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We develop a conservation law for constant mean curvature (CMC) surfacesintroduced by Korevaar, Kusner and Solomon, and provide a converse, so as tocharacterize CMC surfaces by a conservation law. We work with `twizzler'construction, which applies a screw-motion to some base curve. We show that,excluding cylinders, CMC helicoidal surfaces can be completely determined by afirst-order ODE of the base curve. Further, we demonstrate that in R^3 thiscondition is equivalent to the treadmillsled characterization of helicoidal CMCsurfaces given by O. Perdomo.

A conservation approach to helicoidal surfaces of constant mean curvature in R^3, S^3 and H^3