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.