By Andreas Juhl
The relevant item of the booklet is Q-curvature. this significant and sophisticated scalar Riemannian curvature volume was once brought by means of Tom Branson approximately 15 12 months in the past in reference to variational formulation for determinants of conformally covariant differential operators. The booklet reviews structural homes of Q-curvature from an extrinsic perspective via concerning it as a derived volume of convinced conformally covariant households of differential operators that are linked to hypersurfaces. the hot technique is on the leading edge of critical advancements in conformal differential geometry within the final twenty years (Fefferman-Graham ambient metrics, spectral thought on Poincaré-Einstein areas, tractor calculus, Verma modules and Cartan geometry). the idea of conformally covariant households is galvanized through the belief of holography within the AdS/CFT-duality. between different issues, it evidently results in a holographic description of Q-curvature. The equipment admit generalizations in a number of instructions.
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Extra resources for Families of Conformally Covariant Differential Operators, Q-Curvature and Holography
11) it is reasonable to ask whether also n D˙ nres (h; 0)(1) = −(−1) 2 Qn (h). 1). 12) is equivalent to the formula n 2 2(−1) Qn (h) = δn (h)(1) + 2 n n n ! −1 ! 13) 26 Chapter 1. 1). In terms of the operators P2j (h; 0), that identity reads n ! 2(−1) Qn (h) = δn (h)(1) + 2 n 2 n 2 −1 2n−2j j=0 ( n2 −j −1)! ∗ P2j (h; 0)(vn−2j ). 14) j! 14), the nature of the contribution, which is deﬁned by δn (h) = P˙n (h; 0) − P˙n∗ (h; 0) (acting on 1), diﬀers from the remaining terms. But calculations up to n = 6 show that this contribution can be written also as a linear combination of the other terms.
In order to simplify notation, we often write Q2N if the dimension of the underlying space is evident. In the critical case 2N = n, we set Qn = Qn,n . A comment on the nature of these deﬁnitions is in order. These deﬁnitions are extrinsic in the following sense. Q-curvature is derived from GJMS-operators. The GJMS-operators P2N (g) are induced by the powers of the Laplacian of the Feﬀerman-Graham ambient metric associated to g. This is an extrinsic deﬁnition since the interesting object is generated by a construction on a certain ambient space of two dimensions higher.
It is called the holographic anomaly of the asymptotic volume of the Poincar´e-Einstein metric g (see  and the discussion below). , the residue families specialize to GJMS-operators for appropriate values of the family parameters. 1). Dnres (h; λ) is called the critical residue family. It depends on the coeﬃcients h0 = h, h(2) , . . , h(n−2) and the h-trace of h(n) . Since all these terms are determined by h, the family is completely determined by h. 8) continues to hold true. By parity reasons, there is no critical residue family in that case.