Periodic minimal surfaces embedded in R-3 derived from the singly periodic Scherk minimal surface

Abstract

We construct three kinds of periodic minimal surfaces embedded in R-3. We show the existence of a 1-parameter family of minimal surfaces invariant under the action of a translation by 2 pi, which seen from a distance look like m equidistant parallel planes intersecting orthogonally k equidistant parallel planes, m, k is an element of N, mk >= 2. We also consider the case where the surfaces are asymptotic to m is an element of N+ equidistant parallel planes intersecting orthogonally infinitely many equidistant parallel planes. In this case, the minimal surfaces are doubly periodic, precisely they are invariant under the action of two orthogonal translations. Last we construct triply periodic minimal surfaces which are invariant under the action of three orthogonal translations in the case of two stacks of infinitely many equidistant parallel planes which intersect orthogonally.