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A flow approach to Bartnik's static metric extension conjecture in axisymmetry

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Rinne,  Oliver
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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Strehlau,  Markus
Geometric Analysis and Gravitation, AEI-Golm, MPI for Gravitational Physics, Max Planck Society;

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1904.11040.pdf
(Preprint), 632KB

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Citation

Cederbaum, C., Rinne, O., & Strehlau, M. (in press). A flow approach to Bartnik's static metric extension conjecture in axisymmetry. Pure and Applied Mathematics Quarterly. Retrieved from http://arxiv.org/abs/1904.11040.


Cite as: http://hdl.handle.net/21.11116/0000-0003-AD8A-F
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.