Covering instability for the existence of positive scalar curvature metrics
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918080658014208 |
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| author | Li, Chao Zhang, Boyu |
| author_facet | Li, Chao Zhang, Boyu |
| contents | We show that a closed non-orientable $3$-manifold admits a positive scalar curvature metric if and only if its orientation double cover does; however, for each $4\le n\le 7$, there exist infinitely many smooth non-orientable $n$-manifolds $M$ that are mutually non-homotopy equivalent, such that the orientation double cover of $M$ admits positive scalar curvature metrics, but every closed smooth manifold that is homotopy equivalent to $M$ cannot admit positive scalar curvature metrics. These examples were first introduced by Alpert-Balitskiy-Guth in the study of Urysohn widths.
To prove the nonexistence result, we extend the Schoen-Yau inductive descent approach to non-orientable manifolds. We also discuss band width estimates and the notion of enlargeability for non-orientable PSC manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13885 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Covering instability for the existence of positive scalar curvature metrics Li, Chao Zhang, Boyu Differential Geometry Geometric Topology Metric Geometry We show that a closed non-orientable $3$-manifold admits a positive scalar curvature metric if and only if its orientation double cover does; however, for each $4\le n\le 7$, there exist infinitely many smooth non-orientable $n$-manifolds $M$ that are mutually non-homotopy equivalent, such that the orientation double cover of $M$ admits positive scalar curvature metrics, but every closed smooth manifold that is homotopy equivalent to $M$ cannot admit positive scalar curvature metrics. These examples were first introduced by Alpert-Balitskiy-Guth in the study of Urysohn widths. To prove the nonexistence result, we extend the Schoen-Yau inductive descent approach to non-orientable manifolds. We also discuss band width estimates and the notion of enlargeability for non-orientable PSC manifolds. |
| title | Covering instability for the existence of positive scalar curvature metrics |
| topic | Differential Geometry Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2506.13885 |