
SchwarzChristoffel Maps 

The SchwarzChristoffel 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 1forms G dh and 1/G dh very often are SchwarzChristoffel integrands.
One of the simplest examples is the ChenGacksttater 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 SchwarzChristoffel 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 ChenGackstatter 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. 