Summary Notes(3) by Chang, Q-curvature on conformal covariant operators

Q-curvature and Conformal Covariant operators Sun-Yung Alice Chang Princeton University January 9, 2004 Table of Contents §1. Introduction §2. Conformal Compact Einstein manifold, Scattering theory §3. Q-curvature, general structure §4. Renormalized volume, n odd §5. Renormalized volume, n even 1 Question: What are the general conformal in- variants? What are the conformal covariant operators and their related curvature invari- ants? 2 • Second order invariants: 1. (∆g, Kg) on (M2, g) satisfying ∆gw = e −2w∆g and −∆gw + Kg = Kgwe 2w. 2. (Lg, Rg) on (Mn, g), n ≥ 3, where Lg = −cn∆g + Rg where Rg the scalar curvature, satisfying Lgw = e −n+22 wLg(e n−2 2 w·), Yamabe equation Lg(e n−2 2 w) = Rgwe n+2 2 w. 3 • 4th order invariants: When n = 4, Paneitz operator: (1983) Pϕ ≡ ∆2ϕ + δ[ ( 2 3 Rg − 2Ric ) dϕ] Satisfying: Pgw = e −4wPg Pgw + 2Qg = 2Qgwe 4w Q = 1 12 (−∆R + R2 − 3|Ric|2) 4 3. On (Mn, g), n 6= 4. Existence of 4-th order conformal Paneitz operator P n4 , Pn4 = ∆ 2 + δ (anRg + bnRic) d + n− 4 2 Qn4. For g¯ = u 4 n−4g: Pn4 u = Q¯ n 4u n+4 n−4 . • Pn4 is conformal covariant of bidegree ( n−4 2 , n+4 2 ). • Qn4 is a fourth order curvature invariants. i.e. under dilation δtg = t −2g, (Qn4)(δtg) = t 4(Qn4)(g). 5 Fefferman-Graham (1985) systematically con- struct (pointwise) conformal inviariants: Example: The Riemann curvature tensor has the decomposition Rijkl = Wijkl + [Ajkgil + Ailgjk −Ajlgik −Aikgjl] where A = 1 n− 2 [Rij − R 2(n− 1) gij] is called the Schouten tensor. The Weyl tensor satisfies Wgw = e −2wWg. Graham-Jenne-Mason-Sparling (1992) applied method of construction to existence results of general conformal covariant operators P n2k for n even. 6 §2. Conformally compact Einstein manifold Given (Mn, g), denote [g] class of conformal metrics gw = e2wg for w ∈ C∞(Mn). Definition: Given (Xn+1, Mn, g+) with smooth boundary ∂X = Mn. Let r be a defining func- tion for Mn in Xn+1 as follows: r > 0 in X; r = 0 on M ; dr 6= 0 on M. • We say g+ is conformally compact, if there exists some r so that (Xn+1, r2g+) is a com- pact manifold. • (Xn+1, Mn, g+) is conformally compact Ein- stein if g+ is Einstein (i.e. Ricg+ = cg + ). • We call g+ a Poincare metric if Ricg+ = −ng+. 7 Example: On (Hn+1, Sn, gH) (Hn+1, ( 2 1− |y|2 )2|dy|2). We can then view (Sn, [g]) as the compactifi- cation of Hn+1 using the defining function r = 2 1− |y| 1 + |y| gH = g + = r−2  dr2 + (1− r2 4 ) 2 g   . 8 Given (Mn, g), consider M+ = Mn × [0,1] and metric g+ with (i) g+ has [g] as conformal infinity, (ii) Ric(g+) = −ng+. In an appropriate coordinate system (ξ, r), where ξ ∈ M with (iii) g+ = r−2 ( dr2 + ∑n i,j=1 g + ij (ξ, r)dξidξj ) , and g+ij is even in r. Theorem: (C. Fefferman- R. Graham, ’85) (a) In case n is odd, up to a diffeomorphism fixing M , there is a unique formal power series solution of g+ to (i)–(iii). (b) In case n is even, there exists a formal power series solution for g+ for which the com- ponents of Ric(g+)+ng+ vanish to order n−2 in power series of r. 9 Remarks: • Conformally compact Einstein manifold is of current interest in the physics literature. The Ads/CFT correspondence proposed by Malda- cena involves string theory and super-gravity on such X. • The construction of the Poincare metric is actually accomplished via the construction of a Ricci flat metric, called the ambient metric on the manifold G˜, where G˜ = G × (−1,1) of dimension n + 2 and G is the metric bundle G = { (ξ, t2g(ξ)) : ξ ∈ Mn, t > 0 } of the bundle of symmetric 2 tensors S2T ∗M on M . The conformal invariants are then con- tractions of (∇˜k1R˜⊗∇˜k2R˜⊗......∇˜klR˜) restricted to TM where R˜ denotes the curvature tensor of the ambient metric. 10 A model example is given by the standard sphere (Sn, g). Denote Sn = {∑n+1 1 ξ 2 k = 1 } . G =   n+1∑ 1 p2k − p 2 0 = 0   under ξk = pk/p0 (1 ≤ k ≤ n + 1). Then the ambient space G˜ is Minkowski space Rn+1,1 = { (p, p0), |p ∈ R n+1, p0 ∈ R } with the Lorentz metric g˜ = |dp|2 − dp20, The standard hyberbolic space is realized as the quadric Hn+1 = { |p|2 − p20 = −1 } ⊂ Rn+1,1. PICTURE 11 Graham, Jenne, Mason and Sparling (1992) The existence of conformal covariant operator Pn2k on (M n, g) with: • Order 2k with leading symbol (−∆)k • Conformal covariant of bi-degree (n−2k2 , n+2k 2 ); where k ∈ N when n is odd, but 2k ≤ n when n is even. • In general,the operators P n2k is not unique, e.g. add |W |k to Pn2k , where W is the Weyl tensor, when k is even. • On Rn , the operator is unique and is equal to (−∆)k. Hence the formula for P n2k on the standard sphere and on Einstein metric. 12 Q curvature associated with P n2k. • When 2k 6= n, then Pn2k(1) = c(n, k)Q n 2k, e.g. when k = 1, 2 < n, P n2 = −cn∆ + R = L, and Qn2 = R = P n 2 (1). • When 2k = n, n even Branson (’93) justified the existence of Qnn by a dimension continua- tion (in n) argument from Qn2k. e.g. When k = 1 and n = 2k = 2, Q22 = K the Gaussian curvature. When k = 2 and n = 4, Q44 = 2Q4. • Graham and Zworski (’02) Existence of Qnn when n even, the analytic continuation of a spectral parameter in scattering theory. 13 Spectral Theory on (Xn+1, Mn, g+), with g+ Poincare metric and (Mn, [g]) as conformal in- finity. • A basic fact is (Mazzeo, Melrose-Mazzeo) σ(−∆g+) = [( n 2 )2,∞) ∪ σpp(−∆g+) the pure point spectrum σpp(−∆g+) ( L 2 eigen- values), is finite. • For s(n− s) /∈ σpp, consider (−∆g+ − s(n− s))u = 0. Given f ∈ C∞(M) , then there is a meromor- phic family of solutions u(s) = ℘(s)f ℘(s)f = Frn−s + Grs if s /∈ n/2 + N with F |M = f Define Scattering matrix to be S(s)f = G|M 14 The relation of f to S(s)f is like that of the Dirichlet to Neumann data. Theorem: (Graham-Zworski 2002) Let (Xn+1, Mn, g+) be a Poincare metric with (Mn, [g]) as conformal infinity. Suppose n is even, and k ∈ N, k ≤ n2 and s(n − s) not in σpp(−∆g+). Then the scattering matrix S(s) has a simple pole at s = n2 + k and ckP n 2k = −Ress=n2+k S(s) When 2k 6= n, Pn2k(1) = c(n, k)Q n 2k When 2k = n, cn 2 Qn = S(n)1. 15 §3. Known facts for Qn, n even: • Qn is a conformal density of weight -n ; i.e. with respect to the dilation δt of metric g given by δt(g) = t 2g, we have (Qn)δtg = t −n(Qn)g. • ∫ Mn(Qn)gdvg is conformally invariant. • For gw = e2wg, we have (Pn)gw + (Qn)g = (Qn)gwe nw. • When (Mn, g) is locally conformally flat, then (Qn)g = cnσn 2 (Ag) + divergence terms, e.g. Q4 = σ2(Ag)− 1 6∆gR. • Alexakis Qn = cnPfaffian + J + div(Tn). where Pfaffian is the Euler class density, which is the integrand in the Gauss-Bonnet formula, J is a pointwise conformal invariant, and div(Tn) is a divergence term. 16 • Alexakis (also Fefferman-Hirachi) has extended the existence of conformal covariant opera- tor to conformal densities of weight γ , where γ 6= (−n2) + k where k is a positive integer and γ not a nonnegative integer. An example of such operator is: 2P (f) = ∇i(||W ||2∇if) + n− 6 n− 2 ||W ||2∆f. with corresponding Q-curvature explicit. • Fefferman and Hirachi have also extended the construction of conformal covariant operator and Q curvature to CR manifolds. • Branson, Eastwood-Gover survey articles, AIM meeting August 2003. 17 §4. Renormalized Volume (Witten, Gubser- Klebanov -Polyakov, Henningson-Skenderis, Gra- ham) On conformal compact (Xn+1, Mn, g+) with defining function r, For n odd, Volg+({r > �}) = c0� −n + c2� −n+2 + · · · · · + cn−1� −1 + V + o(1) For n even, Volg+({r > �}) = c0� −n + c2� −n+2 + · · · + cn−2� −2 + L log 1 � + V + o(1) • For n odd, V is independent of g ∈ [g], and for n even, L is independent of g ∈ [g], and hence are conformal invariants. 18 Theorem: (Graham-Zworski) When n is even, L = −2 ∫ M S(n)1 = 2cn 2 ∫ M Qndvg. Theorem: (Fefferman-Graham ’02) Consider v = dds|s=nS(s)1 then v is a smooth function defined on X solving −∆g+(v) = n and with the asymptotic v = { log x + A + Bxnlogx for n even log x + A + Bxn for n odd where A, B ∈ C∞(X) are even mod O(x∞) and A|M = 0. Moreover (i) If n is even, then B|M = −2S(n)1 = −2cn 2 Qn hence L = 2cn 2 ∫ M Qn. 19 (ii) If n is odd, then B|M = − d ds |s=nS(s)1, and if one defines Qn(g+, [g]) to be Qn(g +, [g]) = knB|M then knV = ∫ M Qn(g +, [g])dvg. Remark: when n is odd, the Q curvature thus defined is not intrinsic, it depends not only on the boundary metric g on M but also on the extension of g+ on X. 20 On compact Riemannian 4-manifold (X4, M3, g+) with boundary, Chang-Qing introduced (Pb)gw = e −3w(Pb)g, on M and (Pb)gw + Tg = Tgwe 3won M. 8pi2χ(X) = ∫ X4 ( 1 4 |W |2+Q4)dv+2 ∫ M3 (L+T )dσ, where L is a point-wise conformal invariant term on the boundary of the manifold. On conformally compact Einstein (X4, M3, g+): (Pb)g = − 1 2 ∂ ∂n ∆g+|M , Tg = 1 12 ∂R ∂n |M , and in this case L vanishes. 21 § When n = 3, on (X4, M3, g+), conformally compact Einstein Theorem: (Chang-Qing-Yang) On (X4, M3, g+) (i) (Q4)e2vg+ = 0, Proof: Recall Q4 = 1 6 (−∆R + R2 − 3|Ric|2). Thus for g+ a Poincare metric with Ric g+ = −3g+, we have (Q4)g+ = 6 and (P4)g+ = (∆) 2 g+ + 2∆g+. We then use the equations −∆g+(v) = n = 3 and (P4)g+(v) + (Q4)g+ = (Q4)e2vg+ to conclude the proof. 22 (i) (Q4)e2vg+ = 0, (ii) Q3(e 2vg+, [e2vg]) = 3B|x=0 = Te2vg. As a consequence we have 6V = ∫ X4 (Q4)e2vg+ + 2 ∫ M3 Te2vg = ∫ X4 σ2(Ae2vg+). Hence (M. Anderson) 8pi2χ(X4) = 1 4 ∫ X4 |W |2dvg¯ + ∫ X4 σ2(Ag¯) = 1 4 ∫ X4 |W |2dvg¯ + 6V, for g¯ = e2vg+ or any conformal compact g¯. 23 Conformal Sphere Theorem: (Chang-Gursky-Yang) On (M4, g) with Y (M4, g) > 0. If∫ M4 |Wg| 2dvg < 16pi 2χ(M4), or equivalently∫ M4 σ2(Ag)dvg > 4pi 2χ(M4) then M4 is diffeomorphic to S4 or R4. Note that on (M4, g), with Y (M4, g) > 0.∫ M4 σ2(Ag)dvg ≤ 16pi 2 with equality if and only if M4 is diffeomorphic to S4. 24 Theorem: (Chang-Qing-Yang ) Suppose (X4, M3, g+) is a conformal compact Einstein manifold, and (M3, [g]) has positive Yamabe constant, then (i) V ≤ 4pi 2 3 , with equality holds if and only if (X4, g+) is the hyperbolic space (H4, gH), and therefore (M3, g) is the standard 3-sphere. (ii) If V > 1 3 ( 4pi2 3 χ(X)), then X is homeomorphic to the 4-ball B4 up to a finite cover. (iii) If V > 1 2 ( 4pi2 3 χ(X)), then X is diffeomorphic to B4 and M is diffeo- morphic to S3. 25 A crucial step in the proof of the theorem above is an earlier result: Theorem: (Qing ’02) Suppose (Xn+1, Mn, g+) is a conformal com- pact Einstein manifold, with Y (Mn, [g]) posi- tive, then there is a positive eigenfunction u satisfying −∆g+u = (n + 1)u on X n+1, so that (Xn+1, u−2g+) is a compact manifold with totally geodesic boundary and the scalar curvature is greater than or equal to n+1n−1Rg, where g ∈ [g] is the Yamabe metric. PICTURE 26 Theorem: (Chang-Qing-Yang, Epstein) On conformally compact Einstein (Xn+1, Mn, g+), when n is odd,∫ Xn+1 Wn+1dvg + cnV (X n+1, g) = χ(Xn+1) for some curvature invariant Wn+1, which is a sum of contractions of Weyl curvatures and/or its covariant derivatives in an Einstein metric. Proof: Use structure equation of Qn; in particular, the result of Alexakis that Qn = anPfaffian + J + div(Tn). 27 §5. Renormalized volume when n is even. The renormalized volume can also be defined via the scattering matrix: V (X3, [g], g+) = − ∫ M2 d ds |s=2S(s)1dvg, for n = 2 V (X5, [g], g+) =− ∫ M4 d ds |s=4S(s)1dvg − 1 32 · 36 ∫ M4 R2[g]dvg, for n = 4 V (Xn+1, [g], g+) =− ∫ Mn d ds |s=nS(s)1dvg + correction terms, for n even 28 Definition: We call a functional F defined on (Mn, g) a conformal primitive of a curvature tensor A if d dα |α=0F[e 2αwg] = −2cn 2 ∫ M wAgdvg. Theorem: On (Xn+1, Mn, g+), n even, the scattering term S(g, g+) = dds|s=nS(s)1(g, g +) is the conformal primitive of (Qn)g. Corollary: (Henningson-Skenderis, Graham) On (X3, M2, g+), V is the conformal primitive of K, the Gaussian curvature. On (X5, M4, g+), V is the conformal primitive of 116σ2, where σ2 = 1 6(R 2 − 3|Ric|2). 29 • Qing established the rigidity result that any conformal compact Einstein manifold with con- formal infinity the standard n-sphere is the hy- perbolic n + 1 space extending prior results of L. Andersson. • X. Wang proved that on (Xn+1, Mn, g+) with λ0(g +) > n− 1, then Hn(X, Z) = 0. In particu- lar, the conformal infinity M is connected; thus extending an earlier result of Witten-Yau. Given (Mn, [g]) in general, both the existence and uniqueness problem of a conformal com- pact Einstein manifold with (Mn, [g]) as con- formal infinity remain open. 30