Schwarz-Christoffel Maps

The Schwarz-Christoffel formulas gives an integral expression which maps the upper half plane conformally onto a polygonal shaped domain. As an example, consider the map to the right which maps the upper half plane conformally to a square: square formula
foliated circle foliated square
What does this have to do with minimal surfaces? Well, in the Weierstraß representation of many important minimal surfaces, the 1-forms G dh and 1/G dh very often are Schwarz-Christoffel integrands. One of the simplest examples is the Chen-Gacksttater surface, a minimal torus with one Enneper type end. Chen-Gackstatter
One quarter of the surface is conformally the upper half plane. If we map this upper half plane not into euclidean space but instead to a polygonal domain, using first G dh as a Schwarz-Christoffel integrand, we get the following zigzag shaped domain: Upper half plane
G dh integral G dh image
Using 1/G dh instead, we obtain a complemantary zigzag. Angles at corresponding vertices fit togther to 360 degrees.
1/G dh integral 1/G dh image
The important feature of these period domains is now that they can help understanding the period problem. For instance, higher genus Chen-Gackstatter surfaces correspond to zigzags with more vertices. The period problem for these surfaces can be reduced to the seemingly unrelated question whether there is a zigzag which divides the plane into two regions which are conformal by a map mapping corners to corners.