Rob Kusner discovered an interesting class of immersed minimal spheres with an even number 2n of planar ends. If n is odd, then the immersion commutes with the antipodal map of S² so that we get an immersion of the projective plane with n planar ends.

Above are the cases n=2 (showing one half of the surface)  and 5, and below two views for n=3 with different cutoffs for the planar ends.

Robert Bryant suggested to apply an inversion in space to the 3-ended surface to obtain a conformal immersion of the projective plane with a single triple point singularity (corresponding to the three planar ends) just like Boys’ surface. This specific parametrization was used to design the metal sculpture in front of the research institute in Oberwolfach:


Below is an animation that varies the cutoff and shows how a triply twisted cylinder can be “closed”.



Mathematica Notebook and CDF

PoVRay Sources

Kusner, R. Conformal geometry and complete minimal surfaces. Bull. Amer. Math. Soc. 17 (291–295), 1987.

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