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 v {\mathrm {v}} in a closed half-space H v ~ = { x ∈ R n + 1 ∣ ⟨ x , v ~ ⟩ ≤ 0 } \mathcal {H}_{\widetilde {{\mathrm {v}}}}= \{ x \in \mathbb {R}^{n+1} \mid \langle x, \widetilde {{\mathrm {v}}}\rangle \leq 0 \} for ⟨ v , v ~ ⟩ > 0 \langle {\mathrm {v}}, \widetilde {{\mathrm {v}}} \rangle > 0 , whereas in a half-space H v ~ \mathcal {H}_{\widetilde {{\mathrm {v}}}} , ⟨ v , v ~ ⟩ ≤ 0 \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 v {\mathrm {v}} in a right circular cone C v , a = { x ∈ R n + 1 ∣ ⟨ x ‖ x ‖ , v ⟩ ≤ a > 1 } C_{ {{\mathrm {v}}}, a}=\{ x \in \mathbb {R}^{n+1} \mid \langle \frac {x}{\|x\|} , {{\mathrm {v}}} \rangle \leq a > 1 \} .