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Symmetry of embedded genus 1 helicoids. (English) Zbl 1228.53008

Authors’ abstract: We use the Lopez-Ros deformation to show that any embedded genus 1 helicoid (or “genus-one helicoid”) must be symmetric with respect to rotation by \(180^\circ\) around a normal line. This partially answers a conjecture of Bobenko. We also show that this symmetry holds for an embedded genus \(k\) helicoid \(\Sigma\), provided that the underlying conformal structure of \(\Sigma\) is hyperelliptic.

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
14H40 Jacobians, Prym varieties
37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.)
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References:

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