Projective compactness and conformal boundaries

Čap, Andreas; Gover, A. Rod

doi:10.1007/s00208-016-1370-9

Cited by 12 publications

(20 citation statements)

References 31 publications

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“…The results in Theorem 5, along with converse results in [3], expose a previously unseen critical role for the scalar curvature in questions of projective compactification. In Proposition 3 we show that if the projective structure of a metric on M extends to M then, surprisingly, its scalar curvature also extends smoothly (as a function) to M .…”

Section: Introductionmentioning

confidence: 66%

“…Here being Einstein is replaced by a much weaker condition on the asymptotics of the scalar curvature of g, but we still can conclude projective compactness of order α = 2. Via the results of [3], this provides a number of further facts about g, including a certain asymptotic form, an asymptotic version of the Einstein property, and the fact that ∂M inherits a canonical conformal structure determined by g. Some of these consequences are summarised Corollary 6.…”

Section: Introductionmentioning

confidence: 95%

“…Knowing that a metric g satisfying the assumptions of Theorem 5 is projectively compact of order α = 2 allows one to apply all the results of [3] to g. We summarise here some of the key points from there (that are mainly drawn from Theorems 7 and 11 in that source):…”

Section: 4mentioning

confidence: 99%

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Scalar curvature and projective compactness

Čap

Gover

2015

Journal of Geometry and Physics

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Abstract. Consider a manifold with boundary, and such that the interior is equipped with a pseudo-Riemannian metric. We prove that, under mild asymptotic non-vanishing conditions on the scalar curvature, if the Levi-Civita connection of the interior does not extend to the boundary (because for example the interior is complete) whereas its projective structure does, then the metric is projectively compact of order 2; this order is a measure of volume growth toward infinity. The result implies a host of results including that the metric satisfies asymptotic Einstein conditions, and induces a canonical conformal structure on the boundary. Underpinning this work is a new interpretation of scalar curvature in terms of projective geometry. This enables us to show that if the projective structure of a metric extends to the boundary then its scalar curvature also naturally and smoothly extends.

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Section: Introductionmentioning

confidence: 66%

Section: Introductionmentioning

confidence: 95%

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Scalar curvature and projective compactness

Čap

Gover

2015

Journal of Geometry and Physics

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“…Further the authors would like to thank A. Rod Gover for pointing our attention towards the results in [Me] and explaining the results of [CG13] and [CG14].…”

Section: Introductionmentioning

confidence: 90%

Completeness of hyperbolic centroaffine hypersurfaces

Cortés

Nardmann

Suhr

2016

Communications in Analysis and Geometry

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This paper is concerned with the completeness (with respect to the centroaffine metric) of hyperbolic centroaffine hypersurfaces which are closed in the ambient vector space. We show that completeness holds under generic regularity conditions on the boundary of the convex cone generated by the hypersurface. The main result is that completeness holds for hyperbolic components of level sets of homogeneous cubic polynomials. This implies that every such component defines a complete quaternionic Kähler manifold of negative scalar curvature.

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“…Moreover, from (17) and (18) one deduces that the tractor connection on A with respect to ∇ may be written as…”

Section: Adjoint Tractorsmentioning

confidence: 99%

Projective geometry of Sasaki–Einstein structures and their compactification

Gover¹,

Neusser²,

Willse³

2019

Dissertationes Math.

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We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the Kähler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail.

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Projective compactness and conformal boundaries

Cited by 12 publications

References 31 publications

Scalar curvature and projective compactness

Scalar curvature and projective compactness

Completeness of hyperbolic centroaffine hypersurfaces

Projective geometry of Sasaki–Einstein structures and their compactification

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