Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 2

Special Issue: In Honor of Robert Bartnik (Part 1 of 2)

Guest Editors: Piotr T. Chruściel, Greg Galloway, Jim Isenberg, Pengzi Miao, Mu-Tao Wang, and Shing-Tung Yau

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

Pages: 611 – 666

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n2.a1


Carla Cederbaum (Department of Mathematics, University of Tübingen, Germany)

Oliver Rinne (HTW Berlin (University of Applied Sciences), Berlin, Germany; and Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany)

Markus Strehlau (Institute of Mathematics, Brandenburgische Technische Universität, Cottbus, Germany; and Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Potsdam, Germany)


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.


general relativity, axisymmetry, Weyl–Papapetrou coordinates, geometric flow, free boundary value problem

C. Cederbaum is indebted to the Baden–Württemberg Stiftung for the financial support of this research project by the Eliteprogramme for Postdocs. The work of C. Cederbaum is supported by the Institutional Strategy of the University of Tübingen (Deutsche Forschungsgemeinschaft, ZUK 63).

The early stages of O. Rinne’s work on this project were supported by a Heisenberg Fellowship and grant RI 2246/2 of the Deutsche Forschungsgemeinschaft.

M. Strehlau is indebted to the Graduate Research School (GRS) of the Brandenburg University of Technology Cottbus–Senftenberg for financial support.

Received 24 April 2019

Received revised 19 July 2019

Accepted 28 July 2019

Published 4 December 2019