A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866914878188421120 |
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| author | Del Prete, Andrea Lee, Hojoo Manzano, José M. |
| author_facet | Del Prete, Andrea Lee, Hojoo Manzano, José M. |
| contents | We develop a conformal duality for spacelike graphs in Riemannian and Lorentzian three-manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is timelike in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz-Minkowski space $\mathbb{L}^3$ with prescribed mean curvature a bounded function $H\in C^\infty(\mathbb{R}^2)$ with bounded gradient. On the other hand, we obtain conditions for existence and non existence of entire graphs which are related to a notion of critical mean curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2306_00562 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions Del Prete, Andrea Lee, Hojoo Manzano, José M. Differential Geometry Primary 53A10, Secondary 53C50 We develop a conformal duality for spacelike graphs in Riemannian and Lorentzian three-manifolds that admit a Riemannian submersion over a Riemannian surface whose fibers are the integral curves of a Killing vector field, which is timelike in the Lorentzian case. The duality swaps mean curvature and bundle curvature and sends the length of the Killing vector field to its reciprocal while keeping invariant the base surface. We obtain two consequences of this result. On the one hand, we find entire graphs in Lorentz-Minkowski space $\mathbb{L}^3$ with prescribed mean curvature a bounded function $H\in C^\infty(\mathbb{R}^2)$ with bounded gradient. On the other hand, we obtain conditions for existence and non existence of entire graphs which are related to a notion of critical mean curvature. |
| title | A duality for prescribed mean curvature graphs in Riemannian and Lorentzian Killing submersions |
| topic | Differential Geometry Primary 53A10, Secondary 53C50 |
| url | https://arxiv.org/abs/2306.00562 |