Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds

Gover, A. Rod

doi:10.3842/sigma.2007.100

Cited by 22 publications

(36 citation statements)

References 47 publications

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“…The operator I·D on tractor bundles determines natural equations on tensor and spinor fields, this is central to the treatments in [33,34,59]. This means our results can be used to deduce results for general tensor and spinor fields in the spirit of the Eastwood curved translation principle, see [27] for an early discussion of this. In physics there is currently considerable interest in understanding higher spin systems via the AdS/CFT machinery [46,24].…”

Section: 1)mentioning

confidence: 87%

“…Away from Σ, and calculated in the metric g o , the equation I·D u = 0 is (∆ g o + s(n−s))u = 0, as in 1.1 with the spectral parameter s a fixed dimensional shift of the conformal weight, see Section 2.5, and also [27]. (Precisely, I·D u = 0 agrees with (∆ g o + s(n − s))u = 0 in the case of g o having constant scalar curvature Sc = −d(d − 1), in general scalar curvature enters explicitly.)…”

Section: 1)mentioning

confidence: 99%

“…However the normal tractor connection is canonical from other points of view, and so this suggests a deep link between conformal geometry and Einstein structures, or more generally to almost Einstein structures [27] as follows. Definition 2.2.…”

Section: Elements Of Tractor Calculusmentioning

confidence: 99%

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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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Section: 1)mentioning

confidence: 87%

Section: 1)mentioning

confidence: 99%

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Boundary calculus for conformally compact manifolds

Gover¹,

Waldron²

2014

Indiana Univ. Math. J.

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“…That result led to an effective approach to certain key problems for these structures, extension to the notion of almost Einstein manifolds [27,28], and also methods for geometrically constructing, and partly characterising, examples of PE manifolds [30]. In [28] it is seen that the almost Einstein class also naturally includes asymptotically locally Euclidean (ALE) structures that admit isolated point conformal compactification; in fact the nature of the compactification is shown to be an easy consequence of the compatibility of Ricci-flatness with the governing conformal PDE.…”

Section: Introductionmentioning

confidence: 99%

Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

Čap

Gover

Hammer

2012

Journal of the London Mathematical Society

109

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Abstract. For curved projective manifolds we introduce a notion of a normal tractor frame field, based around any point. This leads to canonical systems of (redundant) coordinates that generalise the usual homogeneous coordinates on projective space. These give preferred local maps to the model projective space that encode geometric contact with the model to a level that is optimal, in a suitable sense. In terms of the trivialisations arising from the special frames, normal solutions of classes of natural linear PDE (so-called first BGG equations) are shown to be necessarily polynomial in the generalised homogeneous coordinates; the polynomial system is the pull back of a polynomial system that solves the corresponding problem on the model. Thus questions concerning the zero locus of solutions, as well as related finer geometric and smooth data, are reduced to a study of the corresponding polynomial systems and algebraic sets. We show that a normal solution determines a canonical manifold stratification that reflects an orbit decomposition of the model. Applications include the construction of structures that are analogues of Poincaré-Einstein manifolds.

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“…In [32], we see that Poincaré-Einstein manifolds are a special class of directed AE manifolds. On the other hand, from the following result we see that a large class of directed AE manifolds are necessarily boundary gluings of PE manifolds.…”

Section: Introductionmentioning

confidence: 99%

A Class of Compact Poincaré–einstein Manifolds: Properties and Construction

Gover

Leitner

2010

Commun. Contemp. Math.

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We develop a geometric and explicit construction principle that generates classes of Poincaré-Einstein manifolds, and more generally almost Einstein manifolds. Almost Einstein manifolds satisfy a generalization of the Einstein condition; they are Einstein on an open dense subspace and, in general, have a conformal scale singularity set that is a conformal infinity for the Einstein metric. In particular, the construction may be applied to yield families of compact Poincaré-Einstein manifolds, as well as classes of almost Einstein manifolds that are compact without boundary. We obtain classification results which show that the construction essentially exhausts a class of almost Einstein (and Poincaré-Einstein) manifolds. We develop the general theory of fixed conformal structures admitting multiple compatible almost Einstein structures. We also show that, in a class of cases, these are canonically related to a family of constant mean curvature totally umbillic embedded hypersurfaces.

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Conformal Dirichlet-Neumann Maps and Poincaré-Einstein Manifolds

Cited by 22 publications

References 47 publications

Boundary calculus for conformally compact manifolds

Boundary calculus for conformally compact manifolds

Projective BGG equations, algebraic sets, and compactifications of Einstein geometries

A Class of Compact Poincaré–einstein Manifolds: Properties and Construction

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