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

TL;DR

This paper proves that regular algebraic hypersurfaces with non-zero constant mean curvature in Euclidean spaces are either spheres or round cylinders, with restrictions on polynomial degree, addressing a question by Barbosa and do Carmo.

Contribution

It establishes non-existence results for certain algebraic hypersurfaces with constant mean curvature and classifies those of degree up to three.

Findings

No regular algebraic hypersurfaces of odd degree have non-zero constant mean curvature.

Spheres and round cylinders are the only such hypersurfaces of degree ≤ 3.

Results partially answer a question by Barbosa and do Carmo.

Abstract

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