Rigidity and Non-existence Results of $\lambda$-Translating Solitons

A $\lambda$-translating soliton is a hypersurface in ${\mathbb{R}}^{n+1}$ satisfying $H=⟨\mathbf{T},\nu ⟩+\lambda$; equivalently, it has constant weighted mean curvature with respect to the log-linear density $e^{⟨T,X⟩}$, and is an eternal solution of the mean curvature flow with a constant forcing term. In this paper, we first prove that every complete properly immersed $\lambda$-translating soliton with $\lambda >0$ and ${\inf }_{\mathrm{\Sigma }}H>\lambda$ has at least exponential volume growth, in contrast with the linear growth of ordinary translating solitons. We then prove sharp non-existence results for graphic $\lambda$-translating solitons ($\lambda ⩾0$) with bounded gradient, and two rigidity theorems.