Rigidity of compact quasi-Einstein manifolds with boundary
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913016782520320 |
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| author | Costa, Johnatan Ribeiro Jr, Ernani Zhou, Detang |
| author_facet | Costa, Johnatan Ribeiro Jr, Ernani Zhou, Detang |
| contents | In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We show that a $3$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{3}_{+}$, or the cylinder $I\times\mathbb{S}^2$ with the product metric. For dimension $n=4,$ we prove that a $4$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{4}_{+},$ or the cylinder $I\times\mathbb{S}^3$ with the product metric, or the product space $\mathbb{S}^{2}_{+}\times\mathbb{S}^2$ with the product metric. Other related results for arbitrary dimensions are also discussed. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_16929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity of compact quasi-Einstein manifolds with boundary Costa, Johnatan Ribeiro Jr, Ernani Zhou, Detang Differential Geometry Analysis of PDEs In this article, we investigate the geometry of compact quasi-Einstein manifolds with boundary. We show that a $3$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{3}_{+}$, or the cylinder $I\times\mathbb{S}^2$ with the product metric. For dimension $n=4,$ we prove that a $4$-dimensional simply connected compact quasi-Einstein manifold with boundary and constant scalar curvature is isometric, up to scaling, to either the standard hemisphere $\mathbb{S}^{4}_{+},$ or the cylinder $I\times\mathbb{S}^3$ with the product metric, or the product space $\mathbb{S}^{2}_{+}\times\mathbb{S}^2$ with the product metric. Other related results for arbitrary dimensions are also discussed. |
| title | Rigidity of compact quasi-Einstein manifolds with boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2401.16929 |