Half-space type theorem for translating solitons of the mean curvature flow in Euclidean space

Abstract:

In this paper, we determine which half-space contains a complete translating soliton of the mean curvature flow and it is related to the well-known half-space theorem for minimal surfaces. We prove that a complete translating soliton does not exist with respect to the velocity ${\mathrm {v}}$ in a closed half-space $\mathcal {H}_{\widetilde {{\mathrm {v}}}}= \{ x \in \mathbb {R}^{n+1} \mid \langle x, \widetilde {{\mathrm {v}}}\rangle \leq 0 \}$ for $\langle {\mathrm {v}}, \widetilde {{\mathrm {v}}} \rangle > 0$, whereas in a half-space $\mathcal {H}_{\widetilde {{\mathrm {v}}}}$, $\langle {\mathrm {v}}, \widetilde {{\mathrm {v}}} \rangle \leq 0$, a complete translating soliton can be found. In addition, we extend this property to cones: there are no complete translating solitons with respect to ${\mathrm {v}}$ in a right circular cone $C_{ {{\mathrm {v}}}, a}=\{ x \in \mathbb {R}^{n+1} \mid \langle \frac {x}{\|x\|} , {{\mathrm {v}}} \rangle \leq a < 1 \}$.