Periodic minimal surfaces embedded in ℝ3 derived from the singly periodic Scherk minimal surface

Abstract

We construct three kinds of periodic minimal surfaces embedded in [Formula: see text] We show the existence of a [Formula: see text]-parameter family of minimal surfaces invariant under the action of a translation by [Formula: see text] which seen from a distance look like [Formula: see text] equidistant parallel planes intersecting orthogonally [Formula: see text] equidistant parallel planes, [Formula: see text] [Formula: see text] We also consider the case where the surfaces are asymptotic to [Formula: see text] 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.