The rigidity of the Riemannian positive mass theorem for asymptotically flat manifolds states that the total mass of such a manifold is zero if and only if the manifold is isometric to the Euclidean space. This leads us to ask us if an asymptotically flat manifold that has total mass almost zero whether or not it is close in some sense to the Euclidean space. I will review Huang-Lee-Sormani, Allen-Perales and Huang-Lee-Perales's stability result for asymptotically flat graphical manifolds where intrinsic flat distance was used. Motivated by these, and as a part of an ongoing project with A. Cabrera Pacheco, I will also discuss an analogous result for asymptotically hyperbolic graphical manifolds.
Department of Mathematics