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

Gover, A. Rod; Leitner, Felipe

doi:10.1142/s021919971000397x

Cited by 10 publications

(7 citation statements)

References 48 publications

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“…This picture is slightly generalised by the notion of an almost Einstein manifold which means simply any conformal manifold with a similar type of holonomy reduction (meaning essentially the Cartan holonomy group fixes a point), and a programme to study the nature and geometry of the singularity set using the tractor calculus associated to the Cartan connection (see Section 2.3 below) was developed in [Gov07,Gov10]. Further examples and related reductions are constructed and discussed in [GL10]. More subtle examples (including a discussion of the singularity set) and a treatment of decomposable conformal holonomy are presented in the works [Lei10, Lei12, AL12] of Leitner and Armstrong-Leitner.…”

Section: Introductionmentioning

confidence: 99%

Holonomy reductions of Cartan geometries and curved orbit decompositions

Čap¹,

Gover²,

Hammerl³

2014

Duke Math. J.

195

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We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an equivalent description of solutions to these equations as parallel sections of a tractor bundle. In projective geometry we study a third order differential equation that governs the existence of a compatible Einstein metric. In CR-geometry we discuss an invariant equation that governs the existence of a compatible K\"{a}hler-Einstein metric.Comment: v2: major revision; 30 pages v3: final version to appear in Duke Math.

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Section: Introductionmentioning

confidence: 99%

Holonomy reductions of Cartan geometries and curved orbit decompositions

Čap¹,

Gover²,

Hammerl³

2014

Duke Math. J.

195

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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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“…There are examples with Σ totally umbilic and the metric g = σ −2 g Einstein on the complement of Σ. We have seen this above on the sphere in Section 3.1 (following [13]), and there are also examples on suitable products of the sphere with Einstein manifolds [15,16]. This leads to our main questions.…”

Section: Singular Yamabe and Obata Problemsmentioning

confidence: 88%

Singular Yamabe and Obata Problems

Gover¹,

Waldron²

2019

Preprint

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A conformal geometry determines a distinguished, potentially singular, variant of the usual Yamabe problem, where the conformal factor can change sign. When a smooth solution does change sign, its zero locus is a smoothly embedded separating hypersurface that, in dimension three, is necessarily a Willmore energy minimiser or, in higher dimensions, satisfies a conformally invariant analog of the Willmore equation. In any case the zero locus is critical for a conformal functional that generalises the total Q-curvature by including extrinsic data. These observations lead to some interesting global problems that include natural singular variants of a classical problem solved by Obata.

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

Cited by 10 publications

References 48 publications

Holonomy reductions of Cartan geometries and curved orbit decompositions

Holonomy reductions of Cartan geometries and curved orbit decompositions

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

Singular Yamabe and Obata Problems

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