# Almost Rigidity of the Positive Mass Theorem

# Almost Rigidity of the Positive Mass Theorem

The Positive Mass Theorem states that an asymptotically flat Riemannian manifold, $M^3$, with nonegative Scalar curvature has nonnegative ADM mass, $m_{ADM}(M)\ge 0$, and if the ADM mass is 0 then we have rigidity: the manifold is isometric to Euclidean space. It has long been known that if one has a sequence of such manifolds $M^3_j$ with $m_{ADM}(M_j) \to 0$ then $M_j$ need not converge smoothly to Euclidean space. To avoid bubbling, one forbids the sequence of manifolds to have closed interior minimal surfaces, but allows the manifolds to have minimal boundaries. Dan Lee and I conjectured that in this setting the $M_j$ converge to Euclidean space in the pointed intrinsic flat sense for well chosen points. We proved this in the rotationally symmetric setting and provided examples in that setting where smooth and Gromov-Hausdorff convergence fail. Lan-Hsuan Huang, Lee and I have then proven the result in the graph setting using Wenger's Compactness theorem and Arzela-Ascoli Theorems of mine. Iva Stavrov and I have proven it in the geometrostatic setting using Colding-Minnicozzi and Bray's Penrose to locate the minimal surfaces and then joint work of mine with Lakzian to estimate the intrinsic flat distance. I will close with Gromov's conjectures concerning intrinsic flat convergence and scalar curvature. All papers on intrinsic flat convergence may be found at: https://sites.google.com/site/intrinsicflatconvergence/