Constant mean curvature graphs with prescribed asymptotic values in $\mathbb{E}(-1,τ)$
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866909878815031296 |
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| author | Del Prete, Andrea |
| author_facet | Del Prete, Andrea |
| contents | In the homogeneous manifold $\mathbb{E}(-1,τ),$ for $-\tfrac{1}{2}<H<\tfrac{1}{2},$ {we define a new product compactification in which the slices $\left\{t=c\right\}_{c\in\R}$ are rotational $H$-surfaces. This product compatification is the natural setting where it makes sense to study the asymptotic Dirichlet Problem for the constant mean curvature equation. Indeed, for every rectifiable curve $Γ$ projecting bijectively onto $\partial\H^2$ we prove the existence of a unique entire $H$-graph that is asymptotic to $Γ$.} We also find necessary and sufficient conditions for the existence of $H$-graphs over unbounded domains having prescribed, possibly infinite boundary data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07274 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Constant mean curvature graphs with prescribed asymptotic values in $\mathbb{E}(-1,τ)$ Del Prete, Andrea Differential Geometry In the homogeneous manifold $\mathbb{E}(-1,τ),$ for $-\tfrac{1}{2}<H<\tfrac{1}{2},$ {we define a new product compactification in which the slices $\left\{t=c\right\}_{c\in\R}$ are rotational $H$-surfaces. This product compatification is the natural setting where it makes sense to study the asymptotic Dirichlet Problem for the constant mean curvature equation. Indeed, for every rectifiable curve $Γ$ projecting bijectively onto $\partial\H^2$ we prove the existence of a unique entire $H$-graph that is asymptotic to $Γ$.} We also find necessary and sufficient conditions for the existence of $H$-graphs over unbounded domains having prescribed, possibly infinite boundary data. |
| title | Constant mean curvature graphs with prescribed asymptotic values in $\mathbb{E}(-1,τ)$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2402.07274 |