Rigidity and Non-existence Results of $λ$-Translating Solitons

A $λ$-translating soliton is a hypersurface in $\mathbb{R}^{n+1}$ satisfying $H=\langle \mathbf{T},ν\rangle+λ$; equivalently, it has constant weighted mean curvature with respect to the log-linear density $e^{\langle T,X\rangle}$, and is an eternal solution of the mean curvature flow with a constant