The ambient obstruction tensor and conformal holonomy

Leistner, Thomas; Lischewski, Andree

doi:10.2140/pjm.2017.290.403

Cited by 4 publications

(6 citation statements)

References 36 publications

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“…And thus the well-definedness of the Cauchy problem implies the existence of an associated (n + 1)-dimensional Lorentzian with a parallel spinor. Such a relation between families of metrics with special holonomy and solutions of the constraint equations was already conjectured by Leistner and Lischewski, see [27]. We essentially show that the conditions in [27, Table 1] is satified if and only if the divergence condition (31) in our Appendix D is satisfied.…”

Section: Manifold Metric Dimension Typesupporting

confidence: 72%

“…Important progress about Lorentzian manifolds with parallel spinors was recently achieved by H. Baum, Th. Leistner and A. Lischewski [8,28,27], see also [9] for associated lecture notes. In particular, these authors showed the well-posedness of an associated Cauchy problem which we will now describe in more detail and which will be the main topic of the present article.…”

Section: The Cauchy Problem For Parallel Light-like Spinorsmentioning

confidence: 99%

“…In this article we are concerned with the case, that ϕ is a parallel spinor with a light-like Dirac-current V ϕ . This problem was studied in [8], [28], and [27].…”

Section: The Cauchy Problem For Parallel Light-like Spinorsmentioning

confidence: 99%

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Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

Ammann,

Kroencke,

Müller

2019

Preprint

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Lorentzian manifolds with parallel spinors are important objects of study in several branches of geometry, analysis and mathematical physics. Their Cauchy problem has recently been discussed by Baum, Leistner and Lischewski, who proved that the problem locally has a unique solution up to diffeomorphisms, provided that the intial data given on a space-like hypersurface satisfy some constraint equations. In this article we provide a method to solve these constraint equations. In particular, any curve (resp. closed curve) in the moduli space of Riemannian metrics on M with a parallel spinor gives rise to a solution of the constraint equations on M × (a, b) (resp. M × S 1 ). * B.A. has been partially supported by SFB 1085 Higher Invariants, Regensburg, funded by the DFG.

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Section: Manifold Metric Dimension Typesupporting

confidence: 72%

Section: The Cauchy Problem For Parallel Light-like Spinorsmentioning

confidence: 99%

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Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

Ammann,

Kroencke,

Müller

2019

Preprint

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“…Initial data triples are also tightly connected to a Cauchy problem for parallel spinors on Lorentzian manifolds, recently studied by Baum, Leistner and Lischewski [9,8,35,36], see also [44]. Obviously the work by Baum, Leistner, and Lischewski is tightly connected to Leistner's work on holonomy groups of Lorentzian manifolds [34].…”

Section: Results Of This Articlementioning

confidence: 92%

Dominant energy condition and spinors on Lorentzian manifolds

Ammann¹,

Glöckle²

2021

Preprint

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Let (M , g) be a time-and space-oriented Lorentzian spin manifold, and let M be a compact spacelike hypersurface of M with induced Riemannian metric g and second fundamental form K. If (M , g) satisfies the dominant energy condition in a strict sense, then the Dirac-Witten operator of M ⊆ M is an invertible, self-adjoint Fredholm operator. This allows us to use index theoretical methods in order to detect non-trivial homotopy groups in the space of initial on M satisfying the dominant energy condition in a strict sense. The central tool will be a Lorentzian analogue of Hitchin's α-invariant. In case that the dominant energy condition only holds in a weak sense, the Dirac-Witten operator may be non-invertible, and we will study the kernel of this operator in this case. We will show that the kernel may only be non-trivial if π 1 (M ) is virtually solvable of derived length at most 2. This allows to extend the index theoretical methods to spaces of initial data, satisfying the dominant energy condition in the weak sense. We will show further that a spinor ϕ is in the kernel of the Dirac-Witten operator on (M, g, K) if and only if (M, g, K, ϕ) admits an extension to a Lorentzian manifold (N , h) with parallel spinor φ such that M is a Cauchy hypersurface of (N , h), such that g and K are the induced metric and second fundamental form of M , respectively, and ϕ is the restriction of φ to M . DOMINANT ENERGY CONDITION AND SPINORS ONLORENTZIAN MANIFOLDS 1 4.1. The hypersurface spinor bundle 10 4.2. Dirac currents 11 4.3. The Cauchy problem for parallel spinors 12 5. The kernel of the Dirac-Witten operator 15 5.1. Dirac-Witten operators and Schrödinger-Lichnerowicz formula 15 5.2. From the kernel of the Dirac-Witten operator to initial data triples 16 5.3. Examples of lightlike (generalized) initial data triples 20 5.4. Lightlike initial data manifolds and other notation 22 5.5. Lightlike generalized initial data triples with compact leaves 24 5.6. More results for all lightlike generalized initial data triples 25 5.7. Further results in the case of non-compact leaves 29 5.8. Conclusions 32 6. Homotopy groups of I > (M ) and I ≥ (M ) 32 6.1. Initial data sets and positive scalar curvature 33 6.2. The α-index and index difference for psc metrics 35 6.3. The index difference for initial data sets strictly satisfying DEC 37 6.4. Application to general relativity 40 Concluding remark 42 Appendix A. The Taylor development map for Ricci-flat metrics 42 Appendix B. Proof of Lemma 5.17 45 References 46

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“…We will prove this theorem in Section 2.4. Note that the statement about the obstruction tensor can also be obtained from results in [23]. Theorem 1.2 leads us to study the equation (1.2) for g as in (1.1) defined by a Walker metric g with parallel null distribution N and with a tensor h with Im(h) ⊂ N .…”

Section: Introduction and Main Resultsmentioning

confidence: 96%

Conformal Walker metrics and linear Fefferman-Graham equations

Anderson,

Leistner,

Lischewski

et al. 2016

Preprint

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The conformal Fefferman-Graham ambient metric construction is one of the most fundamental constructions in conformal geometry. It embeds a manifold with a conformal structure into a pseudo-Riemannian manifold whose Ricci tensor vanishes up to a certain order along the original manifold. Despite the general existence result of such ambient metrics by Fefferman and Graham, not many examples of conformal structures with Ricci-flat ambient metrics are known. Motivated by previous examples, for which the Fefferman-Graham equations for the ambient metric to be Ricci-flat reduce to a system of linear PDEs, in the present article we develop a method to find ambient metrics for conformal classes of metrics with two-step nilpotent Schouten tensor. Using this method, for metrics for which the image of the Schouten tensor is invariant under parallel transport, i.e., certain types of Walker metrics, we obtain explicit ambient metrics. This includes certain left-invariant Walker metrics as well as pp-waves.

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The ambient obstruction tensor and conformal holonomy

Cited by 4 publications

References 36 publications

Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

Construction of initial data sets for Lorentzian manifolds with lightlike parallel spinors

Dominant energy condition and spinors on Lorentzian manifolds

Conformal Walker metrics and linear Fefferman-Graham equations

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