Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915855425601536 |
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| author | Sui, Zhenan |
| author_facet | Sui, Zhenan |
| contents | We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - κ_i) = \big( (n - 1) σ\big)^n$ for $n = 4$ and has prescribed asymptotic boundary $Γ$ at the infinity of hyperbolic space of dimension 5, where $σ\in (0, 1)$ is a constant and $Γ$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f (κ) = \frac{1}{n - 1} \Big( \prod\limits_{i = 1}^n (H - κ_i) \Big)^{\frac{1}{n}}$ during uniform global curvature estimate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00565 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$ Sui, Zhenan Differential Geometry Analysis of PDEs We prove the existence of a smooth complete $3$-convex hypersurface which satisfies prescribed curvature equation $\prod\limits_{i = 1}^n (H - κ_i) = \big( (n - 1) σ\big)^n$ for $n = 4$ and has prescribed asymptotic boundary $Γ$ at the infinity of hyperbolic space of dimension 5, where $σ\in (0, 1)$ is a constant and $Γ$ is assumed to have nonnegative mean curvature. We introduce Lagrange multiplier method to compute the extreme value of the concavity of $f (κ) = \frac{1}{n - 1} \Big( \prod\limits_{i = 1}^n (H - κ_i) \Big)^{\frac{1}{n}}$ during uniform global curvature estimate. |
| title | Asymptotic Plateau problem for $3$-convex hypersurface in $\mathbb{H}^5$ |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2506.00565 |