Experimental Mathematics

A family of singly periodic minimal surfaces invariant under a screw motion

Michael Callahan, David Hoffman, and Hermann Karcher

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We construct explicitly, using the generalized Weierstrass representation, a complete embedded minimal surface $M_{k,\theta}$ invariant under a rotation of order $k+1$ and a screw motion of angle $2\theta$ about the same axis, where $k \gt 0$ is any integer and $\theta$ is any angle with $|\theta| \lt \pi/(k+1)$. The existence of such surfaces was proved in [Callahan et al. 1990], but no practical procedure for constructing them was given there.

We also show that the same problem for $\theta=\pm\pi/(k+1)$ does not have a solution enjoying reflective symmetry; the question of the existence of a solution without such symmetry is left open.

Article information

Experiment. Math., Volume 2, Issue 3 (1993), 157-182.

First available in Project Euclid: 3 September 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]


Callahan, Michael; Hoffman, David; Karcher, Hermann. A family of singly periodic minimal surfaces invariant under a screw motion. Experiment. Math. 2 (1993), no. 3, 157--182. https://projecteuclid.org/euclid.em/1062620829

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