Derived from Scherk’s Examples

During my last semester as an undergraduate student at the Technical University in Berlin in 1984, Dirk Ferus mentioned in his Algebraic Topology class that there would be a geometry conference over the weekend, which he recommended attending. Stupid me, I didn’t go. I could have met my future advisor (Hermann Karcher), and I could have seen a future collaborator (David Hoffman) present the first images of the Costa surface.

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This conference is also mentioned in the introduction of another paper from my list of highly influential papers with new examples of minimal surfaces, namely Hermann Karcher’s 1988 Embedded Minimal Surfaces Derived from Scherk’s Examples.

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During the academic year 1984/85, I had attended two semesters of Karcher’s Differential Geometry. At the end of the second term he announced that while the third semester would only be for those specializing in geometry, we all should come for the first two weeks, because he intended to spend them with explaining the basics about minimal surfaces, which he had completely neglected. I was a little disappointed, because I was eager to learn about the darker arts – symmetric spaces, Einstein manifolds, Finiteness Theorems…

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Karcher didn’t just spend the first two weeks on minimal surfaces, but about half of the semester, developing and presenting what would become the paper mentioned above.

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The images here represent only a selection of the surfaces described in that paper: There are the saddle towers, the toroidal half plane layers, and the helicoidal saddle towers. Besides all these example Karcher develops a method to derive the complex analytic Enneper-Weierstraß data from geometric features of the surface, which, ultimately, has led to the enormous zoo of examples we are dealing with today.