Positive scalar curvature and isolated conical singularity
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916507044282368 |
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| author | Dai, Xianzhe Sun, Yukai Wang, Changliang |
| author_facet | Dai, Xianzhe Sun, Yukai Wang, Changliang |
| contents | We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_02941 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Positive scalar curvature and isolated conical singularity Dai, Xianzhe Sun, Yukai Wang, Changliang Differential Geometry We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical. |
| title | Positive scalar curvature and isolated conical singularity |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2412.02941 |