractible. Remark 1.5 Here we are endowing R (M) with the C topology. Loosely speaking, two metrics are close i, which we write down their expressions in local coordinates, their derivatives o all orders are close. There is more or less a unique reasonable way to ormalize this when M is compact. However, we won t be using any subtle properties o the C topology here. 2 The Yamabe Problem We now embark on a brie tangent to discus the Yamabe problem. Remark 2.1 The Yamabe problem has a long and interesting history, which we won t discuss here. Let Di + (M) act on C (M) by precomposition, giving rise to the semi-direct product C (M) Di + (M). Recall the uniormization theorem (see [RS] or a brie summary) Theorem 2.2 (Uniormization) For M an oriented connected closed 2-maniold, C (M) Di + (M) acts transitively on R(M). Corollary 2.3 Every metric on M is conormally equivalent to a metric o constant (sectional) curvature. Remark 2.4 The obvious generalization to higher dimensions is too much to ask or, since the ull curvature tensor has too many degrees o reedom. However, we do have: Theorem 2.5 (Yamabe Problem)(Aubin,Schoen,Trudinger,Yamabe) [LP] For (M, g) a closed Riemannian maniold o dimension n 3, there exists a conormally equivalent metric o constant scalar curvature. Why is this relevant to our problem? First observe that the condition π i (R (M)) = 0 means that any map : S i R (M) has an extension to the ball F : R (M). S i F R (M) Our strategy will be as ollows: 2

3 Step 1: Extend to a map F 1 : R(M) using convexity o R(M). S i R(M) F 1 Step 2: By connect-summing (in a continuous way) with a copy o S n with highly negative scalar curvature, alter F 1 to an extension o o the orm F 2 : R av(m). Here R av(m) denotes the space o metrics g on M such that the average scalar curvature R g dv g < 0, where R g denotes the scalar curvature associated to g. M S i F 2 R av(m) Step 3: Conormally rescale the metrics corresponding to F 2 with appropriate unctions on M, to obtain a new map F such that the resulting metrics have honest negative scalar curvature. S i F R (M) Remark 2.6 It is this last step that shares close connections with the Yamabe problem. In order to ind an appropriate scaling unction to take us rom average negative scalar curvature to actual negative scalar curvature, we ll need to analyze a similiar PDE to that o the Yamabe problem. 3 Warm Up As a warm up, we prove the ollowing: Proposition 3.1 [RS] R + (S 2 ) is contractible. 3

4 Proo By uniormization, C (S 2 ) Di + (S 2 ) acts transitively on R(S 2 ). A little thought shows that stabilizer o g 0 (the standard round metric on S 2 ) can be identiied with the conormal equivalences o (S 2, g 0 ), i.e. PSL(2, C) (by a well-known result in complex analysis). Thus by the orbit-stabilizer theorem, we have ( C (S 2 ) Di + (S 2 ) ) /PSL(2, C) = R(S 2 ). Remark 3.2 Note that Di + (S 2 ) preserves R + (S 2 ) (or R (S 2 )) since pulling back a metric under a dieomorphism just shules the curvature around. Remark 3.3 This shows that Di + (S 2 )/PSL(2, C) is contractible, since C (S 2 ) and R(S 2 ) are. Now we will need to understand how scalar curvature changes under conormal transormations g u 4 n 2 g, where u C 4 (M) and the exponent is only to simpliy later n 2 ormulas. The ormula or n 3 is: ( ) n+2 4(n 1) R 4 (M) = u n 2 u n 2 g n 2 gu + R g (M)u (1) Recall that g u = div g g u is the Laplace-Beltrami operator, where div g X or a vector ield X is deined by d(ι X dv g ) = div g XdV g. In our case, or n = 2 and g = g 0, we have a simpler ormula: R e u g 0 = e u g0 (u) + R g0 (S 2 )e u = e u g0 (u) + 2e u. Let C p (S 2 ) denote the space o unctions u C (M) such that R e u g 0 > 0. Claim: C p (S 2 ) is star-shaped about the zero unction. Claim Proo: For R e u g 0 > 0, we have (or 0 t 1) R e tu g 0 = e tu g0 (tu) + 2e tu = e tu ( t g0 (u) + 2) = e tu (t(e u R e u g 0 2) + 2) = e tu (te u R e u g 0 + 2(1 t)) > 0 Thus C p (S 2 ) is contractible. Now we have an identiication ( C p (S 2 ) Di + (S 2 ) ) /PSL(2, C) = R + (S 2 ). Then since C p (S 2 ) and Di + (S 2 )/PSL(2, C) are contractible, so is R + (S 2 ). 4

5 4 Main Proo We now sketch a proo o Theorem 1.3. Proo (Theorem 1.3) Let : S i R (M) be given. We goal is to ind an extension S i 6 F R (M), where it will convenient to use the ball 6 centered at the origin o radius 6. Step 1: Thinking o 6 as (S i [0, 6]) /(S i {0}), we begin by deining F 1 S i R (M) 6 F 1 R(M) by the ormula F 1 (x, t) := or g 0 a ixed metric on M. Step 2: { (1 t)g0 + t(x) on (S i [0, 1]) /S i {0} (x) on S i [1, 6] First we ll need a controlled way o connect-summing with a sphere o negative scalar curvature. We ll then use this procedure to alter F 1 to F 2 with S i R (M). 6 F 2 R av(m) Consider the ollowing data: g M a ixed metric on M with injectivity radius inj(m, g M ) > 5 g S n any metric on S n with inj(s n, g S n) > 5 g any metric on M g neg a ixed negative scalar curvature metric on S n 5

6 Φ M : (T p M, g M ) (R n, g euc ) a ixed linear isometry Φ S n : (T p S n, g M ) (R n, g euc ) a ixed linear isometry Let λ : B λ2 g M 5 \ {p} (0, 5) S n 1 be deined by or That is, we have a composition λ (z) := P (λ 2 (Φ M (exp λ2 g M p ) 1 (z))) P (z) := ( z, z/ z ). (M, λ 2 g M ) ( ) 1 exp λ2 g M p (T p M, λ 2 g M (p)) Φ M (R n, g euc ) P ((0, 5) S n 1, g R g S n 1) Let h C (R) be a unction with h 1 on (, 3] and h 0 on [4, ). G λ := λ (g R + g S n 1). Now deine a metric g 1 (λ, g) on M by Let g 1 (λ, g) := h(d λ 2 g M (p, id M ))G λ + (1 h(d λ 2 g M (p, id M )))λ 2 g. Deine a metric g 2 (µ, g neg ) on S n similarly, using g S n and Φ S n instead o g M and Φ M. Since we ve standardized the metrics in balls, we can perorm the connect-sum (M, g 1 (λ, g))#(s n, g 2 (µ, g neg )) (M#S n, g # (λ, µ, g)). Using a continuous amily o dieomorphisms F (λ, µ) : M M#S n with F (λ, µ) id on M \ B λ2 g M 5 (p), we get metrics G(g, λ, µ) on M deined by We now deine F 2 (λ 0, µ 0, µ 1, x, t) := G(g, λ, µ) := F (λ, µ) g # (λ, µ, g). (x) on S i [4, 6] ((4 t)λ (1 [4 t])) (x) on S i [3, 4] (3 t)g((x), λ 0, µ 0 ) + (1 [3 t])λ 2 0(x) on S i [2, 3] G((x), λ 0, (2 t)µ 1 + (1 (2 t))µ 0 ) on S i [1, 2] G(F 1 (x, t), λ 0, µ 1 ) on (S i [0, 1]) /S i {0} 6

7 Proposition 4.1 There exist λ 0, µ 0, µ 1 continuous extension o with such that F 2 (x, t) := F 2 (λ 0, µ 0, µ 1, x, t) is a R av (M, F 2 (x, t)) < 0 or all x, t. Step 3: Now we would like a (continuous) way o going rom R av(m) to R (M). Speciically, we ll present a mechanism R av(m) R (M) g v(g) 4 n 2 g (a conormal change). Recall ormula (1) or conormal changes: where is the conormal Laplacian. n+2 R 4 (M) = u n 2 Lg u, u n 2 g 4(n 1) L g u := n 2 gu + R g (M)u We will deine v(g) as ollows. We deine the Rayleigh quotient by ( ) 4(n 1) λ 1 (g) := in u C (M), u L 2 (M,g) =1 M n 2 gu 2 + R g u 2 dv g = in u C (M), u L 2 (M,g) =1 u, L gu L 2 (M,g) Proposition 4.2 There is a unique v(g) C (M) with L g v(g) = λ 1 (g)v(g) v(g) > 0 and max v(g) = 1 Remark 4.3 Setting u 1, we get M ( 4(n 1) n 2 gu 2 + R g u 2 so g R av(m) implies that λ 1 (g) < 0. ) dv g = R g dv g, M 7

8 Then or v(g) 4 n 2 g, we have n+2 R 4 = v(g) n 2 Lg v(g) v(g) n 2 g = v(g) n+2 n 2 λ1 (g)v(g) < 0, and hence v(g) 4 n 2 g R (M) as desired. Finally, set (x) on S i [5, 6] F (x, t) = [(5 t)v((x)) + (1 (5 t))] 4 n 2 (x) on S i [4, 5] v(f 2 (x, t)) 4 n 2 F2 (x, t) on (S i [0, 4]) /S i {0} Using (1), one can easily prove Proposition 4.4 F is a continuous extension o with image in R (M). Remark 4.5 Using similar arguments, one can show that the space R 1 (M) o metrics on M with constant scalar curvature 1 is also contractible. Reerences [LM] H.B.A. Lawson and M.L.A. Michelsohn. Spin Geometry (PMS-38). Princeton Mathematical Series. University Press, [LP] J.M. Lee and T.H. Parker. The yamabe problem. Bull. Amer. Math. Soc 17(1987). [Loh] Joachim Lohkamp. The space o negative scalar curvature metrics. Inventiones Mathematicae 110(1992), /BF [Pal] R.S. Palais. Homotopy theory o ininite dimensional maniolds. Topology 5(1966), [RS] Jonathan Rosenberg and Stephan Stolz. Metrics o Positive Scalar Curvature and Connections With Surgery. In Annals o Math. Studies, pages Princeton Univ. Press,