On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866910091824857088 |
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| author | Cucinotta, Alessandro Mondino, Andrea |
| author_facet | Cucinotta, Alessandro Mondino, Andrea |
| contents | We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor.
If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$, then we show that $M$ is contained in a neighbourhood of controlled width of an isometrically embedded $1$-dimensional sub-manifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of $M$ is bounded above by $1$, and there is precise information on elements of infinite order in $π_1(M)$.
If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$, then we show that $M$ has at most $(k-1)$-dimensional behavior at large scales.
If $k=n={\rm dim}(M)$, so that the integral lower bound is on the scalar curvature, assuming in addition that the $(n-2)$-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to $n-2$.
From the above results we deduce topological restrictions, such as upper bounds on the first Betti number. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_06865 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature Cucinotta, Alessandro Mondino, Andrea Differential Geometry Metric Geometry We consider manifolds with almost non-negative Ricci curvature and strictly positive integral lower bounds on the sum of the lowest $k$ eigenvalues of the Ricci tensor. If $(M^n,g)$ is a Riemannian manifold satisfying such curvature bounds for $k=2$, then we show that $M$ is contained in a neighbourhood of controlled width of an isometrically embedded $1$-dimensional sub-manifold. From this, we deduce several metric and topological consequences: $M$ has at most linear volume growth and at most two ends, it has bounded 1-Urysohn width, the first Betti number of $M$ is bounded above by $1$, and there is precise information on elements of infinite order in $π_1(M)$. If $(M^n,g)$ is a Riemannian manifold satisfying such bounds for $k\geq 2$, then we show that $M$ has at most $(k-1)$-dimensional behavior at large scales. If $k=n={\rm dim}(M)$, so that the integral lower bound is on the scalar curvature, assuming in addition that the $(n-2)$-Ricci curvature is non-negative, we prove that the dimension drop at large scales improves to $n-2$. From the above results we deduce topological restrictions, such as upper bounds on the first Betti number. |
| title | On manifolds with almost non-negative Ricci curvature and integrally-positive $k^{th}$-scalar curvature |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2504.06865 |