Completeness of hyperbolic centroaffine hypersurfaces

Cortés, Vicente; Nardmann, Marc; Suhr, Stefan

doi:10.4310/cag.2016.v24.n1.a3

Cited by 12 publications

(23 citation statements)

References 11 publications

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“…Then H := {h = 1} ⊂ U is a smooth hypersurface and U = R >0 · H. We assume that for g U := −∂ 2 h the metric g H := ι * g U is positive definite, where ι : H ֒→ U is the inclusion. The manifold H, 1 k g H is a hyperbolic centroaffine hypersurface in the sense of [CNS16].…”

Section: Completeness Of Hessian Metrics Associated With a Hyperbolicmentioning

confidence: 99%

“…We give a global description of the resulting projective special Kähler manifolds (M c , g c ), where (M 0 , g 0 ) = (M , g) is the manifold in the image of the supergravity r-map. The manifold M c is a do- CNS16]. Recall that the level set is required to be locally strictly convex for H to be a projective special real manifold (with positive definite metric).…”

Section: Introductionmentioning

confidence: 99%

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ASK/PSK-correspondence and the r-map

2017

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We formulate a correspondence between affine and projective special Kähler manifolds of the same dimension. As an application, we show that, under this correspondence, the affine special Kähler manifolds in the image of the rigid r-map are mapped to one-parameter deformations of projective special Kähler manifolds in the image of the supergravity r-map. The above one-parameter deformations are interpreted as perturbative α ′ -corrections in heterotic and type-II string compactifications with N = 2 supersymmetry. Also affine special Kähler manifolds with quadratic prepotential are mapped to one-parameter families of projective special Kähler manifolds with quadratic prepotential. We show that the completeness of the deformed supergravity r-map metric depends solely on the (well-understood) completeness of the undeformed metric and the sign of the deformation parameter.

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Section: Completeness Of Hessian Metrics Associated With a Hyperbolicmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

ASK/PSK-correspondence and the r-map

2017

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“…Moreover, it was proved in [45], that both the supergravity r-map and the supergravity c-map preserve completeness of the Riemannian metrics. While complete projective special real curves and surfaces were classified in [45] and [46] respectively, a necessary and suf-ficient condition for the completeness of a projective special real manifold was obtained more recently in [47]. In fact, it was shown that a projective special real manifold H ⊂ Ê n+1 is complete if and only if it is closed as a subset of Ê n+1 , a condition which can be easily checked in many examples.…”

Section: 1 Background and Motivationmentioning

confidence: 99%

Special geometry of Euclidean supersymmetry IV: the local c-map

Cortés

Dempster

Mohaupt

et al. 2015

J. High Energ. Phys.

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Abstract:We consider timelike and spacelike reductions of 4D, N = 2 Minkowskian and Euclidean vector multiplets coupled to supergravity and the maps induced on the scalar geometry. In particular, we investigate (i) the (standard) spatial c-map, (ii) the temporal c-map, which corresponds to the reduction of the Minkowskian theory over time, and (iii) the Euclidean c-map, which corresponds to the reduction of the Euclidean theory over space. In the last two cases we prove that the target manifold is para-quaternionic Kähler.In cases (i) and (ii) we construct two integrable complex structures on the target manifold, one of which belongs to the quaternionic and para-quaternionic structure, respectively. In case (iii) we construct two integrable para-complex structures, one of which belongs to the para-quaternionic structure.In addition we provide a new global construction of the spatial, temporal and Euclidean c-maps, and separately consider a description of the target manifold as a fibre bundle over a projective special Kähler or para-Kähler base.

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“…Based on the effective necessary and sufficient completeness criterion for projective special real manifolds provided in [CNS,Thm. 2.6], it is easy to construct many more examples of complete projective special Kähler domains with cubic prepotential (see for example [CDL] and work in progress by Jüngling, Lindemann and the first two authors) and corresponding one-loop deformed quaternionic Kähler manifolds by Theorem 27.…”

mentioning

confidence: 99%

“…The question of completeness for a projective special real manifold (H, g H ) reduces to a simple topological question for the hypersurface H ⊂ R n :Theorem 22 [CNS,. Thm.…”

mentioning

confidence: 99%

Completeness of Projective Special Kähler and Quaternionic Kähler Manifolds

Cortés

Dyckmanns

Suhr

2017

Springer INdAM Series

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We prove that every projective special Kähler manifold with regular boundary behaviour is complete and defines a family of complete quaternionic Kähler manifolds depending on a parameter c ≥ 0. We also show that, irrespective of its boundary behaviour, every complete projective special Kähler manifold with cubic prepotential gives rise to such a family. Examples include non-trivial deformations of noncompact symmetric quaternionic Kähler manifolds.

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Completeness of hyperbolic centroaffine hypersurfaces

Cited by 12 publications

References 11 publications

ASK/PSK-correspondence and the r-map

ASK/PSK-correspondence and the r-map

Special geometry of Euclidean supersymmetry IV: the local c-map

Completeness of Projective Special Kähler and Quaternionic Kähler Manifolds

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