Duke Mathematical Journal

Symmetry of embedded genus 1 helicoids

Jacob Bernstein and Christine Breiner

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In this article, 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.

Article information

Duke Math. J. Volume 159, Number 1 (2011), 83-97.

First available in Project Euclid: 11 July 2011

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14H40: Jacobians, Prym varieties [See also 32G20]
Secondary: 37K10: Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies (KdV, KP, Toda, etc.)


Bernstein, Jacob; Breiner, Christine. Symmetry of embedded genus 1 helicoids. Duke Math. J. 159 (2011), no. 1, 83--97. doi:10.1215/00127094-1384791. https://projecteuclid.org/euclid.dmj/1310416363.

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