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

Abstract

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 forcing term. In this paper, we first prove that every complete properly immersed $λ$-translating soliton with $λ>0$ and $\inf_ΣH>λ$ 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 $λ$-translating solitons ($λ\geqslant 0$) with bounded gradient, and two rigidity theorems.