Bartnik's quasi-local mass for a compact manifold with boundary is defined by minimizing the ADM masses among admissible extensions. Among several proposed conjectures, Bartnik's stationary conjecture asserts that a minimizing initial data set must be vacuum and admit a timelike Killing vector. We make partial progress toward this conjecture by showing that a minimizing initial data set must sit in a "null dust" spacetime carrying a global Killing vector. On the other hand, we find pp-wave counterexamples to Bartnik's stationary and strict positivity conjectures in dimensions greater than 8.
In our proof, we introduce the concept of improvability of the dominant energy scalar, and we derive strong consequences of non-improvability using a new infinite-dimensional family of deformation to the Einstein constraint operator. This talk is based on a joint work with Dan Lee.
Lan-Hsuan Huang (Connecticut): Bartnik's quasi-local mass conjectures
