Mathematics
Document Type
Article
Abstract
Given a semi-Riemannian 4-manifold (M, g) with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics gK is constructed, defined on an open set in M, which coincides with M in many typical examples. Under certain conditions g and gK share various properties, such as a Killing vector field or a vector field with a geodesic flow. In some cases the Kähler metrics are complete. The Ricci and scalar curvatures of gK are computed under certain assumptions in terms of data associated to g. Many examples are described, including classical spacetimes in warped products, for instance de Sitter spacetime, as well as gravitational plane waves, metrics of Petrov type D such as Kerr and NUT metrics, and metrics for which gK is an SKR metric. For the latter an inverse ansatz is described, constructing g from the SKR metric.
Publication Title
Complex Manifolds
Publication Date
2020
Volume
7
Issue
1
First Page
36
Last Page
61
ISSN
2300-7443
DOI
10.1515/coma-2020-0002
Keywords
Kähler Manifold, Ricci flow, asymptotics
Repository Citation
Aazami, Amir Babak and Maschler, Gideon, "Kähler metrics via Lorentzian Geometry in dimension four" (2020). Mathematics. 25.
https://commons.clarku.edu/mathematics/25
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Copyright Conditions
Aazami, A. B., & Maschler, G. (2020). Kähler metrics via Lorentzian geometry in dimension four. Complex Manifolds, 7(1), 36-61. https://doi.org/10.1515/coma-2020-0002
