Ambient metrics for $n$-dimensional $pp$-waves

Abstract

We provide an explicit formula for the Fefferman-Graham-ambient metric of an $n$-dimensional conformal $pp$-wave in those cases where it exists. In even dimensions we calculate the obstruction explicitly. Furthermore, we describe all 4-dimensional $pp$-waves that are Bach-flat, and give a large class of Bach-flat examples which are conformally Cotton-flat, but not conformally Einstein. Finally, as an application, we use the obtained ambient metric to show that even-dimensional $pp$-waves have vanishing critical $Q$-curvature.

arXiv comment17 pages, in v2 footnote and references added and typos corrected, in v3 remark in the Introduction about Brinkmann's results corrected and footnote added

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Cited by in corpus (18)

• Explicit formulas for GJMS-operators and $Q$-curvatures
• Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary
• Schrödinger Manifolds
• Isotropic quasi-Einstein manifolds
• Differential equations and para-CR structures
• Conformal structures with $G_{2(2)}$-ambient metrics
• Conformal pure radiation with parallel rays
• On conformally recurrent manifolds of dimension greater than 4
• Bach-flat isotropic gradient Ricci solitons
• Fefferman-Graham ambient metrics of Patterson-Walker metrics
• The ambient obstruction tensor and conformal holonomy
• Convergence of the Fefferman-Graham expansion and complex black hole anatomy
• Conformally flat Lorentzian manifolds with special holonomy groups
• Constructing Bach Flat Manifolds of signature $(2,2)$ using the modified Riemannian extension
• Simple conformally recurrent space-times are conformally recurrent PP-waves
• Bach-flat noncompact steady quasi-Einstein manifolds
• The boundary value problem for Laplacian on differential forms and conformally Einstein infinity
• Conformal geometry of non-reductive four-dimensional homogeneous spaces