Geometric Stability of the Schoen-Yau Zero Mass Theorem
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909014270410752 |
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| author | Sormani, Christina |
| author_facet | Sormani, Christina |
| contents | In 1979, Schoen and Yau proved their famous Positive Mass Theorem which is a combination of a comparison theorem: {\em a three dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature has nonnegative ADM mass}, and a rigidity theorem: {\em if such a manifold has zero ADM mass then it is isometric to Euclidean space}. Here we review results and open questions on the geometric stability of their zero mass rigidity theorem: {\em if such a manifold has almost zero mass, how close is its geometry to that of Euclidean space}? We review the geometry of these spaces, examples of sequences of such spaces with mass approaching zero, and a variety of geometric notions of convergence. Although there has been much progress, it is still an open question (even in dimension three): exactly which geometric notion of convergence works best to capture the geometric stability of this famous rigidity theorem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17599 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Geometric Stability of the Schoen-Yau Zero Mass Theorem Sormani, Christina Differential Geometry Mathematical Physics Metric Geometry 54E35, 83C99, 58Z05, 30L05 In 1979, Schoen and Yau proved their famous Positive Mass Theorem which is a combination of a comparison theorem: {\em a three dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature has nonnegative ADM mass}, and a rigidity theorem: {\em if such a manifold has zero ADM mass then it is isometric to Euclidean space}. Here we review results and open questions on the geometric stability of their zero mass rigidity theorem: {\em if such a manifold has almost zero mass, how close is its geometry to that of Euclidean space}? We review the geometry of these spaces, examples of sequences of such spaces with mass approaching zero, and a variety of geometric notions of convergence. Although there has been much progress, it is still an open question (even in dimension three): exactly which geometric notion of convergence works best to capture the geometric stability of this famous rigidity theorem. |
| title | Geometric Stability of the Schoen-Yau Zero Mass Theorem |
| topic | Differential Geometry Mathematical Physics Metric Geometry 54E35, 83C99, 58Z05, 30L05 |
| url | https://arxiv.org/abs/2604.17599 |