On regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces

Abstract

We prove that there are no regular algebraic hypersurfaces with non-zero constant mean curvature in the Euclidean space ${\mathbb{R}}^{n+1}$, $n\ge 2$, 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 ${\mathbb{R}}^{n+1}$, $n\ge 2$, 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.