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

Cederbaum, Carla; Rinne, Oliver; Strehlau, Markus

doi:10.4310/PAMQ.2019.v15.n2.a1

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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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Citation

Cederbaum, C., Rinne, O., & Strehlau, M. (2019). A flow approach to Bartnik's static metric extension conjecture in axisymmetry. Pure and Applied Mathematics Quarterly, 15(2), 611-666. doi:10.4310/PAMQ.2019.v15.n2.a1.

Cite as: https://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.