Algebraic CMC hypersurfaces of order 3 in Euclidean spaces

Abstract

Understanding and finding of general algebraic constant mean curvature surfaces in the Euclidean spaces is a hard open problem. The basic examples are the standard spheres and the round cylinders, all defined by a polynomial of degree 2. In this paper, we prove that there are no algebraic hypersurfaces of degree 3 in mathbb {R}^n, nge 3, with nonzero constant mean curvature.