We establish nonexistence results for complete spacelike translating solitons immersed in a Lorentzian product space ${ℝ}_1\times {\mathbb{P}}^n$, under suitable curvature constraints on the curvatures of the Riemannian base ${\mathbb{P}}^n$. In particular, we obtain Calabi–Bernstein type results for entire translating graphs constructed over ${\mathbb{P}}^n$. For this, we prove a version of the Omori–Yau’s maximum principle for complete spacelike translating solitons. Besides, we also use other two analytical tools related to an appropriate drift Laplacian: a parabolicity criterion and certain integrability properties. Furthermore, under the assumption that the base ${\mathbb{P}}^n$ is non-positively curved, we close our paper constructing new examples of rotationally symmetric spacelike translating solitons embedded into ${ℝ}_1\times {\mathbb{P}}^n$.

我们建立了沉浸在洛伦兹乘积空间中的完全类空间平移孤子的不存在结果${ℝ}_1\times {\mathbb{P}}^n$, 在对黎曼底曲率的适当曲率约束下${\mathbb{P}}^n$. 特别是，我们获得了整个平移图的 Calabi-Bernstein 类型结果${\mathbb{P}}^n$. 为此，我们证明了完全类空间平移孤子的 Omori-Yau 最大原理的一个版本。此外，我们还使用了与适当的漂移拉普拉斯算子相关的其他两个分析工具：抛物线准则和某些可积性属性。此外，假设基${\mathbb{P}}^n$是非正弯曲的，我们结束了我们的论文，构建了旋转对称的类空间平移孤子嵌入到${ℝ}_1\times {\mathbb{P}}^n$.