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| Format: | Preprint |
| Izdano: |
2024
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| Teme: | |
| Online dostop: | https://arxiv.org/abs/2401.04721 |
| Oznake: |
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| _version_ | 1866909067133321216 |
|---|---|
| author | Barbieri, Aires Eduardo Menani |
| author_facet | Barbieri, Aires Eduardo Menani |
| contents | Given a function $\mathcal{H} \in C^1(\mathbb{S}^2)$, an $\mathcal{H}$-surface $Σ$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_Σ$ satisfies $H_Σ= \mathcal{H} \circ η$, where $η$ is the Gauss map of $Σ$. The purpose of this paper is to use a phase space analysis to give some classification results for helicoidal $\mathcal{H}$-surfaces, when $\mathcal{H}$ is rotationally symmetric, that is, $\mathcal{H} \circ η= \mathfrak{h} \circ ν$, for some $\mathfrak{h} \in C^1([-1,1])$, where $ν$ is the angle function of the surface. We prove a classification theorem for the case where $\mathfrak{h}(t)$ is even and increasing for $t \in [0,1]$. Finally, we provide examples of helicoidal $\mathcal{H}$-surfaces in cases where $\mathfrak{h}$ vanishes at some point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_04721 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Helicoidal surfaces of prescribed mean curvature in $\mathbb{R}^3$ Barbieri, Aires Eduardo Menani Differential Geometry 53A10, 53C42, 34C05, 34C40 Given a function $\mathcal{H} \in C^1(\mathbb{S}^2)$, an $\mathcal{H}$-surface $Σ$ is a surface in the Euclidean space $\mathbb{R}^3$ whose mean curvature $H_Σ$ satisfies $H_Σ= \mathcal{H} \circ η$, where $η$ is the Gauss map of $Σ$. The purpose of this paper is to use a phase space analysis to give some classification results for helicoidal $\mathcal{H}$-surfaces, when $\mathcal{H}$ is rotationally symmetric, that is, $\mathcal{H} \circ η= \mathfrak{h} \circ ν$, for some $\mathfrak{h} \in C^1([-1,1])$, where $ν$ is the angle function of the surface. We prove a classification theorem for the case where $\mathfrak{h}(t)$ is even and increasing for $t \in [0,1]$. Finally, we provide examples of helicoidal $\mathcal{H}$-surfaces in cases where $\mathfrak{h}$ vanishes at some point. |
| title | Helicoidal surfaces of prescribed mean curvature in $\mathbb{R}^3$ |
| topic | Differential Geometry 53A10, 53C42, 34C05, 34C40 |
| url | https://arxiv.org/abs/2401.04721 |