## Closing the Gaps

In 1982, Chi Cheng Chen and Fritz Gackstatter published a paper that described the surface below.

Like some of the classical examples of minimal surfaces, this surface is complete and has finite total curvature. A famous theorem of Osserman from 1964 asserts that any such surface can be defined on a punctured Riemann surface. In the classical examples, this had always been a sphere, but here we have a torus with one puncture.  There were some earlier examples, but this one, while not embedded, was surprisingly simple. From far away, it looks just like the Enneper surface.

How does one make such an example? One problem is illustrated above: While Osserman’s theorem also guarantees that the derivative of a conformal parametrization has a meromorphic extension to the compact surface, the integration of these so-called Weierstrass data might leave gaps.

To close the gap, we use the help of symmetries: Two vertical planes cut the surface into four congruent pieces, each represented by the upper half plane. The Weierstrass forms $\phi_1$ and  $\phi_2$ then turn out to be Schwarz-Christoffel integrands. The corresponding integrals map the upper half plane to (infinite) Euclidean polygons, shown above. The left extends to cover a bit more than a quarter plane, the right a bit less than a three quarter plane.

Incidentally, we can see the torus by fitting four copies of the right polygon together. We obtain the plane with a square missing. Identifying opposite edges of the missing square creates a torus with one puncture.

Now the condition that makes the gaps disappear is just that the two polygons fit together, which can be achieved by scaling. It’s really that simple. Similarly one can have more symmetric versions by just changing the angles in the polygons. Below is an example with sevenfold symmetry.