We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space $\mathbb {R}^{n+1},\,\;n\geq 2,$ defined by polynomials of odd degree. Also we prove that the hyperspheres and the round cylinders are the only regular algebraic hypersurfaces with non-zero constant mean curvature in $\mathbb {R}^{n+1}, n\geq 2,$ defined by polynomials of degree less than or equal to three. These results give partial answers to a question raised by Barbosa and do Carmo.

中文翻译：

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在欧几里得空间中具有非零常数平均曲率的规则代数超曲面

我们证明在奇数次多项式定义的欧几里得空间$\mathbb {R}^{n+1},\,\;n\geq 2,$中不存在具有非零常数平均曲率的正则代数超曲面。我们还证明了超球面和圆柱体是唯一在$\mathbb {R}^{n+1}, n\geq 2,$中具有非零常数平均曲率的正则代数超曲面，由次数小于的多项式定义或等于三。这些结果部分回答了 Barbosa 和 do Carmo 提出的问题。