A survey on extensions of Riemannian manifolds and Bartnik mass estimates

Mantoulidis and Schoen developed a novel technique to handcraft asymptotically flat extensions of Riemannian manifolds $(\Sigma \cong \mathbb{S}^2,g)$, with $g$ satisfying $\lambda_1 = \lambda_1(-\Delta_g + K(g))>0$, where $\lambda_1$ is the first eigenvalue of the operator $-\Delta_g+K(g)$ and $K(g)$ is the Gaussian curvature of $g$, with control on the ADM mass of the extension. Remarkably, this procedure allowed them to compute the Bartnik mass in this so-called minimal case; the Bartnik mass is a notion of quasi-local mass in General Relativity which is very challenging to compute. In this survey, we describe the Mantoulidis-Schoen construction, its impact and influence in subsequent research related to Bartnik mass estimates when the minimality assumption is dropped, and its adaptation to other settings of interest in General Relativity.