Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Juhl, Andreas

doi:10.1007/978-3-7643-9900-9

Cited by 73 publications

(32 citation statements)

References 116 publications

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“…A sample of the subsequent and related progress includes [42,44,45,53,61]. In all of these works the formal asymptotics play a critical role.…”

Section: 1)mentioning

confidence: 99%

Boundary calculus for conformally compact manifolds

Gover¹,

Waldron²

2014

Indiana Univ. Math. J.

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Abstract. On conformally compact manifolds of arbitrary signature, we use conformal geometry to identify a natural (and very general) class of canonical boundary problems. It turns out that these encompass and extend aspects of already known holographic bulk-boundary problems, the conformal scattering description of boundary conformal invariants, and corresponding questions surrounding a range of physical bulk wave equations. These problems are then simultaneously solved asymptotically to all orders by a single universal calculus of operators that yields what may be described as a solution generating algebra. The operators involved are canonically determined by the bulk (i.e. interior) conformal structure along with a field which captures the singular scale of the boundary; in particular the calculus is canonical to the structure and involves no coordinate choices. The generic solutions are also produced without recourse to coordinate or other choices, and in all cases we obtain explicit universal formulae for the solutions that apply in all signatures and to a range of fields. A specialisation of this calculus yields holographic formulae for GJMS operators and Branson's Q-curvature.

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“…A sample of the subsequent and related progress includes [42,44,45,53,61]. In all of these works the formal asymptotics play a critical role.…”

Section: 1)mentioning

confidence: 99%

Boundary calculus for conformally compact manifolds

Gover¹,

Waldron²

2014

Indiana Univ. Math. J.

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“…The nonexistence of GJMS operators with m > n 2 on general even-dimensional manifolds was shown in [13]. The recursive structure of these operators was investigated by Juhl in [21][22][23], and [24].…”

Section: Example 64 (The Paneitz-branson Operator) An Example Of a Cmentioning

confidence: 99%

Asymptotic expansions and conformal covariance of the mass of conformal differential operators

Ludewig

2017

Ann Glob Anal Geom

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We give an explicit description of the full asymptotic expansion of the Schwartz kernel of the complex powers of m-Laplace type operators L on compact Riemannian manifolds in terms of Riesz distributions. The constant term in this asymptotic expansion turns out to be given by the local zeta function of L. In particular, the constant term in the asymptotic expansion of the Green's function L −1 is often called the mass of L, which (in case that L is the Yamabe operator) is an important invariant, namely a positive multiple of the ADM mass of a certain asymptotically flat manifold constructed out of the given data. We show that for general conformally invariant m-Laplace operators L (including the GJMS operators), this mass is a conformal invariant in the case that the dimension of M is odd and that ker L = 0, and we give a precise description of the failure of the conformal invariance in the case that these conditions are not satisfied.

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“…These are the 'trivial' poles of the scattering operator, so-called because their location is independent of the interior geometry; One obtains a holomorphic family of elliptic pseudodifferential operators Pḡ γ (which patently depends on the filling (X, G)). An alternate construction of these operators has been obtained by Juhl, and his monograph [Juh09] describes an intriguing general framework for studying conformally covariant operators, see also [Juh].…”

Section: Xxi-6mentioning

confidence: 99%

Elliptic problems with integral diffusion

Sire¹

2014

Séminaire Laurent Schwartz — EDP Et Applications

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Families of Conformally Covariant Differential Operators, Q-Curvature and Holography

Cited by 73 publications

References 116 publications

Boundary calculus for conformally compact manifolds

Boundary calculus for conformally compact manifolds

Asymptotic expansions and conformal covariance of the mass of conformal differential operators

Elliptic problems with integral diffusion

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