# 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:
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.
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:
Using 1/G dh instead, we obtain a complemantary zigzag. Angles at corresponding vertices fit togther to 360 degrees.
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.