发表机构
Jiangsu University; Wuhan University（江苏大学; 武汉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究$\lambda$-平移孤子，证明当$\lambda>0$且$\inf H>\lambda$时其体积至少指数增长，并给出图形孤子的不存在性与刚性定理。
AI 中文摘要
$\lambda$-平移孤子是$\mathbb{R}^{n+1}$中的超曲面，满足$H=\langle \mathbf{T},\nu\rangle+\lambda$；等价地，它关于对数线性密度$e^{\langle T,X\rangle}$具有常加权平均曲率，并且是带常强迫项的平均曲率流的永恒解。本文首先证明，每个具有$\lambda>0$且$\inf_{\Sigma}H>\lambda$的完备适当浸入$\lambda$-平移孤子至少具有指数体积增长，这与普通平移孤子的线性增长形成对比。然后，我们证明了具有有界梯度的图形$\lambda$-平移孤子（$\lambda\geqslant 0$）的尖锐不存在性结果，以及两个刚性定理。
英文摘要
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.
