Hermann Karcher described in 1988 a surprising deformation of the fully symmetric Scherk surfaces into screw motion invariant surfaces, the first new examples since the helicoid.

This can be done for any number 2k (k≥2) number of ends. The Weierstraß representation becomes multivalued on the 2k-punctured quotient spheres:

G(z) = z^{1-\frac{1}{k}} \left(\frac{R^2-z}{R^2 z+1}\right)^a \quad\text{and}\quad dh=\frac{i\, dz}{2 z \left(-R^2+\frac{1}{R^2}+z-\frac{1}{z}\right)}

Here < a < \frac1k determines the screw motion angle, and R needs to be determined numerically to close a single period.

When the screw motion parameter a reaches its upper limit 1/k, the surfaces converge to a horizontal foliation of R³ by planes with singular vertical lines at the 2k-th roots of unity.

It is still an open question whether these screw-motion invariant Karcher-Scherk surfaces together with the helicoid are the only embedded, screw-motion invariant minimal surfaces with genus 0 in the quotient.

Resources

H. Karcher: Embedded Minimal Surfaces Derived from Scherk’s Examples, manuscripta math. 62, 83-114 (1988).

Mathematica Notebook

Wolfram CDF

PoVRay Sources

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