Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes
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arXiv
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| Natura: | Preprint |
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2019
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| _version_ | 1866911777862713344 |
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| author | Lou, Bendong Wang, Xiaoliu Yuan, Lixia |
| author_facet | Lou, Bendong Wang, Xiaoliu Yuan, Lixia |
| contents | We consider a curvature flow $V=H$ in the band domain $Ω:=[-1,1]\times \R$, where, for a graphic curve $Γ_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Γ_t$ contacts the two boundaries $\partial_\pm Ω$ of $Ω$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $Γ_t$ converges to a {\it grim reaper} contacting $\partial_\pm Ω$ with the same prescribed slopes. In this paper we consider the case where $Γ_t$ contacts $\partial_\pm Ω$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)\times \R)$ topology to the {\it grim reaper} with span $(-1,1)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1907_11535 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes Lou, Bendong Wang, Xiaoliu Yuan, Lixia Differential Geometry 35K93, 53C44, 35C07 We consider a curvature flow $V=H$ in the band domain $Ω:=[-1,1]\times \R$, where, for a graphic curve $Γ_t$, $V$ denotes its normal velocity and $H$ denotes its curvature. If $Γ_t$ contacts the two boundaries $\partial_\pm Ω$ of $Ω$ with constant slopes, in 1993, Altschular and Wu \cite{AW1} proved that $Γ_t$ converges to a {\it grim reaper} contacting $\partial_\pm Ω$ with the same prescribed slopes. In this paper we consider the case where $Γ_t$ contacts $\partial_\pm Ω$ with slopes equaling to $\pm 1$ times of its height. When the curve moves to infinity, the global gradient estimate is impossible due to the unbounded boundary slopes. We first consider a special symmetric curve and derive its uniform interior gradient estimates by using the zero number argument, and then use these estimates to present uniform interior gradient estimates for general non-symmetric curves, which lead to the convergence of the curve in $C^{2,1}_{loc} ((-1,1)\times \R)$ topology to the {\it grim reaper} with span $(-1,1)$. |
| title | Convergence to the Grim Reaper for a Curvature Flow with Unbounded Boundary Slopes |
| topic | Differential Geometry 35K93, 53C44, 35C07 |
| url | https://arxiv.org/abs/1907.11535 |