Stephen McCormick: Estimates for the Bartnik mass of constant mean curvature surfaces

Let \(g\) be a metric on \(S^{2}\) with positive Gaussian curvature and \(H_0\) be a positive constant that is not too large. Based on recent work by Mantoulidis and Schoen, we construct asymptotically flat initial data for Einstein's equations with boundary isometric to \((S^2,g)\) and mean curvature \(H_0\), while controlling the ADM mass.

In particular, we show that (Bray's formulation of) Bartnik's quasilocal mass is approximated by the Hawking mass. Furthermore, the Bartnik mass can be made close to the Hawking mass by choosing the surface data to be close to round, or \(H_0\) small. In this talk, we will briefly review the Bartnik mass before sketching the proof of this result. This is joint work with Armando Cabrera Pacheco, Carla Cederbaum and Pengzi Miao.