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Conformal Structure in Geometry, Analysis, and Physics the American Institute of Mathematics Conformal structure in Geometry, Analysis, and Physics The American Institute of Mathematics This is a hard{copy version of a web page available through http://www.aimath.org Input on this material is welcomed and can be sent to [email protected] Version: Wed Oct 15 14:10:49 2003 1 2 Table of Contents A. A Primer on Q-Curvature . 3 B. Origins, applications, and generalizations of the Q-curvature . 11 C. Open problems . 23 D. Reference list . 30 3 Chapter A: A Primer on Q-Curvature by Michael Eastwood and Jan Slov¶ak Disclaimer: These are rough notes only, aimed at setting the scene and promoting discussion at the American Institute of Mathematics Research Conference Center Work- shop `Conformal Structure in Geometry, Analysis, and Physics,' 12th{16th August 2003. For simplicity, we have omitted all references. Curvature conventions are in an appendix. Conversations with Tom Branson and Rod Gover have been extremely useful. Let M be an oriented even-dimensional Riemannian n-manifold. Branson's Q-curvature is a canonically de¯ned n-form on M. It is not conformally invariant but enjoys certain nat- ural properties with respect to conformal transformations. When n = 2, the Q-curvature is a multiple of the scalar curvature. With conventions 1 2 as in the appendix Q = ¡ 2 R. Under conformal rescaling of the metric, gab 7! gab = ­ gab we have Q = Q + ¢ log ­; b a where ¢ = r ra is the Laplacian. b When n = 4, the Q-curvature is given by 1 2 1 ab 1 Q = 6 R ¡ 2 R Rab ¡ 6 ¢R: (1) Under conformal rescaling, Q = Q + P log ­; where P is the Paneitz operator b a b ab 2 ab P f = ra r r + 2R ¡ 3 Rg rbf: (2) For general even n, the Q-curv£ature transforms as follo¤ ws:{ Q = Q + P log ­; (3) where P is a linear di®erential operatorb from functions to n-forms whose symbol is ¢n=2. It follows from this transformation law that P is conformally invariant. To see this, suppose that 2 2f f 2 gab = ­ gab and gab = e gab = (e ­) gab: Then b b b Q = Q + P log ef = Q + P log ­ + P f but also b b b b Q = Q + P log(ef ­) = Q + P f + P log ­: Therefore, P f = P f. Withb suitable normalisation, P is the celebrated Graham-Jenne- Mason-Sparling operator. Thus, Q may be regarded as more primitive than P and, therefore, is at least asbmysterious. Even when M is conformally flat, the existence of Q is quite subtle. We can reason as follows. When M is actually flat then Q must vanish. Therefore, in the conformally flat 2 case, locally if we write gab = ­ ´ab where ´ab is flat, then (3) implies that Q = ¢n=2 log ­; (4) where ¢ is the ordinary Laplacian in Euclidean space with ´ab as metric. An immediate problem is to verify that this purported construction of Q is well-de¯ned. The problem is 4 2 that there is some freedom in writing gab as proportional to a flat metric. If also gab = ­ ´ab, then we must show that b b ¢n=2 log ­ = ¢n=2 log ­: This easily reduces to two facts:{ b b fact 1:: ¢n=2 is conformally invariant on flat space. n=2 fact 2:: if gab is itself flat, then ¢ log ­ = 0. The second of these is clearly necessary in order that (37) be well-de¯ned. For n = 2 it is immediate from (17). For n ¸ 4 it may be veri¯ed by direct calculation as follows. If gab and ´ab are both flat then 1 c ra¨b = ¨a¨b ¡ 2 ¨ ¨cgab; (5) where ¨a = ra log ­. Therefore, a k a k¡1 a a k rc(¨ ¨a) = 2k(¨ ¨a) ¨ rc¨a = k(¨ ¨a) ¨c and a k 2 a k a k 1 a rbrc(¨ ¨a) = k (¨ ¨a) ¨b¨c + k(¨ ¨a) (¨b¨c ¡ 2 ¨ ¨agbc) whence a k n a k+1 ¢(¨ ¨a) = k(k + 1 ¡ 2 )(¨ ¨a) : (6) Taking the trace of (5) gives a n a ¢ log ­ = r ¨a = (1 ¡ 2 )¨ ¨a and now (6) gives, by induction, k+1 n n n a k+1 ¢ log ­ = k!(1 ¡ 2 )(2 ¡ 2 ) ¢ ¢ ¢ (k + 1 ¡ 2 )(¨ ¨a) : In particular, ¢n=2 log ­ = 0, as advertised. That ¢n=2 is conformally invariant on flat space is well-known. It may also be veri¯ed directly by a rather similar calculation. For example, here is the calculation when n = 4. 2 For general conformally related metrics gab = ­ gab in dimension 4, 2 2 a a ¢ f = ¢ f + 2¨ ¢raf b¡ 2¨ ra¢f a b a a b b + 4(r ¨ )rarbf ¡ 2(r ¨a)¢f ¡ 4¨ ¨ rarbf a a b a b + 2(¢¨ )raf ¡ 4(r ¨ )¨arbf ¡ 4(r ¨a)¨ rbf: If gab is flat then the third order terms cancel leaving 2 2 a b a a b ¢ f = ¢ f + 4(r ¨ )rarbf ¡ 2(r ¨a)¢f ¡ 4¨ ¨ rarbf a a b a b b + 2(¢¨ )raf ¡ 4(r ¨ )¨arbf ¡ 4(r ¨a)¨ rbf: If gab is also flat, then (5) implies ra¨b = ¨a¨b ¡ 1 ¨c¨ gab and ra¨ = ¡¨a¨ b 2 c a a whence the second order terms cancel and the ¯rst order ones simplify:{ 2 2 b a b ¢ f = ¢ f + 2(¢¨ )rbf + 2¨ ¨a¨ rbf: But using (5) again, b b a b 1 c ab ¢¨ = ra(¨ ¨ ¡ 2 ¨ ¨cg ) a b a b b a a b = (r ¨a)¨ + (r ¨ )¨a ¡ (r ¨ )¨a = ¡¨ ¨a¨ 5 and the ¯rst order terms also cancel leaving ¢2f = ¢2f, as advertised. Conundrum: Deduce fact 2 from fact 1 or bvice versa. Both are consequences of (5). Alter- natively, construct a Lie algebraic proof of fact 2. There is a Lie algebraic proof of fact 1. It corresponds to the existence of a homomorphism between certain generalised Verma modules for so(n + 1; 1). What about a formula for Q, even in the conformally flat case? We have a recipe for Q, namely (37), but it is not a formula. We may proceed as follows. If gab = ­gab and gab is flat, then (16) implies that 1 c b ra¨b = ¡Pab + ¨a¨b ¡ 2 gab¨ ¨c: (7) Taking the trace yields b a 1 a ¢ log ­ = r ¨a = ¡P ¡ 2 (n ¡ 2)¨ ¨a: (8) 1 This identity is also valid when n = 2: it is (17).b Dropping the hat gives Q = ¡P = 2 R. This is the simplest of the desired formulae. To proceed further we need two identities. If Á has conformal weight w, then as described in the appendix, raÁ = raÁ + w¨aÁ; which we rewrite as b raÁ = raÁ ¡ w¨aÁ: (9) Similarly, if Áa has weight w, then b a a a r Áa = r Áa ¡ (n + w ¡ 2)¨ Áa (10) and, if Áab is symmetric and has weighbt w, then a a a a r Áab = r Áab ¡ (n + w ¡ 2)¨ Áab + ¨bÁ a: (11) The quantities in (8) have wbeight ¡2. Therefore, applying (9) gives b ra¢ log ­ = ¡raP ¡ 2¨aP ¡ (n ¡ 2)¨ ra¨b wherein we may use (7) to replace ra¨b b bto obtainb b 1 b ra¢ log ­ = ¡raP ¡ 2¨aP + (n ¡ 2)¨ Pab ¡ 2 (n ¡ 2)¨a¨ ¨b: We may now apply ra, usingb (9),b (10),band (11) to replaceb ra by ra on the right hand side and (7) to replace derivatives of ¨ . We obtain an expression involving only complete a b contractions of Pab, its hatted derivatives, and ¨a:{ 2 ab 2 ¢ log ­b = ¡¢P ¡ (n ¡ 2)P Pab + 2P a a b a b+b(n ¡ 6)¨ rb aPb+ (n ¡b2)¨ r Pab + 2(n ¡ 4)¨ ¨aP a b 1 a b ¡ (n ¡ 2)(bn ¡b 4)¨ ¨ Pab +b4 (bn ¡ 2)(n ¡ 4)¨ ¨a¨b ¨b: b Using the Bianchi identity r Pab = raP, we maybrewrite this as 2 ab 2 ¢ log ­ = ¡¢Pb¡b(n ¡ 2)b Pb Pab + 2P (12) a a b+b2(n ¡ 4)¨brabP + 2(nb¡ 4)¨ ¨aP a b 1 a b ¡ (n ¡ 2)(nb¡b4)¨ ¨ Pab + 4 (n ¡b 2)(n ¡ 4)¨ ¨a¨ ¨b b 6 and, in particular, conclude that when n = 4, 2 ab Q = 2P ¡ 2P Pab ¡ ¢P: (13) Though it is only guaranteed that this formula is valid in the conformally flat case, in fact it agrees with the general expression (1) in dimension 4. Of course, we may continue in the vein, further di®erentiating (38) to obtain a formula k for ¢ log ­ expressed in terms of complete contractions of Pab, its hatted derivatives, and ¨a. With increasing k, this gets rapidly out of hand. Moreover, it is only guaranteed to give Q in the conformally flat case. Indeed, when n = 6 this naivebderivation of Q fails for a general metric. Nevertheless, there are already some questions in the conformally flat case. Conundrum: Find a formula for Q in the conformally flat case. Show that the procedure outlined above produces a formula for Q. In fact, there is a tractor formula for the conformally flat Q. This is not the place to explain the tractor calculus but, for those who know it already:{ n ¡ 2 0 ¤ 2 0 3 = 2 0 3 ¡P Q 4 5 4 5 where n ¡ 2 ¤ = D ¢ ¢ ¢ D (¢ ¡ R) DB ¢ ¢ ¢ DA : A B 4(n ¡ 1) (n¡4)=2 Unfortunately, this formula hides a lot of detail and does|not{zseem}to be of much immediate use. It is not valid in the curved case. Recall that, like Q, the Pfa±an is an n-form canonically associated to a Riemannian metric on an oriented manifold in even dimensions.
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