Algebraic constant mean curvature surfaces in Euclidean space

TL;DR

This paper classifies algebraic constant mean curvature surfaces in three-dimensional Euclidean space of low algebraic order, showing they are limited to planes, spheres, and cylinders, using computational algebra techniques.

Contribution

It proves that algebraic CMC surfaces of order less than four are only planes, spheres, and cylinders, and reduces the problem of finding algebraic CMC hypersurfaces to solving polynomial systems.

Findings

Only planes, spheres, and cylinders are algebraic CMC surfaces of order less than four.

The problem of algebraic CMC hypersurfaces reduces to solving polynomial equations.

Computational methods like Groebner Bases are essential for these classifications.

Abstract

In this paper we prove that the only algebraic constant mean curvature (cmc) surfaces in R^3 of order less than four are the planes, the spheres and the cylinders. The method used heavily depends on the efficiency of algorithms to compute Groebner Bases and also on the memory capacity of the computer used to do the computations. We will also prove that the problem of finding algebraic constant mean curvature hypersurfaces in the Euclidean space completely reduces to the problem of solving a system of polynomial equations.