Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910041818267648 |
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| author | Del Prete, Andrea |
| author_facet | Del Prete, Andrea |
| contents | Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\E$. Then, there exists a Killing Submersion from $\E$ onto a connected and orientable surface $M$ whose fibers are the integral curves of the Killing vector field. In this setting, assuming that $M$ is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of $M$, prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2307_09240 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field Del Prete, Andrea Differential Geometry 53A10, 53C30, 53C42 Let $\mathbb{E}$ be a connected and orientable Riemannian 3-manifold with a non-singular Killing vector field whose associated one-parameter group of the isometries of $\mathbb{E}$ acts freely and properly on $\E$. Then, there exists a Killing Submersion from $\E$ onto a connected and orientable surface $M$ whose fibers are the integral curves of the Killing vector field. In this setting, assuming that $M$ is non-compact and the fibers have infinite length, we solve the Dirichlet problem for minimal Killing graphs over certain unbounded domains of $M$, prescribing piecewise continuous boundary values. We obtain general Collin-Krust type estimates. In the particular case of the Heisenberg group, we prove a uniqueness result for minimal Killing graphs with bounded boundary values over a strip. We also prove that isolated singularities of Killing graphs with prescribed mean curvature are removable. |
| title | Minimal graphs over non-compact domains in 3-manifolds fibered by a Killing vector field |
| topic | Differential Geometry 53A10, 53C30, 53C42 |
| url | https://arxiv.org/abs/2307.09240 |