Pure and Applied Mathematics Quarterly

Volume 15 (2019)

Number 3

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

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

On the evolution of the spacetime Bartnik mass

Pages: 897 – 920

DOI: https://dx.doi.org/10.4310/PAMQ.2019.v15.n3.a6


Stephen McCormick (Matematiska institutionen, Uppsala universitet, Uppsala, Sweden)

Pengzi Miao (Department of Mathematics, University of Miami, Coral Gables, Florida, US.A.)


It is conjectured that the full (spacetime) Bartnik mass of a surface $\Sigma$ is realised as the ADM mass of some stationary asymptotically flat manifold with boundary data prescribed by $\Sigma$. Assuming this holds true for a $1$-parameter family of surfaces $\Sigma_t$ evolving in an initial data set with the dominant energy condition, we compute an expression for the derivative of the Bartnik mass along these surfaces. An immediate consequence of this formula is that the Bartnik mass of $\Sigma_t$ is monotone non-decreasing whenever $\Sigma_t$ flows outward.


quasi-local mass, initial data

2010 Mathematics Subject Classification

Primary 83C40, 83C99. Secondary 83-06.

It is our pleasure to dedicate this paper to Robert Bartnik on the occasion of his 60th birthday.

Research partially supported by Simons Foundation Collaboration Grant for Mathematicians #585168.

Received 21 February 2019

Received revised 3 June 2019

Accepted 3 June 2019

Published 2 January 2020