Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

Gover, A. Rod; Šilhan, Josef

doi:10.1007/s00220-008-0572-8

Cited by 8 publications

(23 citation statements)

References 26 publications

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“…. , n/2 if for instance (M, [h 0 ]) contains an Einstein metric in [h 0 ]; this is a result of Gover and Silhan [7]. If n = 4, L n/2−2 = L 0 is the Paneitz operator (up to a constant factor) and using a result of Gursky and Viaclovsky [14], we deduce that if the Yamabe invariant Y (M, [h 0 ]) is positive and…”

Section: Introductionmentioning

confidence: 58%

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Aubry¹,

Guillarmou²

2011

J. Eur. Math. Soc.

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Abstract. For odd-dimensional Poincaré-Einstein manifolds (X n+1 , g), we study the set of harmonic k-forms (for k < n/2) which are C m (with m ∈ N) on the conformal compactification X of X. This set is infinite-dimensional for small m but it becomes finite-dimensional if m is large enough, and in one-to-one correspondence with the direct sum of the relative cohomology H k (X, ∂X) and the kernel of the Branson-Gover [3] differential operators (L k , G k ) on the conformal infinity (∂X, [h 0 ]). We also relate the set of C n−2k+1 ( k (X)) forms in the kernel of d + δ g to the conformal harmonics on the boundary in the sense of [3], providing some sort of long exact sequence adapted to this setting. This study also provides another construction of Branson-Gover differential operators, including a parallel construction of the generalization of Q-curvature for forms.

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Section: Introductionmentioning

confidence: 58%

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Aubry¹,

Guillarmou²

2011

J. Eur. Math. Soc.

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“…Thus we have exactly the situation of the Theorem above, and it follows that the solution space for P k decomposes directly. (See [26] for further details and [27] for applications as well as a similar treatment of operators on differential forms.) In particular, from the Theorem and (15), we have the following.…”

Section: Algebraic Decompositionsmentioning

confidence: 99%

Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds

Gover¹

2007

SIGMA

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Abstract. A conformal description of Poincaré-Einstein manifolds is developed: these structures are seen to be a special case of a natural weakening of the Einstein condition termed an almost Einstein structure. This is used for two purposes: to shed light on the relationship between the scattering construction of Graham-Zworski and the higher order conformal Dirichlet-Neumann maps of Branson and the author; to sketch a new construction of non-local (Dirichlet-to-Neumann type) conformal operators between tensor bundles.

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“…The Q-operators, the operators L k and the spaces H k , and related issues are studied in detail for Einstein manifolds (of any signature) in [34] (partly following [31]). One finds, for example, that on positive scalar curvature compact Riemannian Einstein spaces dim(H k ) = b k for k = 0, 1, .…”

Section: A Generalisation: Maps Like Qmentioning

confidence: 99%

Origins, Applications and Generalisations of the Q-Curvature

Branson¹,

Gover

2008

Acta Appl Math

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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 in Polyakov-type formulae for the conformal variation of the functional determinants of suitable elliptic operators. We indicate how properties, which partly characterise the Q-curvature, identify it as an extreme case of a class of differential operators on closed differential forms; these so-called Q-operators generalise the Q-curvature.

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Conformal Operators on Forms and Detour Complexes on Einstein Manifolds

Cited by 8 publications

References 26 publications

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Conformal harmonic forms, Branson–Gover operators and Dirichlet problem at infinity

Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds

Origins, Applications and Generalisations of the Q-Curvature

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