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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.09784 |
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| _version_ | 1866909635384967168 |
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| author | Cucinotta, Alessandro Mondino, Andrea |
| author_facet | Cucinotta, Alessandro Mondino, Andrea |
| contents | We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, \emph{Duke Math.\;J}.\;(1992)], but instead of asking that one boundary component is compact and mean-convex, we require that it is parabolic and convex. The parabolicity assumption cannot be dropped as, otherwise, the catenoid in ambient dimension four would give a counterexample. The convexity assumption, instead, can be relaxed to mean-convexity, if one requires an additional control on the volume growth at infinity. Among the applications, we establish a half-space theorem for mean-convex sets in product manifolds. Additionally, we prove splitting results for:
3-manifolds with non-negative Ricci curvature and disconnected mean-convex boundary,
4-manifolds with weakly bounded geometry, non-negative 2-Ricci curvature, scalar curvature bounded away from zero, and disconnected mean-convex boundary. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09784 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A splitting theorem for manifolds with a convex boundary component and applications Cucinotta, Alessandro Mondino, Andrea Differential Geometry We prove a warped product splitting theorem for manifolds with Ricci curvature bounded from below in the spirit of [Croke-Kleiner, \emph{Duke Math.\;J}.\;(1992)], but instead of asking that one boundary component is compact and mean-convex, we require that it is parabolic and convex. The parabolicity assumption cannot be dropped as, otherwise, the catenoid in ambient dimension four would give a counterexample. The convexity assumption, instead, can be relaxed to mean-convexity, if one requires an additional control on the volume growth at infinity. Among the applications, we establish a half-space theorem for mean-convex sets in product manifolds. Additionally, we prove splitting results for: 3-manifolds with non-negative Ricci curvature and disconnected mean-convex boundary, 4-manifolds with weakly bounded geometry, non-negative 2-Ricci curvature, scalar curvature bounded away from zero, and disconnected mean-convex boundary. |
| title | A splitting theorem for manifolds with a convex boundary component and applications |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2406.09784 |