On $q$-convex hypersurfaces in Riemannian manifolds
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866913149151608832 |
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| author | Colombo, Giulio Onti, Christos-Raent |
| author_facet | Colombo, Giulio Onti, Christos-Raent |
| contents | We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed $q$-convex immersed hypersurfaces in $(n+1)$-dimensional Riemannian manifolds, under a lower bound on the average of the smallest $(n-p)$ eigenvalues of the curvature operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20695 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On $q$-convex hypersurfaces in Riemannian manifolds Colombo, Giulio Onti, Christos-Raent Differential Geometry 53C40, 53C42, 53C20, 53C21 We prove that any closed, convex hypersurface in an $(n+1)$-dimensional Riemannian manifold with $\lceil \frac{n}{2} \rceil$-positive curvature operator is a rational homology sphere with finite fundamental group. The same conclusion holds for any $\lceil \frac{n}{2} \rceil$-convex hypersurface, provided that the mean curvature satisfies a sharp pinching condition. Both results follow from more general vanishing and estimation theorems for the Betti numbers of closed $q$-convex immersed hypersurfaces in $(n+1)$-dimensional Riemannian manifolds, under a lower bound on the average of the smallest $(n-p)$ eigenvalues of the curvature operator. |
| title | On $q$-convex hypersurfaces in Riemannian manifolds |
| topic | Differential Geometry 53C40, 53C42, 53C20, 53C21 |
| url | https://arxiv.org/abs/2604.20695 |