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 ( Σ ≅ mathvariant="double-struck">S 2 , g stretchy="false">) (\Sigma \cong \mathbb {S}^2,g) , with alttext="g"> encoding="application/x-tex">g satisfying alttext="lamda 1 colon-equal lamda left-parenthesis minus Delta Subscript Baseline plus K right-parenthesis greater-than 0"> λ<!-- λ <mml:mn>1 ≔ −<!-- − mathvariant="normal">Δ<!-- Δ <mml:mo>+ K &gt; 0 encoding="application/x-tex">\lambda _1 ≔\lambda _1(-\Delta _g + K(g))&gt;0 where 1"> _1 is the first eigenvalue operator alttext="minus encoding="application/x-tex">-\Delta _g+K(g) alttext="upper encoding="application/x-tex">K(g) Gaussian curvature control on ADM mass extension. Remarkably, this procedure allowed them compute Bartnik in so-called minimal case; notion quasi-local General Relativity which very challenging compute. In survey, we describe Mantoulidis–Schoen construction, its impact influence subsequent research related estimates when minimality assumption dropped, adaptation other settings interest Relativity.