One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916701299277824 |
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| author | Liu, Jiayin Zhang, Shijin Zhou, Yuan |
| author_facet | Liu, Jiayin Zhang, Shijin Zhou, Yuan |
| contents | Let $(M,g)$ be an asymptotically conical Riemannian manifold having dimension $n\ge 2$, opening angle $α\in (0,π/2) \setminus \{\arcsin \frac{1}{2k+1}\}_{k \in \mathbb{N}}$ and positive asymptotic rate. Under the assumption that the exponential map is proper at each point, we give a solution to the one dimensional asymptotic Plateau problem on $M$. Precisely, for any pair of antipodal points in the ideal boundary $\partial_\infty M = \mathbb S^{n-1}$, we prove the existence of a geodesic line with asymptotic prescribed boundaries and the Morse index $\le n-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_15058 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds Liu, Jiayin Zhang, Shijin Zhou, Yuan Differential Geometry Let $(M,g)$ be an asymptotically conical Riemannian manifold having dimension $n\ge 2$, opening angle $α\in (0,π/2) \setminus \{\arcsin \frac{1}{2k+1}\}_{k \in \mathbb{N}}$ and positive asymptotic rate. Under the assumption that the exponential map is proper at each point, we give a solution to the one dimensional asymptotic Plateau problem on $M$. Precisely, for any pair of antipodal points in the ideal boundary $\partial_\infty M = \mathbb S^{n-1}$, we prove the existence of a geodesic line with asymptotic prescribed boundaries and the Morse index $\le n-1$. |
| title | One Dimensional Asymptotic Plateau Problem in $n$-Dimensional Asymptotically Conical Manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2504.15058 |