A flow approach to Bartnik's static metric extension conjecture in axisymmetry
Abstract
We investigate Bartnik's static metric extension conjecture under the additional assumption of axisymmetry of both the given Bartnik data and the desired static extensions. To do so, we suggest a geometric flow approach, coupled to the Weyl-Papapetrou formalism for axisymmetric static solutions to the Einstein vacuum equations. The elliptic Weyl-Papapetrou system becomes a free boundary value problem in our approach. We study this new flow and the coupled flow--free boundary value problem numerically and find axisymmetric static extensions for axisymmetric Bartnik data in many situations, including near round spheres in spatial Schwarzschild of positive mass.
Keywords
Cite
@article{arxiv.1904.11040,
title = {A flow approach to Bartnik's static metric extension conjecture in axisymmetry},
author = {Carla Cederbaum and Oliver Rinne and Markus Strehlau},
journal= {arXiv preprint arXiv:1904.11040},
year = {2019}
}
Comments
60 pages, 13 figures. Expanded Section 3.3 to address longtime existence and uniqueness of solutions to the linearised flow equations. To appear in Pure and Applied Mathematics Quarterly, special issue in honour of Robert Bartnik