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| Hovedforfatter: | |
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| Format: | Preprint |
| Udgivet: |
2024
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| Fag: | |
| Online adgang: | https://arxiv.org/abs/2401.04721 |
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Indholdsfortegnelse:
- 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.