Scalar and Mean Curvature Comparison on Compact Cylinder
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912473139904512 |
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| author | Xu, Jie |
| author_facet | Xu, Jie |
| contents | Let $ X $ be a closed, oriented Riemannian manifold. Denote by $ (M = X \times I, \partial M = X \times \lbrace 0 \rbrace \cup X \times \lbrace 1 \rbrace, g) $ a compact cylinder with smooth boundary, $ \dim M \geqslant 3 $. In this article, we address the following question: If $ g $ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $ M $ and nonnegative mean curvature on $ \partial M $; and (ii) the $ g $-angle between normal vector field $ ν_{g} $ along $ \partial M $ and $ \partial_ξ \in Γ(TI) $ being less than $ \fracπ{4} $, then there exists a metric $ \tilde{g} $ on $ M $ such that $ \tilde{g} |_{X \times \lbrace 0 \rbrace} $ is a PSC metric on $ X \cong X \times \lbrace 0 \rbrace $. Equivalently, we show that if $ X $ admits no PSC metric, but $ M $ admits a PSC metric $ g $ satisfying the angle condition, then the mean curvature on $ \partial M $ must be negative somewhere. This generalizes a result of Gromov and Lawson (Ann. of Math. (2), 1980) for $ X = \mathbb{T}^{n} $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07005 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Scalar and Mean Curvature Comparison on Compact Cylinder Xu, Jie Differential Geometry 58J05, 35J60, 53C18 Let $ X $ be a closed, oriented Riemannian manifold. Denote by $ (M = X \times I, \partial M = X \times \lbrace 0 \rbrace \cup X \times \lbrace 1 \rbrace, g) $ a compact cylinder with smooth boundary, $ \dim M \geqslant 3 $. In this article, we address the following question: If $ g $ is a Riemannian metric having (i) positive scalar curvature (PSC metric) on $ M $ and nonnegative mean curvature on $ \partial M $; and (ii) the $ g $-angle between normal vector field $ ν_{g} $ along $ \partial M $ and $ \partial_ξ \in Γ(TI) $ being less than $ \fracπ{4} $, then there exists a metric $ \tilde{g} $ on $ M $ such that $ \tilde{g} |_{X \times \lbrace 0 \rbrace} $ is a PSC metric on $ X \cong X \times \lbrace 0 \rbrace $. Equivalently, we show that if $ X $ admits no PSC metric, but $ M $ admits a PSC metric $ g $ satisfying the angle condition, then the mean curvature on $ \partial M $ must be negative somewhere. This generalizes a result of Gromov and Lawson (Ann. of Math. (2), 1980) for $ X = \mathbb{T}^{n} $. |
| title | Scalar and Mean Curvature Comparison on Compact Cylinder |
| topic | Differential Geometry 58J05, 35J60, 53C18 |
| url | https://arxiv.org/abs/2507.07005 |