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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 thecylinders. The method used heavily depends on the efficiency of algorithms tocompute Groebner Bases and also on the memory capacity of the computer used todo the computations. We will also prove that the problem of finding algebraicconstant mean curvature hypersurfaces in the Euclidean space completely reducesto the problem of solving a system of polynomial equations.

Algebraic constant mean curvature surfaces in Euclidean space