Hermann Karcher’s Scherk Towers are translation invariant minimal surface with genus 0 in the quotient. They were first described in 1988.

In their most symmetric form, these surfaces have the Enneper-Weierstraß representation

G(z) = z^{k-1} \qquad\text{and}\qquad dh =\frac1{z^k+z^{-k}} \frac{dz}z \ .

Their simplicity made Karcher suspect that they might have been known already in the 19th century. As with the 4-ended surfaces of Scherk, there is an almost as simple less symmetric (they are lacking the horizontal straight lines) deformation where the angles between the ends vary.

The two extreme cases above with pairwise parallel ends are in fact the same surface, rotated, translated, and with reversed orientation.

For extreme parameters, the ends will intersect, and the surfaces will converge to the Jorge-Meeks k-Noids.

More variations are possible: Using the Jenkins-Serrin construction, one can construct an n-3 dimensional family of n-ended surfaces. Joaquin Perez and  Martin Traizet have proven in 2007 that these are the only singly periodic minimal surfaces of genus 0 with annular ends.

With genus one (and higher), there are many more possibilities, which have not been fully explored.

Resources

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

J. Perez, M. Traizet: The classification of singly periodic minimal surfaces with genus zero and Scherk type ends. Trans. Amer. Math. Soc. 359, 965-990 (2007)

Mathematica Notebook fully symmetric case and less symmetric case

Wolfram CDF files fully symmetric case and less symmetric case

PoVRay sources fully symmetric case and less symmetric case

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