Stability of the Positive Mass Theorem for Rotationally Symmetric Riemannian Manifolds

TL;DR

This paper demonstrates that rotationally symmetric Riemannian manifolds with nonnegative scalar curvature and zero ADM mass converge to Euclidean space in the Intrinsic Flat sense, providing explicit bounds and extending to non-symmetric cases.

Contribution

It establishes the stability of the Positive Mass Theorem for rotationally symmetric manifolds using Intrinsic Flat Distance and offers methods to estimate this distance without symmetry.

Findings

Sequences with ADM mass approaching zero converge to Euclidean space

Explicit bounds on Intrinsic Flat Distance are derived

Propositions for non-symmetric manifolds are included

Abstract

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.