Ends and Handles

Mathematics gets most exciting when new connections between different areas are discovered. In the theory of minimal surfaces, maybe the most fruitful discovery of this sort was Robert Osserman’s theorem from 1964 that a complete minimal surface of finite total curvature has the conformal type of a compact Riemann surface punctured at finitely many points, with the Weierstrass data extending meromorphically to the compact surface. This was a giant step from the much earlier discovery of the Weierstrass representation, that provided a first link between minimal surfaces and complex analysis.

Symfinite

However, at that time, the only known examples where of genus 0, i.e., punctured spheres, and of those only the catenoid (and the plane) were embedded. In fact, it is pretty easy to make examples like the one above: punctured spheres with many ends that will intersect.

This changed dramatically in 1982 when Celso José da Costa constructed a minimal torus with three ends that was proven to be embedded by David Hoffman and Bill Meeks in 1985. Examples with more ends and of higher genus followed rapidly, all nicely embedded. But there was a pattern: It looked like that if you wanted n ends, you needed to have genus at least n-2. This is the Hoffman-Meeks conjecture. For n=2 this follows from a theorem of Rick Schoen. But why can’t we have (say) a torus with four ends?

Fourends

All attempts to produce such an example have failed. In the image above, the ends will intersect, eventually. On the other hand, there is a fair amount of evidence that there are examples of genus g with precisely g+2 ends. Below is such a surface of genus 3 with 5 ends.

Dh11

That such examples should exist for any genus is supported by the Callahan-Hoffman-Meeks surface, a periodic version of the Costa surface. One just needs to chop it into pieces and put catenoidal ends at the top and bottom…

Chm

By recent work of Bill Meeks, Joaquin Perez, and Antonio Ros, the number of ends of a complete, embedded minimal surface of finite total curvature is bounded above by some constant (which is not explicit).

But even the case of tori remains very much open.

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