In Euclidean space, we investigate surfaces whose mean curvature H satisfies the equation \(H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda \), where N is the Gauss map, \({\mathbf {x}}\) is the position vector, and \(\alpha \) and \(\lambda \) are two constants. There surfaces generalize self-shrinkers and self-expanders of the mean curvature flow. We classify the ruled surfaces and the translation surfaces, proving that they are cylindrical surfaces.

中文翻译：

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平均曲率流的广义自相似解的规则曲面

在欧几里得空间中，我们研究平均曲率H满足方程\(H=\alpha \langle N,{\mathbf {x}}\rangle +\lambda \) 的曲面，其中N是高斯图，\({\ mathbf {x}}\)是位置向量，\(\alpha \)和\(\lambda \)是两个常量。这些表面概括了平均曲率流的自收缩和自膨胀。我们对直纹面和平移面进行分类，证明它们是圆柱面。