A conservation approach to helicoidal surfaces of constant mean\n curvature in R³, S³ and H³

2011/10/05 by Nick Edelen, Edelen, Nick
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds

paper · pdf · doi:10.48550/arxiv.1110.1068

openalex publication_date 2011/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We develop a conservation law for constant mean curvature (CMC) surfaces\nintroduced by Korevaar, Kusner and Solomon, and provide a converse, so as to\ncharacterize CMC surfaces by a conservation law. We work with `twizzler'\nconstruction, which applies a screw-motion to some base curve. We show that,\nexcluding cylinders, CMC helicoidal surfaces can be completely determined by a\nfirst-order ODE of the base curve. Further, we demonstrate that in R³ this\ncondition is equivalent to the treadmillsled characterization of helicoidal CMC\nsurfaces given by O. Perdomo.\n

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