We study hypersurfaces in the pseudo-Euclidean space, whose mean curvature vector satisfies the equation: Laplacian of the vector is parallel to the vector (with constant factor), and the second fundamental form has constant norm. We prove that every such hypersurface of diagonalizable shape operator with at most six distinct principal curvatures has constant mean curvature and constant scalar curvature, and if the above factor is zero then the hypersurface is minimal. We classify locally such non-minimal hypersurfaces with extremal value of the norm of the mean curvature vector. Further, we provide some examples of such hypersurfaces.

中文翻译：

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条件$$ \ triangle \ mathbf {H} = \ lambda \ mathbf {H} $$▵H =λH

我们研究伪欧几里德空间中的超曲面，其平均曲率向量满足以下公式：向量的拉普拉斯算子与向量（具有恒定因子）平行，第二基本形式具有恒定范数。我们证明每个具有最多六个不同主曲率的可对角线化形状算子的每个超曲面都具有恒定的平均曲率和恒定的标量曲率，并且如果上述因子为零，则超曲面极小。我们用平均曲率向量范数的极值对此类非最小超曲面进行局部分类。此外，我们提供了此类超曲面的一些示例。