The original publication can be found at www.springerlink.comBranson’s Q-curvature is now recognized as a fundamental quantity in conformal geometry. We outline its construction and present its basic properties.S.-Y. Alice Chang, Michael Eastwood, Bent Ørsted and Paul C. Yan

Bent Ørsted

C. Fefferman

C.R. Graham

K. Okikiolu

M. Gursky

Michael Eastwood

Paul C. Yang

R.J. Riegert

S.-Y. Alice Chang

S.-Y.A. Chang

T.P. Branson

Acta Applicandae Mathematicae

Chang, Alice

Eastwood, Michael George

Orsted, Bent

Yang, Paul

Adelaide Research &amp; Scholarship

What is Q-Curvature?

Crossref

Branson's Q-curvature is now recognized as a fundamental quantity in conformal geometry. We outline its construction and present its basic properties

Chang, S-Y Alice

Eastwood, Michael

The Australian National University

Acta Appl Math (2008) 102: 119–125DOI 10.1007/s10440-008-9229-zWhat is Q-Curvature?S.-Y. Alice Chang · Michael Eastwood · Bent Ørsted ·Paul C. YangReceived: 11 March 2008 / Accepted: 13 March 2008 / Published online: 21 March 2008© Springer Science+Business Media B.V. 2008Abstract Branson’s Q-curvature is now recognized as a fundamental quantity in conformalgeometry. We outline its construction and present its basic properties.Keywords Invariant · Curvature · ConformalMathematics Subject Classification (2000) Primary 53A30 · Secondary 53C21 · 58J601 IntroductionThroughout his distinguished research career, Tom Branson was fascinated by conformaldifferential geometry and made several substantial contributions to this field. There is nodoubt, however, that his favorite was the notion of Q-curvature. In this article we outlinethe construction and basic properties of Branson’s Q-curvature. As a Riemannian invariant,defined on even-dimensional manifolds, there is apparently nothing special about Q. On aIn memory of Thomas P. Branson (1953–2006).The research of S.-Y.A. Chang and P.C. Yang is supported in part by NSF grant DMS-0245266.M. Eastwood is supported by the Australian Research Council.S.-Y.A. Chang · P.C. YangDepartment of Mathematics, Princeton University, Princeton, NJ 08544-1000, USAS.-Y.A. Change-mail: chang@math.princeton.eduP.C. Yange-mail: yang@math.princeton.eduM. Eastwood ()Department of Mathematics, University of Adelaide, Adelaide, SA 5005, Australiae-mail: meastwoo@maths.adelaide.edu.auB. ØrstedDepartment of Mathematical Sciences, University of Aarhus, Aarhus 8000, Denmarke-mail: orsted@imf.au.dk120 S.-Y.A. Chang et al.surface Q is essentially the Gaussian curvature. In 4 dimensions there is a simple but unre-vealing formula (4.1) for Q. In 6 dimensions an explicit formula is already quite difficult.What is truly remarkable, however, is how Q interacts with conformal, i.e. angle-preserving,transformations.We shall suppose that the reader is familiar with the basics of Riemannian differentialgeometry but a few remarks on notation are in order. Sometimes, we shall write gab fora metric and ∇a for the corresponding connection. Let us write Rab and R for the Ricciand scalar curvatures, respectively. We shall use the metric to ‘raise and lower’ indices inthe usual fashion and adopt the summation convention whereby one implicitly sums overrepeated indices. Using these conventions, the Laplacian is the differential operator  ≡gab∇a∇b = ∇a∇a .A conformal structure on a smooth manifold is a metric defined only up to smoothlyvarying scale. Thus, one can measure angles between vectors but not their lengths. Forsimplicity, let us suppose that all manifolds in this article are oriented.2 The Functional DeterminantSpectral geometry is concerned with the interplay between the geometry of a compact Rie-mannian manifold M and objects in the analysis of natural differential and pseudodiffer-ential operators on M ; for the Laplacian  on functions for example, we may study itseigenvalues (actually, those of −)0 = λ0 < λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · ·and ask how they carry information about the curvature of M and other similar invariants.A good tool here is the heat operator exp(t), which for positive t is a kernel operator oftrace-class withTr(exp(t)) =∞∑n=0e−λnt .The main point is that this trace admits an asymptotic expansion in powers of t , for small t ,and that the coefficients in this expansion are integrals of local geometric invariants over M .The coefficient to t0 is of particular interest, and for elliptic complexes, where we considercertain families of operators like  above, it leads to Atiyah-Singer index theorems. Thereis also a connection to the so-called functional determinants, that arise by considering the‘zeta function’ζ(s) =∞∑n=1λ−snsay for the  above, or more generally over the spectrum of a non-negative elliptic operatorA, disregarding the kernel. This function captures (as does the heat kernel) in a marvelousway the spectrum, and it is meromorphic in s with a removable singularity at s = 0. One maydefine the determinant of A (of use for example in string theory) via ζ ′(0), which in somecases, notably in connection with conformal geometry, is a quantity one may compute. It isat this point that the Q-curvature appears in a natural way. It is remarkable, that the samequantity also appears naturally in quite different contexts, as we shall see later.Consider an elliptic differential operator A of order 2 on M , i.e. possibly acting onsections of a vector bundle; we wish to make a number of technical assumptions, e.g. thatWhat is Q-Curvature? 121A is naturally built from the metric and transforms covariantly with respect to conformalchange of the metric (or is a power of such an operator)—suffice it to say, that A could bethe Yamabe operator Y = − + n−24(n−1)R, where n is the dimension (which we assume tobe even) of M , or D2, where D is the Dirac operator (for M spin). Suppose for simplicitythat A has only positive spectrum, and consider its heat expansion, zeta function, and ζ ′(0).Now the value ζ(0) coincides with the coefficient of t0 in the heat expansion, and it is aconformal invariant expressed as an integral over M :ζ(0) =∫MQand for the derivative of ζ ′(0) along a curve of conformal metrics, viz. e2εωgab we getddεζ ′(0)|ε=0 = 2∫Mω Q.Here ω is a smooth real function on M and Q is the Q-curvature, modulo some divergenceterms, described explicitly below. With more work on the local invariants, based on theformulas above and Sobolev inequalities, one derives the fact, that on the standard spheresof dimensions 2, 4, and 6, the determinant of the Yamabe operator has a maximum, resp.minimum, resp. maximum within the standard conformal class (fixing the total volume); forthe square of the Dirac operator the same is true, except now we have a minimum, resp.maximum, resp. minimum. The case of dimension 6 is particularly demanding, and it isdue to Branson, see [1]. It would be very interesting to see if this pattern continues, andprobably it does, owing to some deep properties of the Q-curvature. In fact, we have beenadvised by Okikiolu that she has been able to extend [12] to provide evidence for this patternregarding the Yamabe operator: the standard metrics on even-dimensional spheres providelocal extrema of the predicted type at least when restricted to a suitable finite-codimensionalsubmanifold of metrics.3 Q-Curvature in 2 DimensionsIn 2 dimensions, scalar curvature R or, equivalently, Gaussian curvature K = R/2 are fun-damental. These quantities enjoy a well-known rôle in controlling the topology of a compactsurface M via the Gauss-Bonnet theorem:∫MK dvol = 12∫MR dvol = 2πχ(M),where χ(M) is the Euler characteristic of M . Furthermore, R transforms in a simple fash-ion under conformal rescaling of the metric. Specifically, if ĝab = e2f gab for some smoothfunction f , thenR̂ = e−2f (R − 2f ),where R̂ denotes the scalar curvature of the metric ĝab . A neater viewpoint on these funda-mental properties of scalar curvature is provided by introducingQ ≡ −12R dvol and Pf ≡ f dvol (3.1)122 S.-Y.A. Chang et al.for then∫MQ = −2πχ(M) and Q̂ = Q + Pf,where P is regarded as a differential operator from functions to 2-forms. It follows that Pis conformally invariant. It is also clear, without the Gauss-Bonnet theorem, that∫MQ is aconformal invariant. This is because Pf is a divergence for any f .4 Q-Curvature in 4 DimensionsIn 4 dimensions, the concept of Q-curvature is due to Branson and Ørsted [3] (with a similarquantity being introduced by Riegert [14]):Q ≡[−16R − 12RabRab + 16R2]dvol. (4.1)Its properties are very much akin to those of scalar curvature in 2 dimensions. Under con-formal rescaling of the metric ĝab = e2f gab , we findQ̂ = Q + Pf (4.2)where P is the ‘Paneitz operator’ [13] from functions to 4-forms given byPf = ∇a[∇a∇b + 2Rab − 23Rgab]∇bf dvol.Riemannian invariants generally transform under conformal rescaling gab → e2f gab by dif-ferential operators that are polynomial in f and its derivatives. The transformation law (4.2)is remarkable in being linear. As in 2 dimensions, this law has two immediate consequences.The first is that P is a conformally invariant differential operator. The second is that forany compact Riemannian 4-manifold M , the integral∫MQ is an invariant of the conformalstructure. It is not just a topological invariant. Rather,Q = 116Pfaff −[14WabcdWabcd + 16R]dvol,where Pfaff is the Pfaffian 4-form and Wabcd is the Weyl curvature tensor, a well-knownlocal conformal invariant. It follows that∫MQ = 8π2χ(M) − 14∫MWabcdWabcd dvol.In particular, on a conformally flat 4-manifold, we have∫MQ = 8π2χ(M).5 Q-Curvature in 2m DimensionsThe notion of Q-curvature was extended to arbitrary even dimensions by Tom Branson [1,p. 11]. Its construction and properties have since been studied by many authors besidesWhat is Q-Curvature? 123Branson himself, including S. Alexakis, S.-Y.A. Chang, Z. Djadli, K. El Mehdi, C. Fef-ferman, A.R. Gover, C.R. Graham, M. Gursky, F. Hang, E. Hebey, K. Hirachi, A. Mal-chiodi, B. Ørsted, L. Peterson, J. Qing, F. Robert, M. Struwe, J. Viaclovsky, P. Yang, andM. Zworski. Even this list is surely incomplete and is rapidly growing. The following con-struction of Q is due to Fefferman and Hirachi [7].Let us consider a Lorentzian metric g̃ given in local coordinates (t, xi, ρ) byg̃ = 2ρ dt2 + 2t dt dρ + t2gij (x, ρ) dxi dxj , (5.1)where gij (x, ρ) dxi dxj is a Riemannian metric in the coordinates x = (x1, . . . , x2m) foreach fixed ρ. Let us write ̃ for the ‘Laplacian’, more correctly the wave-operator, of g̃.Then, associated to g̃ is the quantityQ ≡ −̃m log t |ρ=0,t=1 dvolg(x,0), (5.2)depending on x alone. Now suppose that g̃ is Ricci-flat. Then it turns out that Q dependsonly on gij (x) ≡ gij (x,0). In other words, we have associated to the metric gij (x) dxi dxjwhat turns out to be a Riemannian invariant volume-form Q. The only problem with thisrecipe is that Ricci-flat metrics of the form (5.1) generally do not exist for a given ‘initial’metric gij (x) dxi dxj . However, for the purposes of defining Q, this simply does not matter.As a formal power series in ρ, one can show following [9], that gij (x, ρ) is sufficiently well-defined to a high enough order that the formula (5.2) makes perfect sense. (This definitionof Q is a consequence of the Fefferman-Graham ‘ambient metric’ construction.)It is a simple enough exercise to check that (5.2) gives (4.1) in 4 dimensions. The basictransformation law (4.2) holds in general, where in dimension 2m the conformally invariantoperator P is due to Graham, Jenne, Mason, and Sparling [9]. Explicit formulae for Pand Q, however, are only known in low dimensions. Certainly,Q =[− 12(2m − 1)m−1R + · · ·]dvol and Pf = [mf + · · ·] dvolbut the lower order terms represented by the ellipses · · · are difficult to determine from therecipe (5.2). In particular, a link with the Pfaffian is only known in the conformally flat case,from which:∫MQ = (−1)m(m − 1)!22m−1πmχ(M)on a conformally flat 2m-manifold. In general, it is clear only that∫MQ is a conformalinvariant. Though much is known about Q in general dimensions, a characterization in termsof its many special properties remains elusive. An complete analysis of Q in dimension 6 isgiven in [2].6 Q-Curvature and Geometric AnalysisThe analytic and geometric significance of Q-curvature comes from its close relation tothe Pfaffian. The integral of Q-curvature is clearly a conformal invariant, since the trans-formation rule (4.2) and its analogue in general dimensions displays the difference of theQ-curvature forms under conformal change of metric as a divergence. In fact, in low di-mensions its relation to the Pfaffian is explicit, and thus its sign plays an important rôle in124 S.-Y.A. Chang et al.controlling the geometry and topology of the underlying manifold. For example, in dimen-sion four the positivity of the conformal Laplacian and positivity of the conformal invariant∫Q implies the positivity of the Paneitz operator P ; and under these assumptions we havethe vanishing of the first Betti number [11]; also∫Q ≤ 16π2 with equality holding only ifthe manifold is conformally equivalent to the standard 4-sphere—which can be viewed as aneffective version of the positive mass theorem. The study of the “prescribing Q-curvature”equation also leads to the recent development involving fully non-linear equations relatedto the symmetric functions of the Schouten tensor; with stronger topological results. To seethis, we recall that the natural curvature invariants in conformal geometry are the Weyl ten-sor W and the Schouten tensor Ag = 1n−2 [Ric − R2(n−1) g] that occur in the decomposition ofthe Riemann curvature tensor:Riem = W ⊕ A g,where  denotes the Kulkarni-Nomizu product. Since the Weyl tensor W transforms byscaling under conformal change ĝ = e2wg, only the Schouten tensor depends on the deriva-tives of the conformal factor. It is thus natural to consider σk(Ag) the k-th symmetric func-tion of the eigenvalues of the Schouten tensor Ag as a curvature invariant of the conformalmetric. In general, σ1 = traceA = 12(n−1)R. In dimension 4, a simple computation yieldsσ2(A) dvol = Q + 16R dvol.Thus on closed 4-manifolds,∫σ2(A) dvol =∫Q is again conformally invariant. The studyof the fully non-linear equationsσk(Aĝ) = constantis a fully non-linear version of the Yamabe problem. This is a research direction currentlyunder intensive study. The interested reader is referred to the survey [4].In general even dimensions the study of Q-curvature as a local curvature invariant canalso be approached as part of the general program of Fefferman-Graham as introduced in thearticle [6]. This approach has led to a general formulation of the Paneitz operator P , its gen-eralization to higher dimensions, and associated Q-curvature in terms of the ambient metricconstruction explained in the previous section. An equivalent formulation of the ambientmetric construction is to view the conformal structure on Mn as the boundary at infinity ofan asymptotically hyperbolic Einstein manifold (Xn+1, g). This latter alternative approachhas led to the intrinsic realization of the operators P in all even dimensions as residues ofscattering operators studied by Graham and Zworski [10]. Q-curvature is also related to theconcept of “renormalized volume” on (Xn+1, g) in conformal field theory. When n is odd,one can relate the renormalized volume as the integral of the Q-curvature on (Xn+1, g) viathe Gauss-Bonnet formula; when n is even, one can relate the renormalized volume to theconformal primitive of the Q-curvature on Mn (see [8] and [5]).This circle of ideas has clearly placed the notion of Q-curvature at the intersection ofseveral interacting areas of research. It can be expected that Q-curvature will continue toplay a crucial rôle in the study of conformal geometry.References1. Branson, T.P.: The Functional Determinant. Global Analysis Research Center Lecture Notes Series,vol. 4. Seoul National University, Seoul (1993)What is Q-Curvature? 1252. Branson, T.P.: Q-curvature and spectral invariants. In: Proceedings of the 24th Winter School “Geometryand Physics” Srní, 2004. Suppl. Rend. Circ. Mat. Palermo vol. 75, pp. 11–55 (2005)3. Branson, T.P., Ørsted, B.: Explicit functional determinants in four dimensions. Proc. Am. Math. Soc.113, 669–682 (1991)4. Chang, S.-Y.A.: Conformal invariants and partial differential equations. Bull. Am. Math. Soc. 42, 365–393 (2005)5. Chang, S.-Y.A., Qing, J., Yang, P.C.: On the renormalized volumes for conformally compact Einsteinmanifolds. Preprint (2008)6. Fefferman, C., Graham, C.R.: Conformal invariants. Asterisque 95–116 (1985)7. Fefferman, C., Hirachi, K.: Ambient metric construction of Q-curvature in conformal and CR geome-tries. Math. Res. Lett. 10, 819–831 (2003)8. Graham, C.R.: Volume and area renormalizations for conformally compact Einstein metrics. In: Proceed-ings of the 19th Winter School “Geometry and Physics” Srní, 1999. Suppl. Rend. Circ. Mat. Palermo,vol. 63, pp. 31–42 (2000)9. Graham, C.R., Jenne, R., Mason, L.J., Sparling, G.A.J.: Conformally invariant powers of the Laplacian.I: Existence. J. Lond. Math. Soc. 46, 557–565 (1992)10. Graham, C.R., Zworski, M.: Scattering matrix in conformal geometry. Invent. Math. 152, 89–118 (2003)11. Gursky, M.: The principal eigenvalue of a conformally invariant differential operator, with an applicationto semilinear elliptic PDE. Commun. Math. Phys. 207, 131–143 (1999)12. Okikiolu, K.: Critical metrics for the determinant of the Laplacian in odd dimensions. Ann. Math. 153,471–531 (2001)13. Paneitz, S.: A quartic conformally covariant differential operator for arbitrary pseudo-Riemannian man-ifolds. Preprint (1983)14. Riegert, R.J.: A nonlocal action for the trace anomaly. Phys. Lett. B 134, 56–60 (1984)

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What is Q-curvature?

What is Q-Curvature?

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Adelaide Research & Scholarship

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The Australian National University

What is Q-Curvature?

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Adelaide Research &amp; Scholarship

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Adelaide Research & Scholarship