These surfaces can be viewed as one of several ways to symmetrically add handles to the Schwarz P surface. They have the symmetries of a rectangular box, and the divisor of the squared Gauss map looks as follows:
Here, we need a+c=b+d. Existence can be established from the nodal limits that occur for fixed τ when a converges to 0 or to b.
There is a 1-dimensional subfamily with b=d=1/4 and a+c=1/2 that has horizontal straight lines. Limits of this subfamily are shown below. We obtain singly periodic Scherk surfaces with speed 1 and 3, or doubly periodic Karcher-Scherk surfaces of genus 1.
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Minimal Surfaces 