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

## TL;DR

This paper introduces a geometric flow method to address Bartnik's static metric extension conjecture under axisymmetry, combining numerical analysis with the Weyl-Papapetrou formalism to find static extensions.

## Contribution

It proposes a novel flow-based approach to the conjecture, coupling it with the Weyl-Papapetrou formalism and solving a free boundary problem numerically.

## Key findings

- Successfully finds axisymmetric static extensions for various Bartnik data
- Demonstrates the approach near round spheres in Schwarzschild spacetime
- Provides numerical evidence supporting the conjecture in axisymmetric cases

## 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.