Schoen's conjecture for limits of isoperimetric surfaces
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912422796722176 |
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| author | Eichmair, Michael Koerber, Thomas |
| author_facet | Eichmair, Michael Koerber, Thomas |
| contents | Let $(M,g)$ be an $n$-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface $Σ\subset M$. In the case where $n = 3$, O. Chodosh and the first-named author have proven that $(M, g)$ is necessarily isometric to Euclidean space, confirming a conjecture of R. Schoen. In this paper, we extend this result to dimension $3 < n \leq 7$ provided that $Σ$ arises as a limit of isoperimetric surfaces. By contrast, we prove that when $3 < n \leq 7$, there is no such result for general noncompact area-minimizing $Σ\subset M$, even when additional assumptions on the stability of $Σ$ are imposed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12200 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Schoen's conjecture for limits of isoperimetric surfaces Eichmair, Michael Koerber, Thomas Differential Geometry Let $(M,g)$ be an $n$-dimensional asymptotically flat Riemannian manifold with nonnegative scalar curvature that admits a noncompact area-minimizing hypersurface $Σ\subset M$. In the case where $n = 3$, O. Chodosh and the first-named author have proven that $(M, g)$ is necessarily isometric to Euclidean space, confirming a conjecture of R. Schoen. In this paper, we extend this result to dimension $3 < n \leq 7$ provided that $Σ$ arises as a limit of isoperimetric surfaces. By contrast, we prove that when $3 < n \leq 7$, there is no such result for general noncompact area-minimizing $Σ\subset M$, even when additional assumptions on the stability of $Σ$ are imposed. |
| title | Schoen's conjecture for limits of isoperimetric surfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2303.12200 |