Mean surfaces in Half-Pipe space and infinitesimal Teichmüller theory
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914913484537856 |
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| author | Diaf, Farid |
| author_facet | Diaf, Farid |
| contents | We study a correspondence between smooth spacelike surfaces in Half-Pipe space $\mathbb{HP}^3$ and divergence-free vector fields on the hyperbolic plane $\mathbb{H}^2$. We show that a particular case involves harmonic Lagrangian vector fields on $\mathbb{H}^2$, which are related to mean surfaces in $\mathbb{HP}^3$. Consequently, we prove that the infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field that corresponds to a mean surface in $\mathbb{HP}^3$ with prescribed boundary data at infinity.
We establish both existence and, under certain assumptions, uniqueness results for harmonic Lagrangian extension of a vector field on the circle. Finally, we characterize the Zygmund and little Zygmund conditions and provide quantitative bounds in terms of the Half-Pipe width. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_08268 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Mean surfaces in Half-Pipe space and infinitesimal Teichmüller theory Diaf, Farid Differential Geometry Geometric Topology We study a correspondence between smooth spacelike surfaces in Half-Pipe space $\mathbb{HP}^3$ and divergence-free vector fields on the hyperbolic plane $\mathbb{H}^2$. We show that a particular case involves harmonic Lagrangian vector fields on $\mathbb{H}^2$, which are related to mean surfaces in $\mathbb{HP}^3$. Consequently, we prove that the infinitesimal Douady-Earle extension is a harmonic Lagrangian vector field that corresponds to a mean surface in $\mathbb{HP}^3$ with prescribed boundary data at infinity. We establish both existence and, under certain assumptions, uniqueness results for harmonic Lagrangian extension of a vector field on the circle. Finally, we characterize the Zygmund and little Zygmund conditions and provide quantitative bounds in terms of the Half-Pipe width. |
| title | Mean surfaces in Half-Pipe space and infinitesimal Teichmüller theory |
| topic | Differential Geometry Geometric Topology |
| url | https://arxiv.org/abs/2408.08268 |