Stability of the Positive Mass Theorem for Axisymmetric Manifolds

TL;DR

This paper proves the stability of the Positive Mass Theorem for axisymmetric manifolds in a specific function space, providing estimates for geometric quantities under certain conditions.

Contribution

It establishes the stability of the Positive Mass Theorem in the $W^{1,p}$ sense for axisymmetric manifolds with nonnegative scalar curvature, including geometric estimates.

Findings

Stability of the Positive Mass Theorem proven in $W^{1,p}$ sense

Derived estimates for volumes, areas, and distances within manifolds

Results apply to asymptotically flat axisymmetric manifolds with technical assumptions

Abstract

Away from the central axis, we prove the stability of the Positive Mass Theorem in the $W^{1,p}$ sense for asymptotically flat axisymmetric manifolds with nonnegative scalar curvature satisfying some additional technical assumptions. We also derive estimates for the volumes of regions, the areas of axisymmetric surfaces, and the distances between points within the manifolds.