2011/04/13 by Dan A. Lee, Lee, Dan A., Christina Sormani +1 · 1 citation
Mathematics · Physics and Astronomy · #30L05 #58Z05 #83C99 #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #gr-qc #math.DG #math.MG #msc:30L05 #msc:58Z05 #msc:83C99
paper · pdf · doi:10.48550/arxiv.1104.2657
36 pages, 3 figures, open problems in the back, v2: corrected minor typos found after publication. Journal fur die Reine und Angewandte Mathematik Crelle's Journal 2014
openalex publication_date 2011/04/13 · arxiv created 2014/05/04 · arxiv updated 2015/03/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.