An end-to-end construction of doubly periodic minimal surfaces

Using Traizetu0027s regeneration method, we prove that for each positive integer n there is a family of embedded, doubly periodic minimal surfaces with parallel ends in Euclidean space of genus 2n-1 and 4 ends in the quotient by the maximal group of translations. The genus 2n-1 family converges smoothly to 2n copies of Scherku0027s doubly periodic minimal surface. The only previously known doubly periodic minimal surfaces with parallel ends and genus greater than 3 limit in a foliation of Euclidean space by parallel planes.