Existence of Positive Scalar Curvature and Positive Yamabe constant on Hypersurfaces of Noncompact Cylinders
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| Format: | Preprint |
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2025
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| _version_ | 1866909813133279232 |
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| author | Xu, Jie |
| author_facet | Xu, Jie |
| contents | Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. Let $ (Z, \partial Z) $ be an oriented, compact manifold with (possibly empty) smooth boundary and $ \dim Z \geqslant 2 $. In this article, we show that if the noncompact cylinder $ X \times \mathbb{R} $ admits a complete Riemannian metric $ g $ with positive injectivity radius and uniformly positive scalar curvature, and that is of bounded geometry or bounded curvature, then $ X $ admits a positive scalar curvature metric within the same conformal class provided that some $ g $-angle condition is satisfied. This partially answers a conjecture of Rosenberg and Stolz \cite{RosSto} without topological assumptions. With the $ g $-angle condition, we can also show that if $ (Z \times \mathbb{R}, \partial Z \times \mathbb{R}) $ admits a complete metric $ g $ that has positive Yamabe constant and positive injectivity radius, and is of bounded geometry or bounded curvature, then $ (Z, \partial Z) $ has positive Yamabe constant for the conformal class $ [\imath^{*}g] $ with the natural inclusion $ \imath : (Z, \partial Z) \hookrightarrow (Z \times \mathbb{R}, \partial Z \times \mathbb{R}) $. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_24016 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Positive Scalar Curvature and Positive Yamabe constant on Hypersurfaces of Noncompact Cylinders Xu, Jie Differential Geometry 58J05, 58J60, 53C21, 53C42, 57R40, 35B65 Let $ X $ be an oriented, closed manifold with $ \dim X \geqslant 2 $. Let $ (Z, \partial Z) $ be an oriented, compact manifold with (possibly empty) smooth boundary and $ \dim Z \geqslant 2 $. In this article, we show that if the noncompact cylinder $ X \times \mathbb{R} $ admits a complete Riemannian metric $ g $ with positive injectivity radius and uniformly positive scalar curvature, and that is of bounded geometry or bounded curvature, then $ X $ admits a positive scalar curvature metric within the same conformal class provided that some $ g $-angle condition is satisfied. This partially answers a conjecture of Rosenberg and Stolz \cite{RosSto} without topological assumptions. With the $ g $-angle condition, we can also show that if $ (Z \times \mathbb{R}, \partial Z \times \mathbb{R}) $ admits a complete metric $ g $ that has positive Yamabe constant and positive injectivity radius, and is of bounded geometry or bounded curvature, then $ (Z, \partial Z) $ has positive Yamabe constant for the conformal class $ [\imath^{*}g] $ with the natural inclusion $ \imath : (Z, \partial Z) \hookrightarrow (Z \times \mathbb{R}, \partial Z \times \mathbb{R}) $. |
| title | Existence of Positive Scalar Curvature and Positive Yamabe constant on Hypersurfaces of Noncompact Cylinders |
| topic | Differential Geometry 58J05, 58J60, 53C21, 53C42, 57R40, 35B65 |
| url | https://arxiv.org/abs/2509.24016 |