In this article, we study hypersurfaces \(\Sigma \subset {\mathbb {R}}^{n+1}\) with constant weighted mean curvature, also known as \(\lambda \)-hypersurfaces. Recently, Wei-Peng proved a rigidity theorem for \(\lambda \)-hypersurfaces that generalizes Le–Sesum’s classification theorem for self-shrinkers. More specifically, they showed that a complete \(\lambda \)-hypersurface with polynomial volume growth, bounded norm of the second fundamental form, and that satisfies \(|A|^2H(H-\lambda )\le H^2/2\) must either be a hyperplane or a generalized cylinder. We generalize this result by removing the bound condition on the norm of the second fundamental form. Moreover, we prove that under some conditions, if the reverse inequality holds, then the hypersurface must either be a hyperplane or a generalized cylinder. As an application of one of the results proved in this paper, we will obtain another version of the classification theorem obtained by the authors of this article, that is, we show that under some conditions, a complete \(\lambda \)-hypersurface with \(H\ge 0\) must either be a hyperplane or a generalized cylinder.

在本文中，我们研究具有恒定加权平均曲率的超曲面\（\ Sigma \ subset {\ mathbb {R}} ^ {n + 1} \），也称为\（\ lambda \）-超曲面。最近，Wei-Peng证明了\（\ lambda \） -超曲面的一个刚性定理，该定理推广了Le-Sesum的自收缩器分类定理。更具体地说，他们显示出具有多项式体积增长，第二基本形式的有界范数的完整\（\ lambda \）-超曲面，并且满足\（| A | ^ 2H（H- \ lambda）\ le H ^ 2 / 2 \）必须是超平面或广义圆柱。我们通过删除第二种基本形式的范数上的约束条件来概括此结果。此外，我们证明在某些条件下，如果反向不等式成立，则超曲面必须是超平面或广义圆柱体。作为本文证明的结果之一的应用，我们将获得本文作者获得的分类定理的另一个版本，即，我们证明在某些条件下，一个完整的\（\ lambda \）-超曲面与\（H \ GE 0 \）必须是一个超平面或广义圆柱体。