Our aim is to study invariant hypersurfaces immersed in the Euclidean space $\mathbb{R}^{n+1}$, whose mean curvature is given as a linear function in the unit sphere $\mathbb{S}^n$ depending on its Gauss map. These hypersurfaces are closely related with the theory of manifolds with density, since their weighted mean curvature in the sense of Gromov is constant. In this paper we obtain explicit parametrizations of constant curvature hypersurfaces, and also give a classification of rotationally invariant hypersurfaces.

中文翻译：

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具有线性规定平均曲率的不变超曲面

我们的目标是研究沉浸在欧几里得空间 $\mathbb{R}^{n+1}$ 中的不变超曲面，其平均曲率在单位球面 $\mathbb{S}^n$ 中作为线性函数给出，取决于其高斯图。这些超曲面与密度流形理论密切相关，因为它们的加权平均曲率在 Gromov 的意义上是恒定的。在本文中，我们获得了常曲率超曲面的显式参数化，并给出了旋转不变超曲面的分类。