Rigidity theorems of $\lambda$-translating solitons in Euclidean and Lorentz-Minkowski spaces

Abstract

In this paper, we explore certain properties of $\lambda$-translators, which can be regarded as a natural generalization of translators. We first obtain a rigidity result for a complete $\lambda$-translator that is either a hyperplane or ${\mathbb {S}}^{n-1}\times {\mathbb {R}}$, depending on the squared norm of the second fundamental form and the mean curvature. We then obtain another rigidity result in that a $\lambda$-translator is a hyperplane perpendicular to the density vector V under the conditions of $H(H-\lambda )\ge 0$ and $\int _M\vert V^{\top }\vert e^{\langle V,X\rangle }\textrm{d}\mu <\infty$. Furthermore, when a $\lambda$-translator is constant mean curvature (CMC for short), we show that it is either a hyperplane or a product of a CMC hypersurface in ${\mathbb {R}}^n$ and ${\mathbb {R}}$ in the direction of V. We finally prove that a graphical $\lambda$-translator with a bounded gradient and constant norm of the second fundamental form is a hyperplane. These results are all in Euclidean space, and, in addition, the corresponding conclusions can be obtained in the Lorentz-Minkowski space under analogous conditions.