What is Q-Curvature?

Chang, Alice; Eastwood, Michael; Ørsted, Bent; Yang, Paul

doi:10.1007/s10440-008-9229-z

Cited by 37 publications

(25 citation statements)

References 11 publications

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“…In this appendix, we give a Kazdan-Warner type condition for the solvability of (1.4). In the Euclidean case, such condition was obtained in [1], [25], [26] and [6]. Let us identify H n with the set C n × R = R 2n+1 = {(x 1 , y 1 , .…”

Section: By Oddness It Holds Thatmentioning

confidence: 99%

Perturbation of the CR fractional Yamabe problem

Chen

Wang

2016

Mathematische Nachrichten

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Section: By Oddness It Holds Thatmentioning

confidence: 99%

Perturbation of the CR fractional Yamabe problem

Chen

Wang

2016

Mathematische Nachrichten

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“…Significant mathematical progress was made when Fefferman and Graham [FG02] showed that for Poincaré-Einstein structures (Euclidean signature, asymptotically AdS, Einstein manifolds), the renormalized volume anomaly recovered Branson's Q-curvature [B95] for the boundary manifold. This is an important invariant of conformal geometries (see [GJ07,DM08] and the reviews [BG08,CEOY08]). The renormalized volume is usually obtained by computing a Fefferman-Graham coordinate expansion of a bulk metric tensor solving, to some order, a bulk problem with boundary data at a conformal infinity Σ.…”

Section: Introductionmentioning

confidence: 99%

Renormalized Volume

Gover

Waldron

2017

Commun. Math. Phys.

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We develop a universal distributional calculus for regulated volumes of metrics that are singular along hypersurfaces. When the hypersurface is a conformal infinity we give simple integrated distribution expressions for the divergences and anomaly of the regulated volume functional valid for any choice of regulator. For closed hypersurfaces or conformally compact geometries, methods from a previously developed boundary calculus for conformally compact manifolds can be applied to give explicit holographic formulae for the divergences and anomaly expressed as hypersurface integrals over local quantities (the method also extends to non-closed hypersurfaces). The resulting anomaly does not depend on any particular choice of regulator, while the regulator dependence of the divergences is precisely captured by these formulae. Conformal hypersurface invariants can be studied by demanding that the singular metric obey, smoothly and formally to a suitable order, a Yamabe type problem with boundary data along the conformal infinity. We prove that the volume anomaly for these singular Yamabe solutions is a conformally invariant integral of a local Q-curvature that generalizes the Branson Q-curvature by including data of the embedding. In each dimension this canonically defines a higher dimensional generalization of the Willmore energy/rigid string action. Recently Graham proved that the first variation of the volume anomaly recovers the density obstructing smooth solutions to this singular Yamabe problem; we give a new proof of this result employing our boundary calculus. Physical applications of our results include studies of quantum corrections to entanglement entropies.Comment: 31 pages, LaTeX, 5 figures, anomaly formula generalized to any bulk geometry, improved discussion of hypersurfaces with boundar

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“…In particular, the explicit form of the mentioned decomposition of the critical heat kernel coefficients is not known, and the general principles which may connect heat kernel coefficients of P 2 (and other conformally covariant operators) and Q-curvatures have not yet been found. For a brief introduction to the numerous aspects of Q-curvature see [CEOY08].…”

Section: Introduction and Formulation Of The Main Resultsmentioning

confidence: 99%

Heat kernel expansions, ambient metrics and conformal invariants

Juhl

2016

Advances in Mathematics

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The conformal powers of the Laplacian of a Riemannian metric which are known as the GJMS-operators admit a combinatorial description in terms of the Taylor coefficients of a natural second-order one-parameter family H(r; g) of self-adjoint elliptic differential operators. H(r; g) is a non-Laplace-type perturbation of the conformal Laplacian P 2 (g) = H(0; g). It is defined in terms of the metric g and covariant derivatives of the curvature of g. We study the heat kernel coefficients a 2k (r; g) of H(r; g) on closed manifolds. We prove general structural results for the heat kernel coefficients a 2k (r; g) and derive explicit formulas for a 0 (r) and a 2 (r) in terms of renormalized volume coefficients. The Taylor coefficients of a 2k (r; g) (as functions of r) interpolate between the renormalized volume coefficients of a metric g (k = 0) and the heat kernel coefficients of the conformal Laplacian of g (r = 0). Although H(r; g) is not conformally covariant, there is a beautiful formula for the conformal variation of the trace of its heat kernel. Its proof rests on the combinatorial relations between Taylor coefficients of H(r; g) and GJMS-operators. As a consequence, we give a heat equation proof of the conformal transformation law of the integrated renormalized volume coefficients. By refining these arguments, we also give a heat equation proof of the conformal transformation law of the renormalized volume coefficients itself. The Taylor coefficients of a 2 (r) define a sequence of higher-order Riemannian curvature functionals with extremal properties at Einstein metrics which are analogous to those of integrated renormalized volume coefficients. Among the various additional results the reader finds a Polyakov-type formula for the renormalized volume of a Poincaré-Einstein metric in terms of Q-curvature of its conformal infinity and additional holographic terms.The present study of relations between spectral theoretic heat kernel coefficients and geometric quantities like renormalized volume coefficients is stimulated by the holographic perspective of the AdS/CFT-duality (proposing relations between functional determinants and renormalized volumes).

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What is Q-Curvature?

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

Cited by 37 publications

References 11 publications

Perturbation of the CR fractional Yamabe problem

Perturbation of the CR fractional Yamabe problem

Renormalized Volume

Heat kernel expansions, ambient metrics and conformal invariants

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