Universal recursive formulae for Q-curvatures

  title={Universal recursive formulae for Q-curvatures},
  author={Carsten Falk and A. Juhl},
Abstract We formulate and discuss two conjectures concerning recursive formulae for Branson's Q-curvatures. The proposed formulae describe all Q-curvatures on manifolds of all even dimensions in terms of respective lower order Q-curvatures and lower order GJMS-operators. They are universal in the dimension of the underlying space. The recursive formulae are generated by an algorithm which rests on the theory of residue families of [Juhl, Progr. Math. 275, 2009]. We attempt to resolve the… Expand
4 Citations
On conformally covariant powers of the Laplacian
We propose and discuss recursive formulas for conformally covariant powers $P_{2N}$ of the Laplacian (GJMS-operators). For locally conformally flat metrics, these describe the non-constant part ofExpand
Explicit Formulas for GJMS-Operators and Q-Curvatures
We describe GJMS-operators as linear combinations of compositions of natural second-order differential operators. These are defined in terms of Poincaré–Einstein metrics and renormalized volumeExpand
On Branson's $Q$-curvature of order eight
We prove a universal recursive formulas for Branson's $Q$-curvature of order eight in terms of lower-order $Q$-curvatures, lower-order GJMS-operators and holographic coefficients. The results prove aExpand
Conformally covariant differential operators acting on spinor bundles and related conformal covariants
Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spinExpand


Conformally Invariant Powers of the Laplacian, Q-Curvature, and Tractor Calculus
Abstract: We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differentialExpand
The decomposition of Global Conformal Invariants I: On a conjecture of Deser and Schwimmer
This is the first in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global confor- mal invariants"; these are defined to be conformallyExpand
Conformally invariant powers of the Laplacian — A complete nonexistence theorem
Conformally invariant operators and the equations they determine play a central role in the study of manifolds with pseudo-Riemannian, Riemannian, conformai and related structures. This observationExpand
Holographic formula for Q-curvature
In this paper we give a formula for Q-curvature in even-dimensional conformal geometry. The Q-curvature was introduced by Tom Branson in [B] and has been the subject of much research. There are now aExpand
Laplacian Operators and Q-curvature on Conformally Einstein Manifolds
A new definition of canonical conformal differential operators Pk (k = 1,2,...), with leading term a kth power of the Laplacian, is given for conformally Einstein manifolds of any signature. TheseExpand
Origins, Applications and Generalisations of the Q-Curvature
These expository notes sketch the origins of Branson’s Q-curvature. We give an introductory account of the equations governing its prescription, its roles in a conformal action formula as well as inExpand
Sharp inequalities, the functional determinant, and the complementary series
Results in the spectral theory of differential operators, and recent results on conformally covariant differential operators and on sharp inequalities, are combined in a study of functionalExpand
The Functional determinant
Results in the spectral theory of diierential operators, and recent results on conformally covariant diierential operators and on sharp inequalities, are combined in a study of functionalExpand
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality
where de denotes normalized surface measure, V is the conformal gradient and q = (2n)/(n 2). A modern folklore theorem is that by taking the infinitedimensional limit of this inequality, one obtainsExpand
Ambient metric construction of Q-curvature in conformal and CR geometries
We give a geometric derivation of Branson's Q-curvature in terms of the ambient metric associated with conformal structures; it naturally follows from the ambient metric construction of conformallyExpand