#### Vol. 305, No. 1, 2020

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Stability of the positive mass theorem for axisymmetric manifolds

### Edward T. Bryden

Vol. 305 (2020), No. 1, 89–152
##### 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.

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##### Keywords
positive mass theorem, stability, small mass, axisymmetric
##### Mathematical Subject Classification 2010
Primary: 53C24, 53C80, 83C99
##### Milestones
Received: 20 June 2018
Revised: 28 June 2019
Accepted: 14 October 2019
Published: 17 March 2020
##### Authors
 Edward T. Bryden Department of Mathematics Stony Brook University Stony Brook, NY United States Tuebingen University Tuebingen Germany