Dihedral rigidity for submanifolds of warped product manifolds
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866917029565431808 |
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| author | Wang, Jinmin Xie, Zhizhang |
| author_facet | Wang, Jinmin Xie, Zhizhang |
| contents | In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_13492 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Dihedral rigidity for submanifolds of warped product manifolds Wang, Jinmin Xie, Zhizhang Differential Geometry K-Theory and Homology In this paper, we prove a dihedral extremality and rigidity theorem for a large class of codimension zero submanifolds with polyhedral boundary in warped product manifolds. We remark that the spaces considered in this paper are not necessarily warped product manifolds themselves. In particular, the results of this paper are applicable to submanifolds (of warped product manifolds) with faces that are neither orthogonal nor parallel to the radial direction of the warped product metric. Generally speaking, the dihedral rigidity results require the leaf of the underlying warped space to have positive Ricci curvature and the warping function to be strictly log-concave. Nevertheless, we prove a dihedral rigidity theorem for a large class of hyperbolic polyhedra, where the leaf of the underlying warped product space is flat and the warping function is not strictly log-concave. |
| title | Dihedral rigidity for submanifolds of warped product manifolds |
| topic | Differential Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2303.13492 |