We present a new family of embedded doubly periodic minimal surfaces, of which the symmetry group does not coincide with any other example known before.
In this paper we show how to bypass the usual difficulties in the analysis of elliptic integrals that arise when solving period problems for minimal surfaces. The method consists of replacing period problems with ordinary Sturm-Liouville problems inv
olving the support function. We give a practical application by proving existence of the sheared Scherk-Karcher family of surfaces numerically described by Wei. Moreover, we show that this family is continuous, and both of its limit-members are the singly periodic genus-one helicoid.
We get a continuous one-parameter new family of embedded minimal surfaces, of which the period problems are two-dimensional. Moreover, one proves that it has Scherk second surface and Hoffman-Wohlgemuth example as limit-members.
We introduce a new technique to solve period problems on minimal surfaces called limit-method. If a family of surfaces has Weierstrass-data converging to the data of a known example, and this presents a transversal solution of periods, then the origi
nal family contains a sub-family with closed periods.
For an embedded singly periodic minimal surface M with genus bigger than or equal to 4 and annular ends, some weak symmetry hypotheses imply its congruence with one of the Hoffman-Wohlgemuth examples. We give a very geometrical proof of this fact, al
ong which they come out many valuable clues for the understanding of these surfaces.
