Rigidity theorems of λ -translating solitons in Euclidean and Lorentz-Minkowski spaces

In this paper, we explore certain properties of λ -translators, which can be regarded as a natural generalization of translators. We first obtain a rigidity result for a complete λ -translator that is either a hyperplane or 𝕊^n-1×ℝ , depending on the squared norm of the second fundamental form and the mean curvature. We then obtain another rigidity result in that a λ -translator is a hyperplane perpendicular to the density vector V under the conditions of H(H-λ )≥ 0 and ∫ _M| V^⊤| e^⟨ V,X⟩dμ <∞ . Furthermore, when a λ -translator is constant mean curvature (CMC for short), we show that it is either a hyperplane or a product of a CMC hypersurface in ℝ^n and ℝ in the direction of V . We finally prove that a graphical λ -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.