2014/01/01 by Dan A. Lee, Christina Sormani · 1 citation
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · doi:10.1515/crelle-2012-0094
openalex publication_date 2014/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We study the stability of the positive mass theorem using the intrinsic flat distance. In particular we consider the class of complete asymptotically flat rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and no interior closed minimal surfaces whose boundaries are either outermost minimal hypersurfaces or are empty. We prove that a sequence of these manifolds whose ADM masses converge to zero must converge to Euclidean space in the pointed intrinsic flat sense. In fact we provide explicit bounds on the intrinsic flat distance between annular regions in the manifold and annular regions in Euclidean space by constructing an explicit filling manifold and estimating its volume. In addition, we include a variety of propositions that can be used to estimate the intrinsic flat distance between Riemannian manifolds without rotationally symmetry. Conjectures regarding the intrinsic flat stability of the positive mass theorem in the general case are proposed in the final section.