Critical points of the solution to the $H_R=H_L$ surface equation
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866929235331907584 |
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| author | Albujer, Alma L. Caballero, Magdalena |
| author_facet | Albujer, Alma L. Caballero, Magdalena |
| contents | Spacelike surfaces with the same mean curvature in $\mathbb{R}^3$ and $\mathbb{L}^3$ are locally described as the graph of the solutions to the $H_R=H_L$ surface equation, which is an elliptic partial differential equation except at the points at which the gradient vanishes, because the equation degenerates. In this paper we study precisely the critical points of the solutions to such equation. Specifically, we give a necessary geometrical condition for a point to be critical, we obtain a new uniqueness result for the Dirichlet problem related to the $H_R=H_L$ surface equation and we get a Heinz-type bound for the inradius of the domain of any solution to such equation, improving a previous result by the authors. Finally, we also get a bound for the inradius of the domain of any function of class $\mathcal{C}^2$ in terms of the curvature of its level curves. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_04770 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Critical points of the solution to the $H_R=H_L$ surface equation Albujer, Alma L. Caballero, Magdalena Differential Geometry 53C50, 53C42, 35J93 Spacelike surfaces with the same mean curvature in $\mathbb{R}^3$ and $\mathbb{L}^3$ are locally described as the graph of the solutions to the $H_R=H_L$ surface equation, which is an elliptic partial differential equation except at the points at which the gradient vanishes, because the equation degenerates. In this paper we study precisely the critical points of the solutions to such equation. Specifically, we give a necessary geometrical condition for a point to be critical, we obtain a new uniqueness result for the Dirichlet problem related to the $H_R=H_L$ surface equation and we get a Heinz-type bound for the inradius of the domain of any solution to such equation, improving a previous result by the authors. Finally, we also get a bound for the inradius of the domain of any function of class $\mathcal{C}^2$ in terms of the curvature of its level curves. |
| title | Critical points of the solution to the $H_R=H_L$ surface equation |
| topic | Differential Geometry 53C50, 53C42, 35J93 |
| url | https://arxiv.org/abs/2207.04770 |