Complete proper minimal surfaces in convex bodies of $\mathbb{R}^3$ (II): The behavior of the limit set
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2005
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| _version_ | 1866909539134078976 |
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| author | Martin, Francisco Morales, Santiago |
| author_facet | Martin, Francisco Morales, Santiago |
| contents | Let $D$ be a regular strictly convex bounded domain of $\mathbb{R}^3$, and consider a regular Jordan curve $Γ\subset \partial D$. Then, for each $ε>0$, we obtain the existence of a complete proper minimal immersion $ψ_ε:\mathbb{D} \to D$ satisfying that the Hausdorff distance $δ^H(ψ_ε(\partial \mathbb{D}), Γ) < ε,$ where $ψ_ε(\partial \mathbb{D})$ represents the limit set of the minimal disk $ψ_ε(\mathbb{D}).$
This result has some interesting consequences. Among other things, we can prove that any bounded regular domain $R$ in $\mathbb{R}^3$ admits a complete proper minimal immersion $ψ: \mathbb{D} \longrightarrow R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0505501 |
| institution | arXiv |
| publishDate | 2005 |
| record_format | arxiv |
| spellingShingle | Complete proper minimal surfaces in convex bodies of $\mathbb{R}^3$ (II): The behavior of the limit set Martin, Francisco Morales, Santiago Differential Geometry Primary 53A10, Secondary 49Q05, 49Q10, 53C42 Let $D$ be a regular strictly convex bounded domain of $\mathbb{R}^3$, and consider a regular Jordan curve $Γ\subset \partial D$. Then, for each $ε>0$, we obtain the existence of a complete proper minimal immersion $ψ_ε:\mathbb{D} \to D$ satisfying that the Hausdorff distance $δ^H(ψ_ε(\partial \mathbb{D}), Γ) < ε,$ where $ψ_ε(\partial \mathbb{D})$ represents the limit set of the minimal disk $ψ_ε(\mathbb{D}).$ This result has some interesting consequences. Among other things, we can prove that any bounded regular domain $R$ in $\mathbb{R}^3$ admits a complete proper minimal immersion $ψ: \mathbb{D} \longrightarrow R$. |
| title | Complete proper minimal surfaces in convex bodies of $\mathbb{R}^3$ (II): The behavior of the limit set |
| topic | Differential Geometry Primary 53A10, Secondary 49Q05, 49Q10, 53C42 |
| url | https://arxiv.org/abs/math/0505501 |